Recent zbMATH articles in MSC 35https://zbmath.org/atom/cc/352023-11-13T18:48:18.785376ZUnknown authorWerkzeugMini-workshop: Zero-range and point-like singular perturbations: for a spillover to analysis, PDE and differential geometry. Abstracts from the mini-workshop held October 2--8, 2022https://zbmath.org/1521.000152023-11-13T18:48:18.785376ZSummary: The field of contact interactions and perturbations of differential operators supported on subsets with non-trivial co-dimension is an increasingly active mainstream of mathematical physics (in particular, operator and spectral theory and quantum mechanics), with intimately related applications and mathematical challenges in partial differential equations, and neighbouring sectors of analysis, PDEs, and differential geometry. This Mini-Workshop fostered intense and prolific discussions on recent advances and trends in the field.Mathematical advances in geophysical fluid dynamics. Abstracts from the workshop held November 13--19, 2022https://zbmath.org/1521.000162023-11-13T18:48:18.785376ZSummary: The workshop ``Mathematical Advances in Geophysical Fluid Dynamics'' addressed recent advances in modeling, analytical, computational and stochastical studies of geophysical flows. Of particular interest were contributions concerning modeling and analysis of sea ice models, well-posedness results for the primitive equations and boundary layers, stratified flows and models for moist atmospheric dynamics including phase transitions.Working session: Quantitative stochastic homogenization. Abstracts from the working session held October 16--22, 2022https://zbmath.org/1521.000172023-11-13T18:48:18.785376ZSummary: Homogenization means approximating the effective, i. e. macroscopic, behavior of a heterogeneous medium by a homogeneous one, which amounts to a substantial conceptual and practical reduction of complexity. Stochastic homogenization means that one is considering an ensemble of, i. e. a probability measure on, such heterogeneities (typically expressing a lack of knowledge of the details); and that the effective behavior is also deterministic next to being homogeneous. The aim of this Arbeitsgemeinschaft is to present the recent progress in this field.On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s--1950s).https://zbmath.org/1521.010142023-11-13T18:48:18.785376Z"Bogatov, Egor Mikhailovich"https://zbmath.org/authors/?q=ai:bogatov.egor-mikhailovichVariational calculus has a long history beginning with the work of Euler. The author gives a historical account of the development of the related variational methods for nonlinear equations in the period 1920's--1950's and argues that it owes much to the contributions of Soviet mathematicians that include L. A. Lyusternik, L. G. Shnirelman, S. L. Sobolev and others. He is especially focused on results by the Kyiv mathematicians M. A. Krasnoselskii and M. M. Vainberg.
This interesting article begins with the prehistory of Dirichlet's principle, its use by Riemann, critique by Weierstrass and justification by Hilbert. After a short discussion of semicontinuity in variational-type problems, he discusses the Ritz method but does not mention its massive use by Soviet mathematicians such as B. G. Galerkin. Variational methods served as a catalyst for the emergence of Sobolev spaces in functional analysis. The author puts the work of the Soviet mathematicians in the context of the development of the subject and compares their achievements with those of their foreign colleagues in the period 1920's--1950's: L. Tonelli, W. Ritz, L. Lichtenstein, G. D. Birkhoff, M. Morse, A. Hammerstein, M. Golomb and others.
The use of topological methods started by Poincaré and continued by Birkhoff and Morse, has been used by Lyusternik and Shnirelman, who among other things, extended the concept of category to the case of functionals to estimate the number of solutions of variational problems. As another application of the theory of categories, Lyusternik in 1939 estimated the number of critical points of functionals in a Hilbert space.
The author describes in more detail and with an example of buckling of rods the Krasnoselskii principle of linearization in the problem of bifurcation points [\textit{M. A. Krasnosel'skiĭ}, Топологические методы в теорий нелинейных интегральных уравнений (Russian). Moskau: Staatsverlag für technisch-theoretische Literatur (1956; Zbl 0070.33001)]. Then he finishes his article with a list of solved problems by Vainberg.
Reviewer: Martin Lukarevski (Skopje)Laplacians on infinite graphshttps://zbmath.org/1521.050022023-11-13T18:48:18.785376Z"Kostenko, Aleksey"https://zbmath.org/authors/?q=ai:kostenko.aleksey-s"Nicolussi, Noema"https://zbmath.org/authors/?q=ai:nicolussi.noemaSummary: ``The main focus in this memoir is on Laplacians on both weighted graphs and weighted metric graphs. Let us emphasize that we consider infinite locally finite graphs and do not make any further geometric assumptions. Whereas the existing literature usually treats these two types of Laplacian operators separately, we approach them in a uniform manner in the present work and put particular emphasis on the relationship between them. One of our main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides. Our central goal is twofold. First of all, we explore the relationships between these two objects by comparing their basic spectral (self-adjointness, spectral gap, etc.), parabolic (Markovian uniqueness, recurrence, stochastic completeness, etc.), and metric (quasi isometries, intrinsic metrics, etc.) properties. In turn, we exploit these connections either to prove new results for Laplacians on metric graphs or to provide new proofs and perspective on the recent progress in weighted graph Laplacians. We also demonstrate our findings by considering several important classes of graphs (Cayley graphs, tessellations, and antitrees).''
Contents: Chapter 1. Introduction; Chapter 2. Laplacians on graphs; Chapter 3. Connections via boundary triplets; Chapter 4. Connections between parabolic properties; Chapter 5. One-dimensional Schrödinger operators with point interactions; Chapter 6. Graph Laplacians as boundary operators; Chapter 7. From continuous to discrete and back; Chapter 8. Examples.
Appendix A: Boundary triplets and Weyl functions; Appendix B: Dirichlet forms; Appendix C: Heat kernel bounds; Appendix D: Glossary of notation.
Reviewer: Shuchao Li (Wuhan)Flat bands of periodic graphshttps://zbmath.org/1521.051072023-11-13T18:48:18.785376Z"Sabri, Mostafa"https://zbmath.org/authors/?q=ai:sabri.mostafa"Youssef, Pierre"https://zbmath.org/authors/?q=ai:youssef.pierreSummary: We study flat bands of periodic graphs in a Euclidean space. These are infinitely degenerate eigenvalues of the corresponding adjacency matrix, with eigenvectors of compact support. We provide some optimal recipes to generate desired bands and some sufficient conditions for a graph to have flat bands, we characterize the set of flat bands whose eigenvectors occupy a single cell, and we compute the list of such bands for small cells. We next discuss the stability and rarity of flat bands in special cases. Additional folklore results are proved, and many questions are still open.
{\copyright 2023 American Institute of Physics}Darboux transformations for orthogonal differential systems and differential Galois theoryhttps://zbmath.org/1521.120062023-11-13T18:48:18.785376Z"Acosta-Humánez, Primitivo"https://zbmath.org/authors/?q=ai:acosta-humanez.primitivo-belen"Barkatou, Moulay"https://zbmath.org/authors/?q=ai:barkatou.moulay-a"Sánchez-Cauce, Raquel"https://zbmath.org/authors/?q=ai:sanchez-cauce.raquel"Weil, Jacques-Arthur"https://zbmath.org/authors/?q=ai:weil.jacques-arthurSummary: Darboux developed an ingenious algebraic mechanism to construct infinite chains of ``integrable'' second-order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance in quantum mechanics (where they provide useful tools for supersymmetric quantum mechanics), in soliton theory, Lax pairs and many other fields involving hierarchies of equations. In this paper, we propose a method which allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third-order orthogonal systems \((\mathfrak{so}(3,C_K)\) systems) as well as a framework to extend Darboux transformations to any symmetric power of \(\text{SL}(2,\mathbb{C})\)-systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials. All results in this paper have been implemented and tested in the computer algebra system Maple.\( \mathfrak{sl}(2)\)-type singular fibres of the symplectic and odd orthogonal Hitchin systemhttps://zbmath.org/1521.140222023-11-13T18:48:18.785376Z"Horn, Johannes"https://zbmath.org/authors/?q=ai:horn.johannesThe Hitchin fibration played a major role in two recent developments in the theory of Higgs bundle moduli spaces: First, in the study of the asymptotic of the hyperkähler metric [\textit{R. Mazzeo} et al., Commun. Math. Phys. 367, No. 1, 151--191 (2019; Zbl 1409.14024)] and second, in the Langlands duality of Higgs bundle moduli spaces [\textit{R. Donagi} and \textit{T. Pantev}, Invent. Math. 189, No. 3, 653--735 (2012; Zbl 1263.53078)]. Both results were considered on the regular locus of the Hitchin map and it is an interesting question how they extend to the singular locus. The aim of this paper is to do the first steps in this direction. The author defines and parametrizes so-called \(\texttt{sl}(2)\)-type fibres of the \(Sp(2n,\mathbb{C})\)- and \(SO(2n+1,\mathbb{C})\)-Hitchin system. These are singular Hitchin fibres, such that spectral curve establishes a \(2\)-sheeted covering of a second Riemann surface \(Y\). This identifies the \(\texttt{sl}(2)\)-type Hitchin fibres with fibres of an \(SL(2,\mathbb{C})\)-Hitchin, respectively, \(PSL(2,\mathbb{C})\)-Hitchin map on \(Y\).
Building on results of [\textit{J. Horn}, Int. Math. Res. Not. 2022, No. 5, 3860--3917 (2022; Zbl 1482.14040)], he gives a stratification of these singular spaces by semi-abelian spectral data, studies their irreducible components and obtains a global description of the first degenerations. He also compares the semi-abelian spectral data of \(\texttt{sl}(2)\)-type Hitchin fibres for the two Langlands dual groups. This extends the well-known Langlands duality of regular Hitchin fibres to \(\texttt{sl}(2)\)-type Hitchin fibres. Finally, the author constructs solutions to the decoupled Hitchin equation for \(\texttt{sl}(2)\)-type fibres of the symplectic and odd orthogonal Hitchin system. He conjectures these to be limiting configurations along rays to the ends of the moduli space.
This paper is organized as follows: Section 1 is an introduction to the subject and summarizes the main results. In Section 2, the author introduces \(\texttt{sl}(2)\)-type Hitchin fibres of the symplectic Hitchin system. He proves the identification of these Hitchin fibres with \(\textsc{SL}(2,\mathbb{C})\)-Hitchin fibres and gives the parametrization by semi-abelian spectral data using the results of [loc. cit.]. In Section 3, he repeats these considerations for the odd orthogonal group. In Section 4, the author formulates the Langlands correspondence for \(Sp(2n,\mathbb{C})\)- and \(SO(2n+1,\mathbb{C})\)-Hitchin fibres of \(\texttt{sl}(2)\)-type. Finally, in Section 5, the author shows how to use semi-abelian spectral data for \(\texttt{sl}(2)\)-type Hitchin fibres to produce solutions to the decoupled Hitchin equation. He constructs solutions to the decouple Hitchin equation and motivate why he conjectures theses to be limiting configurations.
Reviewer: Ahmed Lesfari (El Jadida)A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional orderhttps://zbmath.org/1521.260152023-11-13T18:48:18.785376Z"Idczak, Dariusz"https://zbmath.org/authors/?q=ai:idczak.dariuszSummary: In the paper, a new Gronwall lemma for functions of two variables with singular integrals is proved. An application to weak relative compactness of the set of solutions to a fractional partial differential equation is given.A note on arclength null quadrature domainshttps://zbmath.org/1521.300172023-11-13T18:48:18.785376Z"Khavinson, Dmitry"https://zbmath.org/authors/?q=ai:khavinson.dmitry-s"Lundberg, Erik"https://zbmath.org/authors/?q=ai:lundberg.erikA domain \(\Omega\subset \mathbb{C}\) is said to be an arclength null quadrature domain if the identity
\[
\int_{\partial \Omega} g(z) ds(z)=0
\]
holds for all functions \(g\) in the Smirnov space \(E^1(\Omega)\). A sufficient condition for a domain \(\Omega\) to be an arclength null quadrature domain is that \(\Omega\) admits a roof function, a positive function \(u\) which is harmonic in \(\Omega\) such that the gradient \(\nabla u\) coincides with the inward-pointing unit normal vector along \(\partial\Omega\).
In this article the authors obtain the existence of a roof function for arclength null quadrature domains having finitely many boundary components.
Reviewer: Marius Ghergu (Dublin)Inverse problem for a planar conductivity inclusionhttps://zbmath.org/1521.300222023-11-13T18:48:18.785376Z"Choi, Doosung"https://zbmath.org/authors/?q=ai:choi.doosung"Helsing, Johan"https://zbmath.org/authors/?q=ai:helsing.johan"Kang, Sangwoo"https://zbmath.org/authors/?q=ai:kang.sangwoo"Lim, Mikyoung"https://zbmath.org/authors/?q=ai:lim.mikyoungSummary: This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.Uniform approximation by polynomial solutions of elliptic systems on boundaries of Carathéodory domains in \(\mathbb{R}^2\)https://zbmath.org/1521.300472023-11-13T18:48:18.785376Z"Fedorovskiy, K."https://zbmath.org/authors/?q=ai:fedorovskiy.k-yu|fedorovskii.konstantin-yurevichSummary: We study the problem on uniform approximation of functions on compact subsets of the complex plane by polynomial solutions of general second-order elliptic systems with constant coefficients. This problem is well-known for the systems corresponding to second order equations with constant complex coefficients and is rather poor studied in the general case. In particular case, when the compact set where the approximation is considered is the boundary of a Carathéodory domain in the plane, we establish some new sufficient approximability conditions. We also discuss new measure orthogonality conditions that appear in the problem under consideration.On metric properties of \(C\)-capacities associated with solutions of second-order strongly elliptic equations in \(\mathbb{R}^2\)https://zbmath.org/1521.300482023-11-13T18:48:18.785376Z"Paramonov, Petr V."https://zbmath.org/authors/?q=ai:paramonov.peter-vSummary: For certain capacities that were used previously to formulate criteria for the uniform approximability of functions by solutions of strongly elliptic equations of the second order on compact subsets of \(\mathbb{R}^2\), a number of metric properties are established. New, more natural criteria for individual approximability are obtained as consequences. Unsolved problems of interest are stated.The Dirichlet problem for the Beltrami equations with sourceshttps://zbmath.org/1521.300562023-11-13T18:48:18.785376Z"Gutlyanskiĭ, Vladimir"https://zbmath.org/authors/?q=ai:gutlyanskii.vladimir-ya"Ryazanov, Vladimir"https://zbmath.org/authors/?q=ai:ryazanov.vladimir-i"Nesmelova, Olga"https://zbmath.org/authors/?q=ai:nesmelova.o-v"Yakubov, Eduard"https://zbmath.org/authors/?q=ai:yakubov.eduard-hSummary: The paper is devoted to the study of the Dirichlet problem \(Re \omega (z) \rightarrow \phi(\zeta )\) as \(z \rightarrow \zeta\), \(z \in D\), \(\zeta \in \partial D\), with continuous boundary data \(\phi: \partial D \rightarrow \mathbb{R}\) for Beltrami equations \({\omega}_{ \overline{z}} = \mu (z) \omega_z + \sigma (z)\), \(| \mu (z)| < 1\) a.e., with sources \(\sigma : D \rightarrow \mathbb{C}\) in the case of locally uniform ellipticity. In this case, we have established a series of effective integral criteria of the BMO, FMO, Calderon-Zygmund, Lehto, and Orlicz types on the singularities of the equations at the boundary for the existence, representation, and regularity of solutions in arbitrary bounded domains \(D\) of the complex plane \(\mathbb{C}\) with no boundary component degenerated to a single point for sources \(\sigma\) in \(L_p (D)\), \(p > 2\), with compact support in \(D\). Moreover, we have proved the existence, representation, and regularity of weak solutions of the Dirichlet problem in such domains for the Poisson-type equation \(\operatorname{div} [A(z) \nabla u(z)] = g(z)\), whose source \(g \in L_p(D)\), \(p > 1\), has compact support in \(D\) and whose matrix-valued coefficient \(A(z)\) guarantees its locally uniform ellipticity.Biharmonic problem for an angle and monogenic functionshttps://zbmath.org/1521.310052023-11-13T18:48:18.785376Z"Gryshchuk, S. V."https://zbmath.org/authors/?q=ai:gryshchuk.serhii-v"Plaksa, S. A."https://zbmath.org/authors/?q=ai:plaksa.sergii-anatoliiovychSummary: We consider a piecewise continuous biharmonic problem in an angle and the corresponding Schwartz-type boundary-value problem for monogenic functions in a commutative biharmonic algebra. These problems are reduced to a system of integral equations on the positive semiaxis. It is shown that, on each segment of this semiaxis, the set of solutions of this system coincides with the set of solutions of a certain system of Fredholm integral equations.Multigrid solvers for isogeometric discretizations of the second biharmonic problemhttps://zbmath.org/1521.310062023-11-13T18:48:18.785376Z"Sogn, Jarle"https://zbmath.org/authors/?q=ai:sogn.jarle"Takacs, Stefan"https://zbmath.org/authors/?q=ai:takacs.stefanSummary: We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch's framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble's framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches.One-radius theorem for harmonic tempered distributionshttps://zbmath.org/1521.310072023-11-13T18:48:18.785376Z"Ben Chrouda, Mohamed"https://zbmath.org/authors/?q=ai:ben-chrouda.mohamed"Hassine, Kods"https://zbmath.org/authors/?q=ai:hassine.kodsSummary: We show that the ``one-radius'' spherical mean value property is sufficient to characterize harmonic tempered distributions subject to the classical Laplace operator.An overdetermined problem of the biharmonic operator on Riemannian manifoldshttps://zbmath.org/1521.310122023-11-13T18:48:18.785376Z"Chen, Fan"https://zbmath.org/authors/?q=ai:chen.fan"Huang, Qin"https://zbmath.org/authors/?q=ai:huang.qin"Ruan, Qihua"https://zbmath.org/authors/?q=ai:ruan.qihuaSummary: Let \((M,g)\) be an \(n\)-dimensional complete Riemannian manifold with nonnegative Ricci curvature. In this paper, we consider an overdetermined problem of the biharmonic operator on a bounded smooth domain \(\Omega\) in \(M\). We deduce that the overdetermined problem has a solution only if \(\Omega\) is isometric to a ball in \(\mathbb{R}^n\). Our method is based on using a \(P\)-function and the maximum principle argument. This result is a generalization of the overdetermined problem for the biharmonic equation in Euclidean space.The triharmonic equation on the Heisenberg grouphttps://zbmath.org/1521.310132023-11-13T18:48:18.785376Z"Izadjoo, Majid"https://zbmath.org/authors/?q=ai:izadjoo.majid"Akbari, Mojgan"https://zbmath.org/authors/?q=ai:akbari.mojganSummary: Consider the equation
\[
\begin{cases}
\begin{aligned}
{-}\Delta_H^3 u &= f(\xi,u ,\nabla_Hu,\nabla_H^2u,\nabla_H^3u,\nabla_H^4u,\nabla_H^5u) &\text{ in }\Omega,\\
u &>0 &\text{ in }\Omega,\\
u &= \frac{\partial}{\partial\nu}(\nabla_H^2u)= \frac{\partial}{\partial\nu}(\nabla_H^3u)=0 &\text{on }\partial\Omega,
\end{aligned}
\end{cases}
\]
where \(\Omega\) is a domain of the finite-dimensional space \(\mathbb{H}^n\) and \(f\) is a positive and bounded function. We prove the existence of a solution for the above equation. In addition, we prove the uniqueness and the cylindrical symmetry of the solution.Second variation formula and stability of exponentially subelliptic harmonic mapshttps://zbmath.org/1521.310152023-11-13T18:48:18.785376Z"Chiang, Yuan-Jen"https://zbmath.org/authors/?q=ai:chiang.yuanjen"Dragomir, Sorin"https://zbmath.org/authors/?q=ai:dragomir.sorin"Esposito, Francesco"https://zbmath.org/authors/?q=ai:esposito.francesco.2Summary: We study the stability of exponentially subelliptic harmonic (e.s.h.)\ maps from a Carnot-Carathéodory complete strictly pseudoconvex pseudohermitian manifold \((M,\theta)\) into a Riemannian manifold \((N, h)\). E.s.h.\ maps are \(C^\infty\) solutions \(\phi : M \rightarrow N\) to the nonlinear PDE system \(\tau_b (\phi) + \phi_* \nabla^H e_b (\phi ) = 0\) [the Euler-Lagrange equations of the variational principle \(\delta E_b (\phi) = 0\) where \(E_b (\phi) = \int_\Omega \exp \left[ e_b (\phi) \right] \Psi\) and \(e_b (\phi) = \frac{1}{2}\text{trace}_{G_\theta} \{ \Pi_H\phi^*h\}\) and \(\Omega \subset M\) is a Carnot-Carathéodory bounded domain]. We derive the second variation formula about an e.s.h. map, leading to a pseudohermitian analog to the Hessian (of an ordinary exponentially harmonic map between Riemannian manifolds)
\[
\begin{multlined}
H(E_b)_\phi (V, W)= \int_\Omega h^\phi \left( J^\phi_{b\exp} V, W \right) \Psi \\
+ \int_M \exp \left[ e_b (\phi) \right] (h^\phi)^*(D^\phi V, \Pi_H \phi_*) (h^\phi )^* (D^\phi W, \Pi_H \phi_*) \Psi,
\end{multlined}
\]
\[
\begin{multlined}
J_{b, \exp}^\phi V \equiv \left(D^\phi \right)^*\left( \exp \left[e_b (\phi) \right] D^\phi V \right) \\
- \exp \left[ e_b (\phi) \right] \text{trace}_{G_\theta} \left\{\Pi_H \left(R^h \right)^\phi \left( V, \phi_* \cdot \right) \phi_*\cdot \right\},
\end{multlined}
\]
\([\Psi = \theta \wedge (d \theta )^n]\). Given a bounded domain \(\Omega \subset M\) and an e.s.h.\ map \(\phi \in C^\infty (\overline{\Omega},N)\) with values in a Riemannian manifold \(N = N^m (k)\) of nonpositive constant sectional curvature \(k \leq 0\), we solve the generalized Dirichlet eigenvalue problem \(J^\phi_{b,\exp} V = \lambda V\) in \(\Omega\) and \(V = 0\) on \(\partial \Omega\) for the degenerate elliptic operator \(J^\phi_{b,\exp}\), provided that \(\Omega\) supports Poincaré inequality
\[
\Vert V \Vert_{L^2} \leq C \Vert D^\phi V \Vert_{L^2}, \quad V \in C^\infty_0 (\Omega, \phi^{-1} TN ),
\]
and the embedding \(\mathring{W}^{1,2}_H (\Omega,\phi^{-1} TN) \hookrightarrow L^2 (\Omega, \phi^{-1} TN)\) is compact.Self-adjoint Laplacians and symmetric diffusions on hyperbolic attractorshttps://zbmath.org/1521.310212023-11-13T18:48:18.785376Z"Alikhanloo, Shayan"https://zbmath.org/authors/?q=ai:alikhanloo.shayan"Hinz, Michael"https://zbmath.org/authors/?q=ai:hinz.michaelSummary: We construct self-adjoint Laplacians and symmetric Markov semigroups on hyperbolic attractors, endowed with Gibbs \(u\)-measures. If the measure has full support, we can also conclude the existence of an associated symmetric diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on negatively curved manifolds the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians. We observe a quasi-invariance property of energy densities in the \(u\)-conformal case and the existence of nonconstant functions of zero energy.Conservativeness and uniqueness of invariant measures related to non-symmetric divergence type operatorshttps://zbmath.org/1521.310222023-11-13T18:48:18.785376Z"Lee, Haesung"https://zbmath.org/authors/?q=ai:lee.haesungSummary: We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by \(L^1\)-uniqueness and imply that a given infinitesimally invariant measure becomes an invariant measure. We explore further conditions on the coefficients of the partial differential operators that ensure the uniqueness of the invariant measure beyond the case where the corresponding semigroups are recurrent. A main observation is that for conservativeness and uniqueness of invariant measures in this article, no growth conditions are required for the partial derivatives related to the anti-symmetric matrix of functions \(C = (C_{ij})_{1 \le i,j \le d}\) that determine a part of the drift coefficient. As stochastic counterparts, our results can be applied to show not only the existence of a pathwise unique and strong solution up to infinity to a corresponding Itô-SDE, but also the existence and uniqueness of invariant measures for the family of strong solutions.Positive solutions of Schrödinger equations in product form and Martin compactifications of the plane. IIhttps://zbmath.org/1521.310242023-11-13T18:48:18.785376Z"Murata, Minoru"https://zbmath.org/authors/?q=ai:murata.minoru"Tsuchida, Tetsuo"https://zbmath.org/authors/?q=ai:tsuchida.tetsuoSummary: We determine the structure of all positive solutions to a Schrödinger equation \((- \Delta + V_1(x_1) + V_2(x_2))u = 0\) on \(\mathbb{R}^2\), where real potentials \(V_j\), \(j = 1\), \(2\), satisfy \(V_j\in{L^1_1} =\{V; (1+|t|)V(t)\in L^1(\mathbb{R})\}\). We also treat the case where \(V_1\in{L^1_1}\) and \(V_2\) belongs to a wide class of functions including model potentials \(V_2(t)\) = |\(t|^a\), \(a > 0\). We show that non-minimal Martin boundary points appear generically. On analysis of asymptotics of the Green functions of the Schrödinger equations, the Jost solutions of one dimensional Schrödinger equations with potential functions in \({L_1^1}\) play a central role.
For Part I see [the authors, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 23, No. 3, 1141--1194 (2022; Zbl 1504.31024)].The \(\bar\partial\) Neumann problem and Schrödinger operatorshttps://zbmath.org/1521.320012023-11-13T18:48:18.785376Z"Haslinger, Friedrich"https://zbmath.org/authors/?q=ai:haslinger.friedrichPublisher's description: This book's subject lies in the nexus of partial differential equations, operator theory, and complex analysis. The spectral analysis of the complex Laplacian and the compactness of the d-bar-Neumann operator are primary topics.
The revised 2nd edition explores updates to Schrödinger operators with magnetic fields and connections to the Segal Bargmann space (Fock space), to quantum mechanics, and the uncertainty principle.
\begin{itemize}
\item Explores the Segal Bargmann Space (Fock space) connection to quantum mechanics.
\item Updates to Schrödinger operators with magnetic fields
\item New results about compactness, in particular a necessary condition of compactness
\end{itemize}
See the review of the first edition in [Zbl 1316.32001].Spherical cones: classification and a volume minimization principlehttps://zbmath.org/1521.320282023-11-13T18:48:18.785376Z"Nghiem, Tran-Trung"https://zbmath.org/authors/?q=ai:nghiem.tran-trungSummary: Using a variational approach, we establish the equivalence between a weighted volume minimization principle and the existence of a conical Calabi-Yau structure on horospherical cones with mild singularities. This allows us to do explicit computations on the examples arising from rank-two symmetric spaces, showing the existence of many irregular horospherical cones.Monge-Ampère type equations on almost Hermitian manifoldshttps://zbmath.org/1521.320322023-11-13T18:48:18.785376Z"Zhang, Jiao Gen"https://zbmath.org/authors/?q=ai:zhang.jiaogenSummary: In this paper we consider the Monge-Ampère type equations on compact almost Hermitian manifolds. We derive \(C^\infty\) a priori estimates under the existence of an admissible \(\mathcal{C} \)-subsolution. Finally, we obtain an existence result if there exists an admissible supersolution.Weighted Green functions for complex Hessian operatorshttps://zbmath.org/1521.320352023-11-13T18:48:18.785376Z"El Aini, Hadhami"https://zbmath.org/authors/?q=ai:el-aini.hadhami"Zeriahi, Ahmed"https://zbmath.org/authors/?q=ai:zeriahi.ahmedLet \(\Omega\subset\subset\mathbb C^n\) be an \(m\)-hyperconvex domain, let \(A\subset\Omega\) be finite, and let \(\nu:\Omega\longrightarrow[0,+\infty)\) be a weight function. Put \(\mathcal A:=\{(a,\nu(a)): a\in A\}\). Let \(\mathcal{SH}_m(\Omega)\) denote the set of all \(m\)-subharmonic functions on \(\Omega\) and let \[\mathcal{SH}_m^b(\Omega):=\big\{u\in\mathcal{SH}_m(\Omega): \exists_{E\subset\subset\Omega}: u\in L^\infty(\Omega\setminus E)\big\}.\] Set
\[
\Phi_m(z):=\begin{cases}-|z|^{-2s}, &\text{if } 1\leq m<n\\
\log(|z|/R_0), &\text{if } m=n,\end{cases}
\]
where \(R_0\geq1\) is such that \(\Omega\subset B(R_0/2)\) and \(s:=n/m-1\). Put \(\varphi_m(z,\mathcal A):=\inf_{a\in A}\nu(a)\Phi_m(z-a)\), \(\psi_m(z,\mathcal A):=\sum_{a\in A}\nu(a)\Phi_m(z-a)\). Let
\[(\mathcal G_m(\Omega,\mathcal A):=\big\{u\in\mathcal{SH}_m^-(\Omega): \exists_{C_u>0}: u(z)\leq\varphi_m(z,\mathcal A)+C_u,\;z\in\Omega\big\},\]
\(\mathcal G_m(z,\mathcal A):=\sup\{u(z): u\in\mathcal G_m(\Omega,\mathcal A)\}\). Define
\[\mathcal E(\delta_0,\gamma_0,\gamma_1):=\big\{\mathcal A\subset\Omega\times\mathbb R^+: \delta_{\mathcal A}\geq\delta_0,\;\inf_{a\in A}\nu(a)\geq\gamma_0,\;\sum_{a\in A}\nu(a)\leq\gamma_1\big\},\] where
\[
\delta_{\mathcal A}=\begin{cases}\inf_{(a,\nu)\in\mathcal A}\nu^{-1/2s}d(a), &\text{if } 1\leq m<n\\
\inf_{(a,\nu)\in\mathcal A}R_0(d(a)/rR_0)^\nu, &\text{if } m=n,\end{cases}
\]
\(d\) stands for the Euclidean distance to \(\partial\Omega\). Finally, let
\[\mathcal F(\delta_0,\gamma_0,\gamma_1,\sigma_0):=\big\{\mathcal A\in\mathcal E(\delta_0,\gamma_0,\gamma_1): \sigma_{\mathcal A}\geq\sigma_0\big\},\] where \(\sigma_{\mathcal A}:=\tfrac12\min\{|a=b|: (a,b)\in A^2,;a\neq b\}\).
The main results of the paper are the following theorems.
The associated Green function \(G_m(\cdot,\mathcal A,\Omega)\) is \(m\)-subharmonic, negative in \(\Omega\), and has the following properties:
\begin{itemize}
\item[(i)] for any \(z\in\Omega\) we have \(\psi_m(z,A)\leq G_m(z,\mathcal A,\Omega)\leq\varphi_m(z,A)-\Phi_m(\delta_\mathcal A)\);
\item[(ii)] \(\lim_{z\to\partial\Omega}(\inf_{\mathcal A\in\mathcal E(\delta_0,\gamma_0,\gamma_1)}G_m(z,\mathcal A,\Omega))=0\);
\item[(iii)] \(G_m(\cdot,\mathcal A,\Omega))\in\mathcal{SH}^b_m(\Omega)\) and it satisfies the Hessian equation
\[(*)\qquad (dd^cG_m(\cdot,\mathcal A,\Omega))^m\wedge\beta^{n-m}=c_{m,n}\sum_{a\in A}\nu(a)^m\delta_a\beta^n\]in the sense of currents on \(\Omega\);
\item[(iv)] \(G_m(\cdot,\mathcal A,\Omega)\) is a unique \(m\)-subharmonic function in \(G_m(\Omega,\mathcal A)\) with boundary values \(0\) satisfying (*).
\end{itemize}
Assume that \(\Omega\) is of Lipschitz type.
\begin{itemize}
\item[(i)] If \(1\leq m<n\), then for any \(0<\tau<1-\frac{m}{2n-m}\) there exist constants \(M_m\), \(r_1>0\) depending on \((\tau,m,n,\delta_0,\gamma_0,\gamma_1)\) such that for any \((z',\mathcal A')\in\overline\Omega\times\mathcal F(\delta_0,\gamma_0,\gamma_1,\sigma_0)\) and any \((z,\mathcal A)\in\overline\Omega\times\mathcal F(\delta_0,\gamma_0,\gamma_1,\sigma_0)\) with \(|z'-z|+d_H(\mathcal A',\mathcal A)\leq r\leq r_1\) we have \(\exp G_m (z',\mathcal A',\Omega)-\exp G_m (z',\mathcal A',\Omega)\leq M_mr^\tau\).
\item[(ii)] If \(m=n\), then for any \(0<\alpha<1\) there exist constants \(r_1>0\), \(M_n>0\) depending on \((n, \delta_0, \gamma_0, \gamma_1,\sigma_0)\) such that for any \((z',\mathcal A')\in\overline\Omega\times\mathcal F(\delta_0,\gamma_0,\gamma_1,\sigma_0)\) and any \((z,\mathcal A)\in\overline\Omega\times\mathcal F(\delta_0,\gamma_0,\gamma_1,\sigma_0)\) with \(|z'-z|+d_H(\mathcal A',\mathcal A)\leq r\leq r_1\) we have \(\exp G_m (z',\mathcal A',\Omega)-\exp G_m (z',\mathcal A',\Omega)\leq\frac{M_n}{(\log R_0^{1/\gamma_0}/r)^\alpha}\).
\end{itemize}
Reviewer: Marek Jarnicki (Kraków)Explicit Carleman formulas for the Dolbeault cohomology in concave domainshttps://zbmath.org/1521.320402023-11-13T18:48:18.785376Z"Shestakov, Ivan V."https://zbmath.org/authors/?q=ai:shestakov.ivan-vSummary: In [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 45, 253--262 (1999; Zbl 1005.32024)] \textit{M. Nacinovich} et al. suggested an abstract method for constructing Carleman formulas for the Dolbeault complex. What has been lacking are simple and explicit examples. In this article we present a Carleman formula for Dolbeault cohomology classes given on a part of the boundary whose comlement is concave. As corollary we derive a uniqueness theorem for the Dolbeault cohomology.Existence of ground state solution for a class of one-dimensional Kirchhoff-type equations with asymptotically cubic nonlinearitieshttps://zbmath.org/1521.340282023-11-13T18:48:18.785376Z"Khoutir, Sofiane"https://zbmath.org/authors/?q=ai:khoutir.sofianeIn this paper, the author considers the following Kirchoff-type equation \[-\Big(1+\int_{\mathbb{R}} |u'|^2dx\Big)u''+p(x)u=l(x)u^3+f(x,u),~~ x\in \mathbb{R},\] where \(p,l\in C(\mathbb{R})\) and \(f\in C(\mathbb{R}\times\mathbb{R},\mathbb{R}).\) By using the Non-Nehari manifold method in combination with the Mountain Pass Theorem and concentration-compactness argument, the existence of a ground state solution to the above equation is established, in the case when the nonlinearity is asymptotically cubic with respect to the unknown function.
Reviewer: Sotiris K. Ntouyas (Ioannina)Slow travelling wave solutions of the nonlocal Fisher-KPP equationhttps://zbmath.org/1521.340462023-11-13T18:48:18.785376Z"Billingham, John"https://zbmath.org/authors/?q=ai:billingham.johnThe article is concerned with the spike behaviors of slow travelling wave solutions of the nonlocal Fisher-KPP equation. Here slow travelling wave solutions mean that the speed of the travelling wave solutions is sufficiently small. This is a more difficult problem compared to the fast travelling wave solutions. In fact, the fast travelling wave solutions are small perturbations of the travelling wave solution of the local Fisher-KPP equation.
By using the formal method of matched asymptotic expansions and numerical methods, this article is devoted to revealing the number and spacing of the spikes associated with the slow travelling wave solutions, which depend crucially on the behaviour of the kernel in the nonlocal Fisher-KPP equation. Under certain kernels, finite and infinite number of spikes can be generated.
Reviewer: Jianhe Shen (Fuzhou)Stationary fronts and pulses for multistable equations with saturating diffusionhttps://zbmath.org/1521.340472023-11-13T18:48:18.785376Z"Garrione, Maurizio"https://zbmath.org/authors/?q=ai:garrione.maurizio"Sovrano, Elisa"https://zbmath.org/authors/?q=ai:sovrano.elisaAuthors' abstract: We deal with stationary solutions of a reaction-diffusion equation with flux-saturated diffusion and multistable reaction term, in dependence on a positive parameter \(\varepsilon\). Motivated by previous numerical results obtained by \textit{A. Kurganov} and \textit{P. Rosenau} [Nonlinearity 19, No. 1, 171--193 (2006; Zbl 1094.35063)], we investigate stationary solutions of front and pulse-type and discuss their qualitative features. We study the limit of such solutions for \(\varepsilon\rightarrow0\), showing that, in spite of the wide variety of profiles that can be constructed, there is essentially a unique configuration in the limit for both stationary fronts and pulses. We finally discuss some variational features that include the case where the solutions having continuous energy may not be global minimizers of the associated action functional.On the ``Traveling pulses'' of the limit of the FitzHugh-Nagumo equation when \(\varepsilon \to 0\)https://zbmath.org/1521.340482023-11-13T18:48:18.785376Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaConsider the existence of traveling pulses for the FitzHugh-Nagumo system
\[
u_t =u_{xx}+u(u-a)(1-u) +w, \quad w_t=\varepsilon(u-\gamma w)
\]
in case \(\varepsilon =0 \). This problem is equivalent to the existence of homoclinic orbits of the planar autonomous system
\[
\frac{dx}{ds} = y, \quad \frac{dy}{ds} =-cy-x(x-a)(1-x)+w \tag{1}
\]
where \(c\) is the speed of the pulse, \(a\) and \(w\) are real parameters. The authors determine the phase potrait of system (1) in the Poincaré disc for different parameter values by investigating the equilibria of (1) at infinity.
Reviewer: Klaus R. Schneider (Berlin)Stability and Turing patterns of a predator-prey model with Holling type II functional response and Allee effect in predatorhttps://zbmath.org/1521.340542023-11-13T18:48:18.785376Z"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.lu.4"Yang, Feng"https://zbmath.org/authors/?q=ai:yang.feng"Song, Yong-li"https://zbmath.org/authors/?q=ai:song.yongliSummary: In this paper, we are concerned with a predator-prey model with Holling type II functional response and Allee effect in predator. We first mathematically explore how the Allee effect affects the existence and stability of the positive equilibrium for the system without diffusion. The explicit dependent condition of the existence of the positive equilibrium on the strength of Allee effect is determined. It has been shown that there exist two positive equilibria for some modulate strength of Allee effect. The influence of the strength of the Allee effect on the stability of the coexistence equilibrium corresponding to high predator biomass is completely investigated and the analytically critical values of Hopf bifurcations are theoretically determined. We have shown that there exists stability switches induced by Allee effect. Finally, the diffusion-driven Turing instability, which can not occur for the original system without Allee effect in predator, is explored, and it has been shown that there exists diffusion-driven Turing instability for the case when predator spread slower than prey because of the existence of Allee effect in predator.Traveling pulses and their bifurcation in a diffusive Rosenzweig-MacArthur system with a small parameterhttps://zbmath.org/1521.340662023-11-13T18:48:18.785376Z"Hou, Xiaojie"https://zbmath.org/authors/?q=ai:hou.xiaojie"Li, Yi"https://zbmath.org/authors/?q=ai:li.yiSummary: Conditions for the long term coexistence of the prey and predator populations of a diffusive Rosenzweig-MacArthur model are studied. The coexistence is represented by traveling pulses, which approach a coexistence equilibrium state as the moving coordinate approaches to infinities. Three different pulses, according to their speeds, are analyzed by regular perturbation as well as geometric singular perturbation methods. We further show that the pulses are connected by a bifurcation curve (surface) in parametric space. The paper concludes with several numerical simulations.Introduction to reaction-diffusion equations. Theory and applications to spatial ecology and evolutionary biologyhttps://zbmath.org/1521.350012023-11-13T18:48:18.785376Z"Lam, King-Yeung"https://zbmath.org/authors/?q=ai:lam.king-yeung"Lou, Yuan"https://zbmath.org/authors/?q=ai:lou.yuanThis book gives an exposition of some mathematical theories for reaction-diffusion equations with applications to biology. While there is a large number of literature for reaction-diffusion equations, the materials are often scattered in the literature, and some areas are still under development. The aim of the book is to present some of these theories and tools to interested readers, in a self-contained manner that is accessible to beginners.
This book is divided into four parts. Part I discusses the basic theory of linear elliptic and parabolic equations. Parts II and III address applications to ecological dynamics and evolutionary game theory, respectively. Part IV gives several useful tools from nonlinear functional analysis and dynamical systems.
Part I consists of 4 chapters. Chapter 1 begins with scalar parabolic equations with oblique boundary conditions. The book introduces a concept of super- and subsolutions in a generalized sense which enables the gluing of classical super- or subsolutions. They derive the existence of the principal eigenvalue by applying the Krein-Rutman Theorem. Chapter 2 is devoted to the principal eigenvalue of periodic-parabolic problems, where various asymptotic estimates and monotonicity properties of the principal eigenvalues are established. Chapter 3 concerns the theory of principal eigenvalues for the cooperative systems. Chapter 4 presents the basic theory of the principal Floquet bundle and some recent result concerning smooth dependence of the principal Floquet bundle on the coefficients of parabolic operators.
Part II consists of Chapters 5--8. In Chapter 5, the authors present some general theory for semilinear equations modeling a single population, including the monotone iteration method and the relationship of linear and nonlinear stability of equilibria. In Chapter 6, the book discusses the Fisher-KPP equation on the real line and some results concerning the shifting habitat. Chapter 7 is devoted to the diffusive Lotka-Volterra model of two competing species in bounded domains, with particular attention on the problem of evolution of dispersal. In Chapter 8, the authors discuss the dynamics of phytoplankton populations in a water column for two or more species. The authors show that in the two-species case the nonlocal PDE model is in fact order-preserving, albeit with respect to a nonstandard cone. When the number of species is greater than two, the authors present a new approach to analyze the population dynamics, based on the theory of normalized principal Floquet bundles.
Part III consists of Chapters 9 to 10. In Chapter 9, the authors discuss the framework of adaptive dynamics in the context of a river population model, hoping to offer to the readers a PDE viewpoint of the theory. In Chapter 10, the authors discuss the selection-mutation models, which describe population structured by a continuous trait, and they connect the selection-mutation models with the framework of adaptive dynamics.
Part IV consists of Appendices A to E. In Appendix A, the authors derive the fixed point index from the Leray-Schauder degree. In Appendix B, they present a self-contained proof of the Krein-Rutman Theorem for positively homogeneous maps of degree one that are monotone with respect to positive cone, and then derive the classical Krein-Rutman Theorem for compact positive linear operators. In Appendix C, the authors discuss dynamical systems in ordered Banach spaces that are generated by monotone and subhomogeneous maps, and show that such a system has a globally attracting fixed point. An analogous result for continuous-time semiflows is also presented. In Appendix D, the authors consider general monotone dynamical systems and prove the Dancer-Hess Lemma concerning the existence of a connecting orbit between two ordered fixed points, and prove the limit set trichotomy. In Appendix E, the authors present the theory of abstract competitive systems, and develop the trichotomy result due to Hsu, Smith and Waltman. In the case when the mapping (or semiflow) is continuously differentiable, the authors present a new condition to achieve a stronger trichotomy result.
This book serves as a good reference for some modern theories of reaction-diffusion equations and its applications to population dynamics. It is written in a self-contained manner that are friendly to beginners, hoping to bring graduate students or beginning young researchers quickly to research frontlines.
Reviewer: Wan-Tong Li (Lanzhou)Infinite time blow-up solutions to the energy critical wave maps equationhttps://zbmath.org/1521.350022023-11-13T18:48:18.785376Z"Pillai, Mohandas"https://zbmath.org/authors/?q=ai:pillai.mohandasSummary: We consider the wave maps problem with domain \(\mathbb{R}^{2+1}\) and target \(\mathbb{S}^2\) in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from \(\mathbb{R}^2\) to \(\mathbb{S}^2 \), with polar angle equal to \(Q_1(r) = 2 \arctan (r)\). By applying the scaling symmetry of the equation, \(Q_{\lambda }(r) = Q_1(r \lambda )\) is also a harmonic map, and the family of all such \(Q_{\lambda }\) are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps. In this work, we construct infinite time blowup solutions along the \(Q_{\lambda }\) family. More precisely, for \(b>0\), and for all \(\lambda_{0,0,b} \in C^{\infty }([100,\infty ))\) satisfying, for some \(C_l, C_{m,k}>0\), \[ \frac{C_l}{\log^b(t)} \leq \lambda_{0,0,b}(t) \leq \frac{C_m}{\log^b(t)}, \quad |\lambda_{0,0,b}^{(k)}(t)| \leq \frac{C_{m,k}}{t^k \log^{b+1}(t) }, k\geq 1 \quad t \geq 100\] there exists a wave map with the following properties. If \(u_b\) denotes the polar angle of the wave map into \(\mathbb{S}^2 \), we have \[u_b(t,r) = Q_{\frac{1}{\lambda_b(t)}}(r) + v_2(t,r) + v_e(t,r), \quad t \geq T_0\] where \[-\partial_{tt}v_2+\partial_{rr}v_2+\frac{1}{r}\partial_rv_2-\frac{v_2}{r^2}=0 ||\partial_t(Q_{\frac{1}{\lambda_b(t)}}+v_e)||_{L^2(r dr)}^2+||\frac{v_e}{r}||_{L^2(r dr)}^2 + ||\partial_rv_e||_{L^2(r dr)}^2 \leq \frac{C}{t^2 \log^{2b}(t)}, \quad t \geq T_0\] and \[\lambda_b(t) = \lambda_{0,0,b}(t) + O\left (\frac{1}{\log^b(t) \sqrt{\log (\log (t))}}\right ) \]Nonlinear PDE in the presence of singular randomnesshttps://zbmath.org/1521.350032023-11-13T18:48:18.785376Z"Tzvetkov, Nikolay"https://zbmath.org/authors/?q=ai:tzvetkov.nikolaySummary: This paper describes results concerning the construction of lowregularity solutions of nonlinear partial differential equations that depend on a random parameter. The motivations for this study are very varied. However, in the end, the results obtained and the methods used are conceptually very similar.A regularized system for the nonlinear variational wave equationhttps://zbmath.org/1521.350042023-11-13T18:48:18.785376Z"Grunert, Katrin"https://zbmath.org/authors/?q=ai:grunert.katrin"Reigstad, Audun"https://zbmath.org/authors/?q=ai:reigstad.audunSummary: We present a new generalization of the nonlinear variational wave equation. We prove existence of local, smooth solutions for this system. As a limiting case, we recover the nonlinear variational wave equation.Space-like quantitative uniqueness for parabolic operatorshttps://zbmath.org/1521.350052023-11-13T18:48:18.785376Z"Arya, Vedansh"https://zbmath.org/authors/?q=ai:arya.vedansh"Banerjee, Agnid"https://zbmath.org/authors/?q=ai:banerjee.agnidSummary: We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a \(C^1\) potential \(V\). Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.Uniqueness in weighted Lebesgue spaces for an elliptic equation with drift on manifoldshttps://zbmath.org/1521.350062023-11-13T18:48:18.785376Z"Meglioli, Giulia"https://zbmath.org/authors/?q=ai:meglioli.giulia"Roncoroni, Alberto"https://zbmath.org/authors/?q=ai:roncoroni.albertoSummary: We investigate the uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of elliptic equations with a drift posed on a complete, noncompact, Riemannian manifold \(M\) of infinite volume and dimension \(N\geq 2\). Furthermore, in the special case of a model manifold with polynomial volume growth, we show that the conditions on the drift term are sharp.Generalized curvature for the optimal transport problem induced by a Tonelli Lagrangianhttps://zbmath.org/1521.350072023-11-13T18:48:18.785376Z"Yang, Yuchuan"https://zbmath.org/authors/?q=ai:yang.yuchuanSummary: We propose a generalized curvature that is motivated by the optimal transport problem on \(\mathbb{R}^d\) with cost induced by a Tonelli Lagrangian \(L\). We show that non-negativity of the generalized curvature implies displacement convexity of the generalized entropy functional on the \(L\)-Wasserstein space along \(C^2\) displacement interpolants.Integral representation of superoscillations via complex Borel measures and their convergencehttps://zbmath.org/1521.350082023-11-13T18:48:18.785376Z"Behrndt, Jussi"https://zbmath.org/authors/?q=ai:behrndt.jussi"Colombo, Fabrizio"https://zbmath.org/authors/?q=ai:colombo.fabrizio"Schlosser, Peter"https://zbmath.org/authors/?q=ai:schlosser.peter"Struppa, Daniele C."https://zbmath.org/authors/?q=ai:struppa.daniele-carloSummary: In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some convergence to a plane wave is the standard characterizing feature of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of superoscillations either as generalized Fourier series, as certain integrals or via special functions. The aim of this work is to close these gaps and give a general definition of superoscillations, covering the well-known examples in the existing literature. Superoscillations will be defined as sequences of holomorphic functions, which admit integral representations with respect to complex Borel measures and converge to a plane wave in the space \(\mathcal{A}_1(\mathbb{C})\) of entire functions of exponential type.Darboux integrability for diagonal systems of hydrodynamic typehttps://zbmath.org/1521.350092023-11-13T18:48:18.785376Z"Agafonov, Sergey I."https://zbmath.org/authors/?q=ai:agafonov.sergey-iSummary: We prove that (1) diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable, (2) systems for commuting flows are Darboux integrable if and only if the Laplace transformation sequences terminate, (3) Darboux integrable systems are necessarily semihamiltonian. We give geometric interpretation for Darboux integrability of such systems in terms of congruences of lines and in terms of solution orbits with respect to symmetry subalgebras, discuss known and new examples.An identification problem of source function for one semievolutionary systemhttps://zbmath.org/1521.350102023-11-13T18:48:18.785376Z"Belov, Yuriĭ Ya."https://zbmath.org/authors/?q=ai:belov.yurii-yaSummary: An identification problem of source function for the semievolutionary system of two partial differential equations, one of which is parabolic, and the second -- elliptic are investigated. The Cauchy problem and the first boundary-value problem are considered. Initial problems are approximated by problems in which the elliptic equation is replaced with the parabolic equation containing the small parameter \(\varepsilon>0\) at a derivative with respect to time.Lie group analysis for a \((2+1)\)-dimensional generalized modified dispersive water-wave system for the shallow water waveshttps://zbmath.org/1521.350112023-11-13T18:48:18.785376Z"Liu, Fei-Yan"https://zbmath.org/authors/?q=ai:liu.feiyan"Gao, Yi-Tian"https://zbmath.org/authors/?q=ai:gao.yitian"Yu, Xin"https://zbmath.org/authors/?q=ai:yu.xin.1"Ding, Cui-Cui"https://zbmath.org/authors/?q=ai:ding.cui-cui"Li, Liu-Qing"https://zbmath.org/authors/?q=ai:li.liu-qingSummary: Shallow water waves refer to the waves with the bottom boundary affecting the movement of water quality points when the ratio of water depth to wavelength is small. Under investigation in this paper is a \((2+1)\)-dimensional generalized modified dispersive water-wave (GMDWW) system for the shallow water waves. We obtain the Lie point symmetry generators and Lie symmetry groups for the GMDWW system via the Lie group method. Optimal system of the one-dimensional subalgebras is derived. According to that optimal system, we obtain certain symmetry reductions. Hyperbolic-function, trigonometric-function and rational solutions for the GMDWW system are derived via the polynomial expansion, Riccati equation expansion and \(\left(\frac{G^\prime}{G}\right)\) expansion methods.Subsonic time-periodic solution to the compressible Euler equations triggered by boundary conditionshttps://zbmath.org/1521.350122023-11-13T18:48:18.785376Z"Zhang, Xiaomin"https://zbmath.org/authors/?q=ai:zhang.xiaomin"Sun, Jiawei"https://zbmath.org/authors/?q=ai:sun.jiawei"Yu, Huimin"https://zbmath.org/authors/?q=ai:yu.huiminSummary: In this paper, we consider the one-dimensional isentropic compressible Euler equations with source term \(\beta(t, x)\rho|u|^\alpha u\) in a bounded domain, which can be used to describe gas transmission in a nozzle. The model is imposed a subsonic time-periodic boundary condition. The main results reveal that the time-periodic boundary can trigger an unique subsonic time-periodic smooth solution and this unique periodic solution is stable under small perturbations on initial and boundary data. To get the existence of subsonic time-periodic solution, we use the linear iterative skill and transfer the boundary value problem into two initial value ones by using the hyperbolic property of the system, then the corresponding linearized system can be decoupled. The uniqueness is a direct by-product of the stability. There is no small assumptions on coefficient \(\beta(t, x)\).Hyperbolic limit for a biological invasionhttps://zbmath.org/1521.350132023-11-13T18:48:18.785376Z"Hilhorst, Danielle"https://zbmath.org/authors/?q=ai:hilhorst.danielle"Kim, Yongjung"https://zbmath.org/authors/?q=ai:kim.yongjung"Nguyen, Thanh Nam"https://zbmath.org/authors/?q=ai:nguyen.thanh-nam"Park, Hyunjoon"https://zbmath.org/authors/?q=ai:park.hyunjoonSummary: In a spatially heterogeneous environment the propagation speed of a biological invasion varies in space. The traveling wave theory in a homogeneous case is not extended to a heterogeneous case. Taking a singular limit in a hyperbolic scale is a good way to study such a wave propagation with constant speed. The goal of this project is to understand the effect of biological diffusion on the wave speed in a spatial heterogeneous environment. For this purpose, we consider \[U_t=\varepsilon(\gamma(s)U)_{xx}+\frac{1}{\varepsilon}U(1-U/m(x)),\] where \(m\) is a nonconstant carrying capacity, \(s=\frac{U}{m}\) is a starvation measure and \(\gamma(s)=s^{\tilde{k}}\), \(\widetilde{k}\ge 1\). The diffusion is a starvation driven diffusion. We show that the diffusion speed is constant even if \(m\) is nonconstant.Singular perturbation approach to a plankton model generating harmful algal bloomhttps://zbmath.org/1521.350142023-11-13T18:48:18.785376Z"Ikeda, Hideo"https://zbmath.org/authors/?q=ai:ikeda.hideoSummary: Spatially localized blooms of toxic plankton species have negative effects on other organisms owing to the production of toxins, mechanical damage, or other means. A two-prey (toxic and nontoxic phytoplankton) one-predator (zooplankton) Lotka-Volterra system with diffusion has been presented to understand the mechanism underlying the formation of spatial blooms of toxic plankton. In this study, we consider a one-dimensional (1D) system in which the ratio \(D\) of the diffusion rates of the predator and two prey, and the length of the spatial interval \(L\) are both sufficiently large while maintaining \(L^2/D\) as a constant. This system is reduced to a singularly perturbed system. The existence and stability properties of the 1D blooming stationary solutions are demonstrated by improving the previous results.Concentrated solution of Kirchhoff-type equationshttps://zbmath.org/1521.350152023-11-13T18:48:18.785376Z"Lan, Enhao"https://zbmath.org/authors/?q=ai:lan.enhaoSummary: In this paper, we study the Kirchhoff-type equations \[\begin{cases}-h^2A(h^{2-n}\|\nabla u\|^2)\Delta u+V(x)u=u^p \\ 0<u(x)\in H^1(\mathbb{R}^n),\ \lim_{|x|\to\infty}u(x)=0,\end{cases}\] where \(n\ge 1,1<p<2^*-1\), \(h>0\) is a small parameter, \(A\) and \(V\) are continuous functions. Under suitable conditions on \(A\) and \(V\), we show the existence of solution which concentrate at non-degenerate critical point of \(V\).Existence and multiplicity of solutions for perturbed fractional \(p\)-Laplacian equations with critical nonlinearity in \(\mathbb{R}^N\)https://zbmath.org/1521.350162023-11-13T18:48:18.785376Z"Li, Qin"https://zbmath.org/authors/?q=ai:li.qin.2"Yang, Zuodong"https://zbmath.org/authors/?q=ai:yang.zuodongSummary: In this paper, we consider the existence and multiplicity of solutions for the following perturbed fractional \(p\)-Laplacian equation \[\begin{cases}\epsilon^{sp}(-\Delta)^s_pu+V(x)|u|^{p-2}u=A(x)|u|^{p^*_s-2}u+h(x,u),\ x\in\mathbb{R}^N,\\ u(x)\to 0,\quad\text{as }|x|\to 0.\end{cases}\] Under some mild conditions on \(V,A\) and \(h\), we show that the problem has at least one positive weak solution provided \(\epsilon\le\varepsilon_\sigma\), and for any \(m^*\in\mathbb{N}\), it has \(m^*\) pairs of solutions if \(\epsilon\le\varepsilon_{m^*\sigma}\), where \(\varepsilon_\sigma\) and \(\varepsilon_{m^*\sigma}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(W^{s,p}(\mathbb{R}^N)\) as \(\epsilon\to 0\)Small divisors effects in some singularly perturbed initial value problem with irregular singularityhttps://zbmath.org/1521.350172023-11-13T18:48:18.785376Z"Malek, Stephane"https://zbmath.org/authors/?q=ai:malek.stephaneSummary: We examine a nonlinear initial value problem both singularly perturbed in a complex parameter and singular in complex time at the origin. The study undertaken in this paper is the continuation of a joined work with Lastra published in 2015. A change of balance between the leading and a critical subdominant term of the problem considered in our previous work is performed. It leads to a phenomenon of coalescing singularities to the origin in the Borel plane with respect to time for a finite set of holomorphic solutions constructed as Fourier series in space on horizontal complex strips. In comparison to our former study, an enlargement of the Gevrey order of the asymptotic expansion for these solutions relatively to the complex parameter is induced.A singular perturbation problem for mean field games of acceleration: application to mean field games of controlhttps://zbmath.org/1521.350182023-11-13T18:48:18.785376Z"Mendico, Cristian"https://zbmath.org/authors/?q=ai:mendico.cristianSummary: The singular perturbation of mean field game systems arising from minimization problems with control of acceleration is addressed, that is, we analyze the behavior of solutions as the acceleration costs vanishes. In this setting, the Hamiltonian fails to be strictly convex and coercive w.r.t. the momentum variable and, so, the classical results for Tonelli Hamiltonian systems cannot be applied. However, we show that the limit system is of MFG type in two different cases: we first study the convergence to the classical MFG system and, then, by a finer analysis of the Euler-Lagrange flow associated with the control of acceleration, we prove the convergence to a class of MFG systems, known as, MFG of control.Non-degeneracy of single-peak solutions to a Kirchhoff equationhttps://zbmath.org/1521.350192023-11-13T18:48:18.785376Z"Yan, Jiahong"https://zbmath.org/authors/?q=ai:yan.jiahong"Yang, Jing"https://zbmath.org/authors/?q=ai:yang.jing.1|yang.jing.5|yang.jing.4|yang.jing.6|yang.jing|yang.jing.2|yang.jing.7In this paper, the authors study the following singular perturbation Kirchhoff equation
\[
-\left(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2\mathrm{d}y \right)\Delta u+V(y)u=|u|^{p-1}u, \ u\in H^1(\mathbb{R}^3),
\]
where \(a,b,\varepsilon>0\), \(1<p<5\), and \(V(y):\mathbb{R}^3\rightarrow\mathbb{R}\) satisfies the following assumption:
\begin{itemize}
\item [\((V_0)\)] \(V(y)\) is a bounded \(\mathcal{C}^1\) function with
\[
V(y_0)=\inf_{y\in\mathbb{R}^3}V(y)>0 \ \mathrm{and} \ V(y_0)<V(y) \ \mathrm{for} \ y\in\mathbb{R}^3\backslash \{y_0\}.
\]
\end{itemize}
By using local Pohozaev identity and blow-up analysis, if \((V_0)\) holds and \(y_0\) is a non-degenerate critical point of \(V\), they derive the non-degeneracy of the single-peak solutions to the above equation for small \(\varepsilon\).
Reviewer: Chun-Lei Tang (Chongqing)On homogenization problems with oscillating Dirichlet conditions in space-time domainshttps://zbmath.org/1521.350202023-11-13T18:48:18.785376Z"Zhang, Yuming Paul"https://zbmath.org/authors/?q=ai:zhang.yuming-paulSummary: We prove the homogenization of fully nonlinear parabolic equations with periodic oscillating Dirichlet boundary conditions on certain general prescribed space-time domains. It was proved in
[\textit{W. M. Feldman}, J. Math. Pures Appl. (9) 101, No. 5, 599--622 (2014; Zbl 1293.35109); \textit{W. M. Feldman} and \textit{I. C. Kim}, Ann. Sci. Éc. Norm. Supér. (4) 50, No. 4, 1017--1064 (2017; Zbl 1381.35039)]
that for elliptic equations, the homogenized boundary data exist at boundary points with irrational normal directions, and it is generically discontinuous elsewhere. However, for parabolic problems, on a flat moving part of the boundary, we prove the existence of continuous homogenized boundary data \(\bar{g}\). We also show that, unlike the elliptic case, \(\bar{g}\) can be discontinuous even if the operator is rotation/reflection invariant.Homogenization for both oscillating operator and Neumann boundary value: \(W^{1, p}\) convergence ratehttps://zbmath.org/1521.350212023-11-13T18:48:18.785376Z"Zhao, Jie"https://zbmath.org/authors/?q=ai:zhao.jie.1"Wang, Juan"https://zbmath.org/authors/?q=ai:wang.juanSummary: In this paper, we will study the \(W^{1, p}\) convergence rate for homogenization problems of solutions for both oscillating operator and Neumann boundary data. By introducing the auxiliary periodic problems as well as Neumann correctors, we reduce the setting of both oscillating operator and Neumann boundary data to a fixed operator, which utilize mostly the fact that the Neumann function as well as its gradient pointwise convergence results, respectively. Boundary layer phenomena in periodic homogenization is also considered.Nonintrusive reduced-order models for parametric partial differential equations via data-driven operator inferencehttps://zbmath.org/1521.350222023-11-13T18:48:18.785376Z"McQuarrie, Shane A."https://zbmath.org/authors/?q=ai:mcquarrie.shane-a"Khodabakhshi, Parisa"https://zbmath.org/authors/?q=ai:khodabakhshi.parisa"Willcox, Karen E."https://zbmath.org/authors/?q=ai:willcox.karen-eSummary: This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model.Alien invasion into the buffer zone between two competing specieshttps://zbmath.org/1521.350232023-11-13T18:48:18.785376Z"Ei, Shin-Ichiro"https://zbmath.org/authors/?q=ai:ei.shin-ichiro"Ikeda, Hideo"https://zbmath.org/authors/?q=ai:ikeda.hideo"Ogawa, Toshiyuki"https://zbmath.org/authors/?q=ai:ogawa.toshiyukiSummary: Bifurcation of non-monotone traveling wave solutions of the three-species Lotka-Volterra competition diffusion system under strong competition is studied. The well-known front and back traveling wave formed by two species may lose its stability by the effect of third species and, as a result, allows the invasion. To discuss how the invasion is possible, stability change with respect to the intrinsic growth rate for the alien species are studied. Both numerical and theoretical bifurcation analysis around the bifurcation point reveal how the invasion affects the segregation of the original two species.Short note on the position of bifurcation points for the limiting system arising from the two competing species modelhttps://zbmath.org/1521.350242023-11-13T18:48:18.785376Z"Kan-On, Yukio"https://zbmath.org/authors/?q=ai:kan-on.yukioSummary: In this paper, we treat the competition-diffusion system with nonlinear diffusion term, which was proposed by \textit{N. Shigesada} et al. [J. Theor. Biol. 79, No. 1, 83--99 (1979; \url{doi:10.1016/0022-5193(79)90258-3})] to model the segregation of interacting species, and discuss the bifurcation structure of nonnegative solution for the limiting system arising from the competition-diffusion system as the interspecific competition rate tends to \(+\infty\). To do this, we employ the comparison principle and the property of the Bessel function, and study the position of bifurcation points, at which a nonconstant solution bifurcates from the constant solution, for the limiting system.Stability and bifurcation in a reaction-diffusion model with nonlinear boundary conditionshttps://zbmath.org/1521.350252023-11-13T18:48:18.785376Z"Li, Shangzhi"https://zbmath.org/authors/?q=ai:li.shangzhi"Guo, Shangjiang"https://zbmath.org/authors/?q=ai:guo.shangjiangSummary: In this paper, we investigate the existence, stability, local and global bifurcation of the steady state solutions for a diffusive logistic model with nonlinear boundary conditions. Supercritical and subcritical bifurcation of steady state solutions are obtained by virtue of the Crandall-Rabinowitz theorem and the Lyapunov-Schmidt reduction. The local stability and the possibility of obtaining an Allee effect are also analyzed. Furthermore, the result on global bifurcation is obtained as well by employing the Rabinowitz theorem.Hopf bifurcation in a reaction-diffusion-advection two species model with nonlocal delay effecthttps://zbmath.org/1521.350262023-11-13T18:48:18.785376Z"Li, Zhenzhen"https://zbmath.org/authors/?q=ai:li.zhenzhen"Dai, Binxiang"https://zbmath.org/authors/?q=ai:dai.binxiang"Han, Renji"https://zbmath.org/authors/?q=ai:han.renjiSummary: The dynamics of a general reaction-diffusion-advection two species model with nonlocal delay effect and Dirichlet boundary condition is investigated in this paper. The existence and stability of the positive spatially nonhomogeneous steady state solution are studied. Then by regarding the time delay \(\tau\) as the bifurcation parameter, we show that Hopf bifurcation occurs near the steady state solution at the critical values \(\tau_n\) (\(n=0,1,2,\dots\)). Moreover, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to a Lotka-Volterra competition-diffusion-advection model with nonlocal delay.Extinction behavior for the fast diffusion equations with critical exponent and Dirichlet boundary conditionshttps://zbmath.org/1521.350272023-11-13T18:48:18.785376Z"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannick"Wei, Juncheng"https://zbmath.org/authors/?q=ai:wei.juncheng"Zheng, Youquan"https://zbmath.org/authors/?q=ai:zheng.youquanSummary: For a smooth bounded domain \(\Omega \subseteq \mathbb{R}^n\), \(n\geqslant 3\), we consider the fast diffusion equation with critical sobolev exponent
\[
\frac{\partial w}{\partial \tau} =\Delta w^{\frac{n-2}{n+2}}
\]
under Dirichlet boundary condition \(w(\cdot, \tau) = 0\) on \(\partial \Omega \). Using the parabolic gluing method, we prove existence of an initial data \(w_0\) such that the corresponding solution has extinction rate of the form
\[
\Vert w(\cdot, \tau)\Vert_{L^\infty (\Omega)} = \gamma_0(T-\tau)^{\frac{n+2}{4}}{\left|\ln (T-\tau)\right|}^{\frac{n+2}{2(n-2)}}(1+o(1))
\]
as \(t\rightarrow T^-\), here \(T > 0\) is the finite extinction time of \(w(x, \tau)\). This generalizes a result of \textit{V. A. Galaktionov} and \textit{J. R. King} [Nonlinearity 15, No. 1, 173--188 (2002; Zbl 0988.35088)] for the radially symmetric case \(\Omega =B_1(0) : = \lbrace x\in \mathbb{R}^n||x| < 1\rbrace \subset \mathbb{R}^n\).Spectral instability of small-amplitude periodic waves of the electronic Euler-Poisson systemhttps://zbmath.org/1521.350282023-11-13T18:48:18.785376Z"Noble, Pascal"https://zbmath.org/authors/?q=ai:noble.pascal"Miguel Rodrigues, Luis"https://zbmath.org/authors/?q=ai:rodrigues.luis-miguel"Sun, Changzhen"https://zbmath.org/authors/?q=ai:sun.changzhenSummary: The present work shows that essentially all small-amplitude periodic traveling waves of the electronic Euler-Poisson system are spectrally unstable. This instability is neither modulational nor co-periodic, and thus requires an unusual spectral analysis and, beyond specific computations, newly devised arguments. The growth rate with respect to the amplitude of the background waves is also provided when the instability occurs.Spatial heterogeneity localizes Turing patterns in reaction-cross-diffusion systemshttps://zbmath.org/1521.350292023-11-13T18:48:18.785376Z"Gaffney, Eamonn A."https://zbmath.org/authors/?q=ai:gaffney.eamonn-a"Krause, Andrew L."https://zbmath.org/authors/?q=ai:krause.andrew-l"Maini, Philip K."https://zbmath.org/authors/?q=ai:maini.philip-k"Wang, Chenyuan"https://zbmath.org/authors/?q=ai:wang.chenyuanSummary: Motivated by bacterial chemotaxis and multi-species ecological interactions in heterogeneous environments, we study a general one-dimensional reaction-cross-diffusion system in the presence of spatial heterogeneity in both transport and reaction terms. Under a suitable asymptotic assumption that the transport is slow over the domain, while gradients in the reaction heterogeneity are not too sharp, we study the stability of a heterogeneous steady state approximated by the system in the absence of transport. Using a WKB ansatz, we find that this steady state can undergo a Turing-type instability in subsets of the domain, leading to the formation of localized patterns. The boundaries of the pattern-forming regions are given asymptotically by `local' Turing conditions corresponding to a spatially homogeneous analysis parameterized by the spatial variable. We developed a general open-source code which is freely available, and show numerical examples of this localized pattern formation in a Schnakenberg cross-diffusion system, a Keller-Segel chemotaxis model, and the Shigesada-Kawasaki-Teramoto model with heterogeneous parameters. We numerically show that the patterns may undergo secondary instabilities leading to spatiotemporal movement of spikes, though these remain approximately within the asymptotically predicted localized regions. This theory can elegantly differentiate between spatial structure due to background heterogeneity, from spatial patterns emergent from Turing-type instabilities.Asymptotics of the radiation field for the massless Dirac-Coulomb systemhttps://zbmath.org/1521.350302023-11-13T18:48:18.785376Z"Baskin, Dean"https://zbmath.org/authors/?q=ai:baskin.dean"Booth, Robert"https://zbmath.org/authors/?q=ai:booth.robert-i|booth.robert-k"Gell-Redman, Jesse"https://zbmath.org/authors/?q=ai:gell-redman.jesseSummary: We consider the long-time behavior of solutions to the massless Dirac equation coupled to a Coulomb potential. For nice enough initial data, we find a joint asymptotic expansion for solutions near the null and future infinities and characterize explicitly the decay rates seen in the expansion. This paper can be viewed as a successor to previous work on asymptotic expansions for the radiation field
[the first author et al., Am. J. Math. 137, No. 5, 1293--1364 (2015; Zbl 1332.58009); Adv. Math. 328, 160--216 (2018; Zbl 1387.35048); the first author and \textit{J. L. Marzuola}, ``The radiation field on product cones'', Preprint, \url{arXiv:1906.04769}].
The key new elements are propagation estimates near the singularity of the potential, building on work of the first author with \textit{J. Wunsch} [``Diffraction for the Dirac-Coulomb propagator'', Preprint, \url{arXiv:2011.08890}] and an explicit calculation with hypergeometric functions to determine the rates of decay.Asymptotic behavior of nonlocal bistable reaction-diffusion equationshttps://zbmath.org/1521.350312023-11-13T18:48:18.785376Z"Besse, Christophe"https://zbmath.org/authors/?q=ai:besse.christophe"Capel, Alexandre"https://zbmath.org/authors/?q=ai:capel.alexandre"Faye, Grégory"https://zbmath.org/authors/?q=ai:faye.gregory"Fouilhé, Guilhem"https://zbmath.org/authors/?q=ai:fouilhe.guilhemSummary: In this paper, we study the asymptotic behavior of the solutions of nonlocal bistable reaction-diffusion equations starting from compactly supported initial conditions. Depending on the relationship between the nonlinearity, the interaction kernel and the diffusion coefficient, we show that the solutions can either: propagate, go extinct or remain pinned. We especially focus on the latter regime where solutions are pinned by thoroughly studying discontinuous ground state solutions of the problem for a specific interaction kernel serving as a case study. We also present a detailed numerical analysis of the problem.Asymptotic stability of viscous contact wave to a radiation hydrodynamic limit modelhttps://zbmath.org/1521.350322023-11-13T18:48:18.785376Z"Fan, Lili"https://zbmath.org/authors/?q=ai:fan.lili"Li, Kaiqiang"https://zbmath.org/authors/?q=ai:li.kaiqiangSummary: This paper is concerned with the large time behavior of the solutions for 1D radiation hydrodynamic limit model without viscosity and its asymptotic stability of the viscous contact discontinuity wave under the smallness assumption of the strength of the contact wave and initial perturbations. The present pressure includes a fourth-order term about the absolute temperature from radiation effect which brings the main difficulty. Furthermore, the dissipative of the system is weaker for the lack of viscosity. All these make the problem more challenging. The prove is mainly based on the energy method, including normal and radial directions energy estimates.Symmetry properties of sign-changing solutions to nonlinear parabolic equations in unbounded domainshttps://zbmath.org/1521.350332023-11-13T18:48:18.785376Z"Földes, Juraj"https://zbmath.org/authors/?q=ai:foldes.juraj"Saldaña, Alberto"https://zbmath.org/authors/?q=ai:saldana.alberto"Weth, Tobias"https://zbmath.org/authors/?q=ai:weth.tobiasSummary: We study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlinear parabolic problems in the whole space or in the exterior of a ball with Dirichlet boundary conditions. We show that, under suitable regularity and stability assumptions, solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega-limit set are axially symmetric with respect to a common axis passing through the origin and are nonincreasing in the polar angle. We also obtain symmetry results for solutions of Hénon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions.Convergence to a terrace solution in multistable reaction-diffusion equation with discontinuitieshttps://zbmath.org/1521.350342023-11-13T18:48:18.785376Z"Giletti, Thomas"https://zbmath.org/authors/?q=ai:giletti.thomas"Kim, Ho-Youn"https://zbmath.org/authors/?q=ai:kim.ho-younSummary: In this paper we address the large-time behavior of solutions of bistable and multistable reaction-diffusion equation with discontinuities around the stable steady states. We show that the solution always converges to a special solution, which may either be a traveling wave in the bistable case, or more generally a terrace (i.e. a collection of stacked traveling waves with ordered speeds) in the multistable case.Exponential stability for wave equation with delay and dynamic boundary conditionshttps://zbmath.org/1521.350352023-11-13T18:48:18.785376Z"Hao, Jianghao"https://zbmath.org/authors/?q=ai:hao.jianghao"Huo, Qiuyu"https://zbmath.org/authors/?q=ai:huo.qiuyuSummary: In this paper, we consider a multi-dimensional wave equation with delay and dynamic boundary conditions, related to the Kelvin-Voigt damping. By using the Faedo-Galerkin approximations together with some priori estimates, we prove the local existence of solution. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. Exponential stability result of system is also established by constructing suitable Lyapunov functionals.Optimal convergence rate to nonlinear diffusion waves for bipolar Euler-Poisson equations with critical overdampinghttps://zbmath.org/1521.350362023-11-13T18:48:18.785376Z"Li, Haitong"https://zbmath.org/authors/?q=ai:li.haitongSummary: This paper is concerned with the large time behaviors of smooth solutions to the Cauchy problem of the one dimensional bipolar Euler-Poisson equations with the time dependent critical overdamping. We show that in this critical overdamping case the bipolar Euler-Poisson system admits a unique global smooth solution that asymptotically converges to the nonlinear diffusion wave. In particular, the optimal convergence rate in logarithmic form is derived when the initial perturbations are \(L^2\) sense by using the technical time-weighted energy method.On a parabolic-ODE chemotaxis system with periodic asymptotic behaviorhttps://zbmath.org/1521.350372023-11-13T18:48:18.785376Z"Negreanu, M."https://zbmath.org/authors/?q=ai:negreanu.mihaela"Tello, J. I."https://zbmath.org/authors/?q=ai:tello.jose-ignacio"Vargas, A. M."https://zbmath.org/authors/?q=ai:vargas.antonio-manuelSummary: We consider a system of differential equations modeling chemotaxis, the habillity of some living organisms to move towards a higher concentration of a chemical signal. The system consists of two differential equations, a parabolic one describing the behavior of a biological species ``\(u\)'' coupled to second equation modeling the concentration of a chemical substance ``\(v\)''. The growth of the biological species is limited by a logistic-like term where the carrying capacity presents a time-periodic asymptotic behavior. The production of the chemical species is described in terms of a regular function \(h\), which increases as ``\(u\)'' increases. The system is presented in a regular bounded domain \(\Omega\subset\mathbb{R}^n\), with positive constant chemotaxis coefficient \(\chi\) in the following way
\[
\begin{cases}
u_t=\Delta{u}-\operatorname{div}(\chi u\nabla{v})+\mu u(1-u+f), \quad & x\in\Omega, \\
\epsilon v_t-D_v\Delta v=h(u,v), \quad & x\in\Omega,\; t>0,
\end{cases}
\]
with initial data \((u_0,v_0)\) and appropiate boundary conditions for \(u\). The function \(f\), in the reaction term, is a bounded given function fulfilling
\[
\|f(x,t)-f^*(t)\|_{L^{\infty}(\Omega)}\rightarrow 0, \quad \text{ as }\quad t\rightarrow\infty,
\]
with \(f^*(t)\) being a time-periodic function independent of the space variable ``\(x\)''.
Three different cases may occur:
\begin{itemize}
\item If in the equation of \(v\), the diffusion is dominant in the time scale we are working, then, the system is simplify to a Parabolic-Elliptic equation
\item If there is not diffusion of \(v\), the problem is a Parabolic-ODE system.
\item When diffusion is not dominant and neither neglectable in the time scales we are studying the the problem.
\end{itemize}
In the chapter we present results of existence of solutions and its asymptotic behavior under suitable assumptions on the initial data for given functions \(f\) and \(h\).
For the entire collection see [Zbl 07725138].Dynamics of a diffusive mussel-algae system in closed advective environmentshttps://zbmath.org/1521.350382023-11-13T18:48:18.785376Z"Qu, Anqi"https://zbmath.org/authors/?q=ai:qu.anqi"Tong, Xue"https://zbmath.org/authors/?q=ai:tong.xue"Wang, Jinfeng"https://zbmath.org/authors/?q=ai:wang.jinfengSummary: In this paper, we are concerned with a diffusive system modeling the interaction of mussel and algae in the water layer overlying the mussel bed, where the algae are the main food source for mussels, and the advection pushes algaes in one direction but not out of the domain. We present a threshold result on the global extinction and persistence of mussels. It shows that the condition for persistence depends on the principal eigenvalue of a scalar boundary value problem, which is related to the diffusion, the speed of the tidal flow, the conversion rate of algae to mussel production, and mussel mortality rate. We further investigate the asymptotic profile of the positive steady state when it exists.Memory effects on the stability of viscoelastic Timoshenko systems in the whole \(1D\)-spacehttps://zbmath.org/1521.350392023-11-13T18:48:18.785376Z"Silva, Marcio Antonio Jorge"https://zbmath.org/authors/?q=ai:jorge-silva.marcio-antonio"Ueda, Yoshihiro"https://zbmath.org/authors/?q=ai:ueda.yoshihiroSummary: In this paper, we investigate new classes of viscoelastic Timoshenko-Ehrenfest systems under the presence of full or partial memory effects. Our achievements rely on recent approaches to the theory of dissipative structure for systems of differential equations, by featuring optimal pointwise estimates in the Fourier space, \(L^2\)-estimates for the solutions, and explicit energy decay rates depending on the viscoelastic damping coupling. Therefore, under a complete stability analysis, original results as well as improvements of previous work in the literature are our main findings.Indirect stabilization of a coupled system by memory effectshttps://zbmath.org/1521.350402023-11-13T18:48:18.785376Z"Tyszka, Guilherme F."https://zbmath.org/authors/?q=ai:tyszka.guilherme-f"Oquendo, Higidio Portillo"https://zbmath.org/authors/?q=ai:portillo-oquendo.higidioSummary: We consider an abstract model of two coupled elastic materials. One of the materials has conservative characteristics, whereas the other one has dissipative properties. The dissipative effect is caused by the presence of a memory term that depends on the fractional stationary operator with exponent \(\theta\in[0,1]\). In this paper, we study the asymptotic behavior of the solutions for this system. We show that the solutions decay polynomially with the rate \(t^{-1/(4-2\theta)}\). For problems with that level of generality, we show that the above rate is the best. We also study the asymptotic behavior when the wave propagation speeds of both materials coincide. For this case, we find that the decay rate is so fast as \(t^{-1/(2-2\theta)}\) for \(\theta\ne 1\). For completeness, we also approach the case \(\theta=1\), where an exponential decay of solutions is obtained.Acceleration of propagation in a chemotaxis-growth system with slowly decaying initial datahttps://zbmath.org/1521.350412023-11-13T18:48:18.785376Z"Wang, Zhi-An"https://zbmath.org/authors/?q=ai:wang.zhian"Xu, Wen-Bing"https://zbmath.org/authors/?q=ai:xu.wenbing.1|xu.wenbingSummary: In this paper, we study the spatial propagation dynamics of a parabolic-elliptic chemotaxis system with logistic source which reduces to the well-known Fisher-KPP equation without chemotaxis. It is known that for fast decaying initial functions, this system has a finite spreading speed. For slowly decaying initial functions, we show that the accelerating propagation will occur and chemotaxis does not affect the propagation mode determined by slowly decaying initial functions if the logistic damping is strong, that is, the system has the same upper and lower bounds of the accelerating propagation as for the classical Fisher-KPP equation. The main new idea of proving our results is the construction of auxiliary equations to overcome the lack of comparison principle due to chemotaxis.Global well-posedness of wave equation with weak and strong damping terms p-Laplacian and logarithmic nonlinearity source termhttps://zbmath.org/1521.350422023-11-13T18:48:18.785376Z"Wu, Xiulan"https://zbmath.org/authors/?q=ai:wu.xiulan"Yang, Xiaoxin"https://zbmath.org/authors/?q=ai:yang.xiaoxin"Cheng, Libo"https://zbmath.org/authors/?q=ai:cheng.liboSummary: We consider the following damped p-laplacian type wave equation with logarithmic nonlinearity \(u_{tt} - \varDelta u_t - \varDelta_p u + u_t = |u|^{q - 2} u \ln |u|\), \(x\in\varOmega\), \(t > 0\) in a bounded domain with homogeneous Dirichlet boundary condition. Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and when the initial energy is subcritical, a sufficient condition for the solutions to blow up in finite time is derived, by combining the Nehari manifold with concavity argument. Then we parallelly extend the conclusions of global existence and energy decay for the subcritical case to the critical case by scaling technique. Besides, When the initial energy is supercritical, some new skills are invented to establish another finite time blow-up criterion for this problem.Asymptotic behavior for stochastic plate equations on unbounded domainshttps://zbmath.org/1521.350432023-11-13T18:48:18.785376Z"Yao, Xiao Bin"https://zbmath.org/authors/?q=ai:yao.xiaobin"Feng, Deng Juan"https://zbmath.org/authors/?q=ai:feng.deng-juanSummary: In this paper, we investigate the long-time behavior of the solutions for stochastic plate equations with memory and additive noise. First we establish a continuous cocycle for the equation and the pullback asymptotic compactness of solutions. Second we prove the existence of random attractors for the equation.Stability of a viscoelastic Timoshenko system with non-monotonic kernelhttps://zbmath.org/1521.350442023-11-13T18:48:18.785376Z"Zhang, Hai-E."https://zbmath.org/authors/?q=ai:zhang.haie"Xu, Gen-Qi"https://zbmath.org/authors/?q=ai:xu.gen-qi"Chen, Hao"https://zbmath.org/authors/?q=ai:chen.hao.9Summary: In this paper, the stability of a linear Timoshenko beam system involved with infinite memory is considered. Different from the previous results on where the monotony of kernel is always fulfilled, the memory kernel under consideration is assumed to be non-monotonic. The well-posedness of the system is obtained by means of resolvent family theory and the exponential stability is proved under certain conditions. Numerical simulations are also presented to verify the main results.Boundedness and asymptotic behavior of solutions to one-dimensional urban crime system with nonlinear diffusionhttps://zbmath.org/1521.350452023-11-13T18:48:18.785376Z"Zhao, Xiangdong"https://zbmath.org/authors/?q=ai:zhao.xiangdongSummary: This paper deals with a one-dimensional cross-diffusion system
\[
\begin{cases}
u_t = (D(u)u_x)_x - \chi(\frac{u}{v}v_x)_x - uv + B_1(x, t), & x\in\varOmega, t > 0, \\
v_t = v_{xx} - v + uv + B_2(x, t), & x\in\varOmega, t > 0,
\end{cases}
\]
which is proposed by \textit{M. B. Short} et al. [Math. Models Methods Appl. Sci. 18, 1249--1267 (2008; Zbl 1180.35530)] to describe the dynamics of urban crime. If \(D(u) \geq D_0(u + 1)^{m - 1}\) with \(D_0, m > 0\), it is proved for arbitrary \(\chi > 0\) that the system possesses a globally bounded classical solution provided \(m > \frac{1}{4}\) with some mild assumptions on nonnegative functions \(B_1\), \(B_2\). In addition, if \(B_2 \equiv 0\), the attractiveness value of \(v\) and its derivative \(v_x\) decay to zero in the long time limit.Upper semicontinuity of pullback attractors for nonlinear full von Kármán beamhttps://zbmath.org/1521.350462023-11-13T18:48:18.785376Z"Aouadi, Moncef"https://zbmath.org/authors/?q=ai:aouadi.moncef"Guerine, Souad"https://zbmath.org/authors/?q=ai:guerine.souadSummary: In this paper we study the long-time dynamics of pullback attractors for non-autonomous and nonlinear full von Kármán beam and its upper-semicontinuity property. The one-dimensional full von Kármán beam equations constitutes a basic model to describe the nonlinear oscillations with large displacements due the existence of nonlinear terms in the motion equations. Under quite general assumptions on nonlinear damping and sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions. We prove the existence of pullback attractors in the natural space energy. Finally, we prove the regularity of the family of pullback attractors and its upper semicontinuity with respect to non-autonomous perturbations.Limiting behavior of random attractors of stochastic supercritical wave equations driven by multiplicative noisehttps://zbmath.org/1521.350472023-11-13T18:48:18.785376Z"Chen, Zhang"https://zbmath.org/authors/?q=ai:chen.zhang"Wang, Bixiang"https://zbmath.org/authors/?q=ai:wang.bixiangSummary: This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero.Global attractors for porous-elasticity system from second spectrum viewpointhttps://zbmath.org/1521.350482023-11-13T18:48:18.785376Z"Feng, B."https://zbmath.org/authors/?q=ai:feng.baowei"Freitas, M. M."https://zbmath.org/authors/?q=ai:freitas.mirelson-m"Almeida, D. S."https://zbmath.org/authors/?q=ai:almeida.dilberto-s-jun|almeida-junior.dilberto-da-silva"Ramos, A. J. A."https://zbmath.org/authors/?q=ai:ramos.anderson-j-a"Caljaro, R. Q."https://zbmath.org/authors/?q=ai:caljaro.r-qSummary: This paper is concerned with the global attractors for a new partially damped porous-elasticity system by taking a truncated version which is free of blow-up on second wave speed. We establish the global well-posedness of the system via Faedo-Galerkin method. By considering only one damping term acting on the volume fraction in the system we prove the existence of absorbing set for the solution semigroup regardless any relationship between coefficients of the system. Finally, by using Lyapunov and recent quasi-stability methods we prove the existence of smooth global attractors with finite fractal dimension.Large time behavior of deterministic and stochastic 3D convective Brinkman-Forchheimer equations in periodic domainshttps://zbmath.org/1521.350492023-11-13T18:48:18.785376Z"Kinra, Kush"https://zbmath.org/authors/?q=ai:kinra.kush"Mohan, Manil T."https://zbmath.org/authors/?q=ai:mohan.manil-tSummary: The large time behavior of deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations
\[
\partial_t{\boldsymbol{u}}-\mu \Delta{\boldsymbol{u}}+({\boldsymbol{u}}\cdot \nabla){\boldsymbol{u}}+\alpha{\boldsymbol{u}}+\beta |{\boldsymbol{u}}|^{r-1}{\boldsymbol{u}}+\nabla p={\boldsymbol{f}}, \nabla \cdot{\boldsymbol{u}}=0,
\]
for \(r\ge 3\) (\(\mu,\beta >0\) for \(r>3\) and \(2\beta \mu \ge 1\) for \(r=3\)), in periodic domains is carried out in this work. Our first goal is to prove the existence of global attractors for the 3D deterministic CBF equations. Then, we show the existence of random attractors for the 3D stochastic CBF equations perturbed by small additive smooth noise. Furthermore, we establish the upper semicontinuity of random attractors for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero. Finally, we address the existence and uniqueness of invariant measures of 3D stochastic CBF equations.Structural stability and rate of convergence of global attractorshttps://zbmath.org/1521.350502023-11-13T18:48:18.785376Z"Lee, Jihoon"https://zbmath.org/authors/?q=ai:lee.jihoon.1"Pires, Leonardo"https://zbmath.org/authors/?q=ai:pires.leonardoSummary: In this paper, we prove that a Lipschitz perturbed map of a gradient Morse-Smale diffeomorphism \(T\) on \(\mathbb{R}^m\) has the Lipschitz shadowing property on a neighborhood of the global attractor \(\mathcal{A}_T\), and apply the result to get the structural stability and the rate of convergence of global attractors of Chafee-Infante equations under Lipschitz perturbations of the domain and equation.Long time behavior of semilinear wave equation with localized interior damping term under acoustic boundary conditionhttps://zbmath.org/1521.350512023-11-13T18:48:18.785376Z"Sen, Zehra"https://zbmath.org/authors/?q=ai:sen.zehra"Yayla, Sema"https://zbmath.org/authors/?q=ai:yayla.semaIn this interesting article, the authors consider the semilinear wave equation with localized interior damping in a bounded domain \(\Omega\subset\mathbb{R}^3\), for the unknown \(u\),
\[
u_{tt} + a(x)u_t - \Delta u + u + f(u) = h(x) \quad\text{in}\ (0,+\infty)\times\Omega,
\]
equipped with the acoustic boundary conditions, where the boundary's ``displacement'' from equilibrium is denoted by \(\delta\),
\begin{align*}
& \delta_{tt} + \rho\delta_t + \delta + g(\delta) = -u_t \quad\text{on}\ (0,+\infty)\times\partial\Omega, \\
& \delta_t = \partial_\nu u \quad\text{on}\ (0,+\infty)\times\partial\Omega.
\end{align*}
Of course, \(\rho>0\) and \(a\in L^\infty(\Omega)\) with \(a(x)\ge0\) a.e. in \(\Omega\). The existence of a global attractor for the associated semigroup is proved when \(f,g\in C^1(\mathbb{R})\) satisfy certain growth assumptions. Under further assumptions, the regularity and finite dimensionality of the attractor are also achieved.
Reviewer: Joseph Shomberg (Providence)Pullback attractors of the Bingham modelhttps://zbmath.org/1521.350522023-11-13T18:48:18.785376Z"Zvyagin, V. G."https://zbmath.org/authors/?q=ai:zvyagin.viktor-grigorevich"Ustiuzhaninova, A. S."https://zbmath.org/authors/?q=ai:ustiuzhaninova.a-sSummary: Based on the theory of trajectory pullback attractors, we study the qualitative behavior of weak solutions for the Bingham model with periodic conditions in the space variables. For the model under consideration, a family of trajectory spaces is introduced and the existence of pullback attractors is proved.A critical exponent for blow-up in a two-dimensional chemotaxis-consumption systemhttps://zbmath.org/1521.350532023-11-13T18:48:18.785376Z"Ahn, Jaewook"https://zbmath.org/authors/?q=ai:ahn.jaewook"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelThe authors study the following repulsive chemotaxis-consumption system
\[
\begin{cases}
\partial_t u = \nabla \cdot (D(u)\nabla u) + \nabla \cdot (u\nabla v)\\
0= \Delta v - uv \end{cases}\tag{1}
\]
in a disk \(B_R(0) \subset \mathbb{R}^2\) along with the boundary conditions
\[
(D(u)\nabla u + u \nabla v)\cdot \nu = 0, \ v = M (\mbox{a positive constant}), \ \ \ x \in \partial B_R(0), \ t >0,
\]
where \(D(\xi)\) suitably generalizes the function \((\xi + 1)^{-\alpha}, \xi \geq 0.\) Here \(u\) represents the population density and \(v\) is the signal concentration.
In a recent article, the second author et al. [Proc. R. Soc. Edinb., Sect. A, Math. 153, No. 4, 1150--1166 (2023; Zbl 1518.35155)] observed that for an admirably strong enhancement of chemotactic motion (i.e. if the term \(\nabla \cdot (u\nabla v)\) replaced by \(\nabla \cdot (u\frac{\nabla v}{v})\)), the system (1) exhibits finite time blow-up of solutions for \(\alpha >0\) within a considerably large set of choices of the initial data. The main goal of the present article is to illustrate certain phenomena of criticality regarding the occurrence of singularity formation. In particular, the authors observed that finite time blow-up of \textit{radial solutions} do not require strengthening of the cross-diffusion at small signal concentrations. The main results of the article are:
\begin{itemize}
\item Assume \(D(\xi) \leq \kappa(\xi + 1)^{-\alpha}\) for some \(\alpha>0,\kappa>0.\) For each radially symmetric initial data \(u_0 \in W^{1,\infty}(B_R(0))\) there exists \(M_{\star}(u_0)\) such that if \(M> M_{\star}(u_0)\) then (1) admits a classical solution \textit{blowing up in finite time}.
\item On the other hand, if \(D(\xi) \geq \kappa,\) then for each radially symmetric initial data \(u_0 \in W^{1,\infty}(B_R(0))\) and for each \(M>0,\) (1) admits a global bounded classical solution.
\end{itemize}
The proof of finite time blow-up achieved through an analysis of the cumulated density function \(w(s,t) := \int_0^{\sqrt{s}} \rho u(\rho,t) \ d\rho, \ s \in [0,R^2]\) and a moment like functional \(\int_0^{R^2} s^{-\gamma}w(s,t) \ ds, \gamma>0.\) The proof of global existence consists of first achieving a uniform \(L^2\)-smallness of the signal gradient locally around the origin, following an idea recently employed in [\textit{J. Ahn} et al., Math. Models Methods Appl. Sci. 33, No. 11, 2337--2360 (2023; \url{doi:10.1142/S0218202523400055})], and then by analyzing the evolution of the entropy by an appropriate domain splitting method.
Reviewer: Debabrata Karmakar (Bangalore)Instantaneous blow-up for evolution inequalities of Sobolev type with nonlinear convolution termshttps://zbmath.org/1521.350542023-11-13T18:48:18.785376Z"Alazman, Ibtehal"https://zbmath.org/authors/?q=ai:alazman.ibtehal"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairiSummary: We consider evolution inequalities of Sobolev type involving nonlinearities of the form \(|x|^{\sigma-N}*|u|^p\) and \(|x|^{\sigma-N}*|\nabla u|^p \), where \(*\) is the convolution product in \(\mathbb{R}^N\), \(p>1\) and \(0<\sigma<N\). For each case, we prove the existence of a critical exponent \(p_{cr}(\sigma,N)\in(1,\infty]\) depending on the parameter \(\sigma\) and the dimension \(N\), in the following sense: if \(1<p\le p_{cr}(\sigma,N)\), then there is no local weak solutions; if \(p>p_{cr}(\sigma,N)\), then local weak solutions exist for some initial data.Nonexistence criteria for a generalized Boussinesq-type equation in bounded and unbounded domainshttps://zbmath.org/1521.350552023-11-13T18:48:18.785376Z"Aldawish, Ibtisam"https://zbmath.org/authors/?q=ai:aldawish.ibtisam"Alazman, Ibtehal"https://zbmath.org/authors/?q=ai:alazman.ibtehal"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairi"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: In this paper, we consider a generalized Boussinesq-type equation posed in \((0,\infty)\times\Omega\), where \(\Omega\subset\mathbb{R}^N\). The considered equation arises in many physical models including the description of nonstationary processes in crystalline semiconductors. We will handle two cases: \(\Omega=\overline{\mathbb{R}^N\backslash B_1}\) and \(\Omega=B_1\backslash\{0\}\), where \(B_1\) is the closed unit ball in \(\mathbb{R}^N\). Using a unified approach, we establish nonexistence criteria for each case. Moreover, no restriction on the sign of solutions is imposed.Unstable ground state and blow up result of nonlocal Klein-Gordon equationshttps://zbmath.org/1521.350562023-11-13T18:48:18.785376Z"Carrião, Paulo Cesar"https://zbmath.org/authors/?q=ai:carriao.paulo-cesar"Lehrer, Raquel"https://zbmath.org/authors/?q=ai:lehrer.raquel"Vicente, André"https://zbmath.org/authors/?q=ai:vicente.andreSummary: In this paper we study the behaviour of solutions for a nonlocal hyperbolic equation. We use the Pohozaev manifold combined with a new technique to explicit two invariant regions in the space of initial data. On the first one the solution blows up (in finite or infinite time) and in the second one the solution exists globally. Additionally, we prove that the ground state solution of the elliptic problem associated to the original problem is unstable. The main goal of this paper is to present a new technique which allows us to consider nonlocal problems and to extend the classical result proved by \textit{J. Shatah} [Trans. Am. Math. Soc. 290, 701--710 (1985; Zbl 0617.35072)].Instabilities appearing in cosmological effective field theories: when and how?https://zbmath.org/1521.350572023-11-13T18:48:18.785376Z"Eckmann, Jean-Pierre"https://zbmath.org/authors/?q=ai:eckmann.jean-pierre"Hassani, Farbod"https://zbmath.org/authors/?q=ai:hassani.farbod"Zaag, Hatem"https://zbmath.org/authors/?q=ai:zaag.hatemSummary: Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories originated from cosmological studies. In particular, we are interested in the equation \(\partial^2_tu(x,t)=\alpha(\partial_xu(x,t))^+\beta\partial^2_xu(x,t)\) in \(1+1\) dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when \(\alpha>0\). We study the nature of this divergence as a function of the parameters \(\alpha>0\) and \(\beta\geqslant 0\). The divergence does not disappear even when \(\beta\) is very large contrary to what one might believe (note that since we consider fixed initial data, \(\alpha\) and \(\beta\) cannot be scaled away). But it will take longer to appear as \(\beta\) increases when \(\alpha\) is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to \(3+1\) dimensions.Blow-up on metric graphs and Riemannian manifoldshttps://zbmath.org/1521.350582023-11-13T18:48:18.785376Z"Punzo, Fabio"https://zbmath.org/authors/?q=ai:punzo.fabio"Tesei, Alberto"https://zbmath.org/authors/?q=ai:tesei.albertoSummary: We study blow-up versus global existence of solutions to a model semilinear parabolic equation in metric measure spaces. Applications to metric graphs and Riemannian manifolds are considered, pointing out the occurrence of the Fujita phenomenon.Estimates on blow-up time of a parabolic system with nonlinear boundary fluxhttps://zbmath.org/1521.350592023-11-13T18:48:18.785376Z"Zhang, Lingling"https://zbmath.org/authors/?q=ai:zhang.lingling.1|zhang.lingling"Zhang, Xiaoyue"https://zbmath.org/authors/?q=ai:zhang.xiaoyueSummary: A topic about a class of coupled parabolic equations under nonlinear boundary conditions is studied in this paper. Our attention is primarily on its upper and lower bounds for blow-up time. Necessary assumptions and auxiliary functions are established in the process of our proof. At the same time, we notice that the usual method to approaching the bounds is taking advantage of the different auxiliary functions. The divergence term and the cross term lead us to apply different methods to deal with the upper and lower bounds for blow-up time. Specifically, on the one hand, we seek a lower bound by taking advantage of establishing the auxiliary function and using the differential inequality technique. On the other hand, we derive an upper bound with the assistance of constructing the sub-solution and the comparison principle. In addition, two corollaries are proposed to complete our proof. Finally, we give an instance to state our results.Sharp operator-norm asymptotics for thin elastic plates with rapidly oscillating periodic propertieshttps://zbmath.org/1521.350602023-11-13T18:48:18.785376Z"Cherednichenko, Kirill"https://zbmath.org/authors/?q=ai:cherednichenko.kirill-d"Velčić, Igor"https://zbmath.org/authors/?q=ai:velcic.igorSummary: We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order. Assuming that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response, such as the case in, for example, acoustic wave propagation, we derive a class of `hybrid', homogenisation dimension-reduction, norm-resolvent estimates for the plate, under different energy scalings with respect to the plate thickness.A local energy estimate for 2-dimensional Dirichlet wave equationshttps://zbmath.org/1521.350612023-11-13T18:48:18.785376Z"Hepditch, Kellan"https://zbmath.org/authors/?q=ai:hepditch.kellan"Metcalfe, Jason"https://zbmath.org/authors/?q=ai:metcalfe.jason-lSummary: We examine a variant of the integrated local energy estimate for \((1{+}2)\)-dimensional Dirichlet wave equations exterior to star-shaped obstacles. The classical bound on the solution, rather than the derivative, is not typically available in two spatial dimensions. Using an argument inspired by the \(r^p\)-weighted method of Dafermos and Rodnianski and taking advantage of the Dirichlet boundary conditions allow for the recovery of such a term when the initial energy is appropriately weighted.Gradient higher integrability for degenerate parabolic double-phase systemshttps://zbmath.org/1521.350622023-11-13T18:48:18.785376Z"Kim, Wontae"https://zbmath.org/authors/?q=ai:kim.wontae"Kinnunen, Juha"https://zbmath.org/authors/?q=ai:kinnunen.juha"Moring, Kristian"https://zbmath.org/authors/?q=ai:moring.kristianSummary: We prove a local higher integrability result for the gradient of a weak solution to degenerate parabolic double-phase systems of \(p\)-Laplace type. This result comes with reverse Hölder type estimates. The proof is based on a careful phase analysis, estimates in the intrinsic geometries and stopping time arguments.Interior estimates of derivatives and a Liouville type theorem for parabolic \(k \)-Hessian equationshttps://zbmath.org/1521.350632023-11-13T18:48:18.785376Z"Bao, Jiguang"https://zbmath.org/authors/?q=ai:bao.jiguang"Qiang, Jiechen"https://zbmath.org/authors/?q=ai:qiang.jiechen"Tang, Zhongwei"https://zbmath.org/authors/?q=ai:tang.zhongwei"Wang, Cong"https://zbmath.org/authors/?q=ai:wang.congSummary: In this paper, we establish the gradient and Pogorelov estimates for \(k \)-convex-monotone solutions to parabolic \(k \)-Hessian equations of the form \(-u_t\sigma_k(\lambda(D^2u)) = \psi(x, t, u) \). We also apply such estimates to obtain a Liouville type result, which states that any \(k \)-convex-monotone and \(C^{4, 2}\) solution \(u\) to \(-u_t\sigma_k(\lambda(D^2u)) = 1\) in \(\mathbb{R}^n\times(-\infty, 0]\) must be a linear function of \(t\) plus a quadratic polynomial of \(x \), under some growth assumptions on \(u \).Fractional-order operators on nonsmooth domainshttps://zbmath.org/1521.350642023-11-13T18:48:18.785376Z"Abels, Helmut"https://zbmath.org/authors/?q=ai:abels.helmut"Grubb, Gerd"https://zbmath.org/authors/?q=ai:grubb.gerdSummary: The fractional Laplacian \((-\Delta)^a\), \(a\in (0,1)\), and its generalizations to variable-coefficient \(2a\)-order pseudodifferential operators \(P\), are studied in \(L_q\)-Sobolev spaces of Bessel-potential type \(H^s_q\). For a bounded open set \(\Omega \subset \mathbb{R}^n\), consider the homogeneous Dirichlet problem: \(Pu =f\) in \(\Omega\), \(u=0\) in \(\mathbb{R}^n\setminus \Omega\). We find the regularity of solutions and determine the exact Dirichlet domain \(D_{a,s,q}\) (the space of solutions \(u\) with \(f\in H_q^s(\overline{\Omega}))\) in cases where \(\Omega\) has limited smoothness \(C^{1+\tau}\), for \(2a<\tau <\infty\), \(0\leqslant s<\tau -2a\). Earlier, the regularity and Dirichlet domains were determined for smooth \(\Omega\) by the second author, and the regularity was found in low-order Hölder spaces for \(\tau =1\) by Ros-Oton and Serra. The \(H_q^s\)-results obtained now when \(\tau <\infty\) are new, even for \((-\Delta)^a\). In detail, the spaces \(D_{a,s,q}\) are identified as \(a\)-transmission spaces \(H_q^{a(s+2a)}(\overline{\Omega})\), exhibiting estimates in terms of \(\operatorname{dist}(x,\partial \Omega)^a\) near the boundary.
The result has required a new development of methods to handle nonsmooth coordinate changes for pseudodifferential operators, which have not been available before; this constitutes another main contribution of the paper.On the existence and Hölder regularity of solutions to some nonlinear Cauchy-Neumann problemshttps://zbmath.org/1521.350652023-11-13T18:48:18.785376Z"Audrito, Alessandro"https://zbmath.org/authors/?q=ai:audrito.alessandroSummary: We prove \textit{uniform} parabolic Hölder estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals
\[
{\mathcal{E}}_\varepsilon (W) = \int_0^\infty \frac{e^{- t/\varepsilon}}{\varepsilon} \bigg\{\int_{\mathbb{R}_+^{N+1}} y^a \Big(\varepsilon |\partial_t W|^2 + |\nabla W|^2 \Big) \mathrm{d}X + \int_{\mathbb{R}^N \times \{0\}} \Phi (w) \,\mathrm{d}x \bigg\} \,\mathrm{d}t, \qquad \varepsilon \in (0,1)
\]
where \(a \in (-1, 1)\) is a fixed parameter, \(\mathbb{R}_+^{N+1}\) is the upper half-space and \(\mathrm{d}X = \mathrm{d}x \mathrm{d}y\). As a consequence, we deduce the existence and Hölder regularity of weak solutions to a class of weighted nonlinear Cauchy-Neumann problems arising in combustion theory and fractional diffusion.Boundary regularity for parabolic systems in convex domainshttps://zbmath.org/1521.350662023-11-13T18:48:18.785376Z"Bögelein, Verena"https://zbmath.org/authors/?q=ai:bogelein.verena"Duzaar, Frank"https://zbmath.org/authors/?q=ai:duzaar.frank"Liao, Naian"https://zbmath.org/authors/?q=ai:liao.naian"Scheven, Christoph"https://zbmath.org/authors/?q=ai:scheven.christophSummary: In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the lateral boundary, provided homogeneous Dirichlet data are assumed on that part of the lateral boundary.Improved local smoothing estimates for the fractional Schrödinger operatorhttps://zbmath.org/1521.350672023-11-13T18:48:18.785376Z"Gao, Chuanwei"https://zbmath.org/authors/?q=ai:gao.chuanwei"Miao, Changxing"https://zbmath.org/authors/?q=ai:miao.changxing"Zheng, Jiqiang"https://zbmath.org/authors/?q=ai:zheng.jiqiangSummary: In this paper, we consider local smoothing estimates for the fractional Schrödinger operator \(e^{it(-\Delta )^{\alpha /2}}\) with \(\alpha >1\). Using the \(k\)-broad `norm' estimates of \textit{L. Guth} et al. [Acta Math. 223, No. 2, 251--376 (2019; Zbl 1430.42016)], we improve the previously best-known results of local smoothing estimates of \textit{S. Guo} et al. [Anal. PDE 13, No. 5, 1457--1500 (2020; Zbl 1452.42010)] and \textit{K. M. Rogers} and \textit{A. Seeger} [J. Reine Angew. Math. 640, 47--66 (2010; Zbl 1191.35078)].Partially overlapping travelling waves in a parabolic-hyperbolic systemhttps://zbmath.org/1521.350682023-11-13T18:48:18.785376Z"Bertsch, Michiel"https://zbmath.org/authors/?q=ai:bertsch.michiel"Izuhara, Hirofumi"https://zbmath.org/authors/?q=ai:izuhara.hirofumi"Mimura, Masayasu"https://zbmath.org/authors/?q=ai:mimura.masayasu"Wakasa, Tohru"https://zbmath.org/authors/?q=ai:wakasa.tohruSummary: We study the existence of travelling wave solutions of a one-dimensional parabolic-hyperbolic system for \(u(x,t)\) and \(v(x,t)\), which arises as a model for contact inhibition of cell growth. Compared to the scalar Fisher-KPP equation, the structure of the travelling wave solutions is surprisingly rich and strongly parameter-dependent. In the present paper we consider a parameter regime where the minimal wave speed is positive. We show that there exists a branch of travelling wave solutions for wave speeds which are larger than the minimal one. But the main result is more surprising: for certain values of the parameters the travelling wave with minimal wave speed is not segregated (a solution is called segregated if the product \(uv\) vanishes almost everywhere) and in that case there exists a second branch of ``partially overlapping'' travelling wave solutions for speeds between the minimal one and that of the (unique) segregated travelling wave.Distributional profiles for traveling waves in the Camassa-Holm equationhttps://zbmath.org/1521.350692023-11-13T18:48:18.785376Z"Boto, Miguel"https://zbmath.org/authors/?q=ai:boto.miguel"Sarrico, C. O. R."https://zbmath.org/authors/?q=ai:sarrico.carlos-orlando-rSummary: In this paper travelling waves with distributional profiles for the Camassa-Holm equation are studied. Using a product of distributions a new solution concept is introduced which extends the classical one. As a consequence, necessary and sufficient conditions for the propagation of a distributional profile are established and we present examples with discontinuous solutions, measures and even distributions that are not measures. One of these examples may be interpreted as a simple model for a ``tsunami'' in the setting of shallow water theory. We also prove that, under natural assumptions, profiles belonging to the Sobolev space \(H_{loc}^1(\mathbb{R})\) usually considered in the classical weak formulation can be seen as particular cases of our distributional solution concept.Spreading speeds for time heterogeneous prey-predator systems with diffusionhttps://zbmath.org/1521.350702023-11-13T18:48:18.785376Z"Ducrot, Arnaud"https://zbmath.org/authors/?q=ai:ducrot.arnaud"Jin, Zhucheng"https://zbmath.org/authors/?q=ai:jin.zhuchengSummary: We investigate the large time behavior for two components reaction-diffusion systems of prey-predator type in a time varying environment. Here we assume that these variations in time exhibit an averaging property, which will be called mean value in this work. This framework includes in particular time periodicity, almost periodicity and unique ergodicity. We describe the spreading behavior of the prey and the predator, wherein the two populations are able to co-invade the empty space. Our analysis is based the parabolic strong maximum principle for scalar equation and on the derivation of local pointwise estimates that are used to compare the solutions of the prey-predator problem with those of a KPP scalar equation on suitable spatio-temporal domains.Modeling bacterial traveling wave patterns with exact cross-diffusion and population growthhttps://zbmath.org/1521.350712023-11-13T18:48:18.785376Z"Kim, Yong-Jung"https://zbmath.org/authors/?q=ai:kim.yongjung"Yoon, Changwook"https://zbmath.org/authors/?q=ai:yoon.changwookSummary: Keller-Segel equations are widely employed to explain chemotaxis-induced bacterial traveling band phenomena. In this system, the dispersal of bacteria is modeled by independently given diffusion and advection terms, and the growth of cell population is neglected. In the paper, we develop a chemotaxis model which consists of cross-diffusion and population growth. In particular, we consider the case that the diffusion and advection terms form an exact cross-diffusion. The developed mathematical models are based on the conversion dynamics between active and inactive cells with different dispersal rates. The process consists of three steps and the performance of each step is complemented by comparing numerical simulations and experimental data.Two model equations with a second degree logarithmic nonlinearity and their Gaussian solutionshttps://zbmath.org/1521.350722023-11-13T18:48:18.785376Z"Liu, Cheng-Shi"https://zbmath.org/authors/?q=ai:liu.chengshiSummary: In the paper, we try to study the mechanism of the existence of Gaussian waves in high degree logarithmic nonlinear wave motions. We first construct two model equations which include the high order dispersion and a second degree logarithmic nonlinearity. And then we prove that the Gaussian waves can exist for high degree logarithmic nonlinear wave equations if the balance between the dispersion and logarithmic nonlinearity is kept. Our mathematical tool is the logarithmic trial equation method.Front propagation and blocking for the competition-diffusion system in a domain of half-lines with a junctionhttps://zbmath.org/1521.350732023-11-13T18:48:18.785376Z"Morita, Yoshihisa"https://zbmath.org/authors/?q=ai:morita.yoshihisa"Nakamura, Ken-Ichi"https://zbmath.org/authors/?q=ai:nakamura.ken-ichi"Ogiwara, Toshiko"https://zbmath.org/authors/?q=ai:ogiwara.toshikoSummary: The two-component Lotka-Volterra competition-diffusion system is well accepted as a model describing the invasion of a superior species into a new habitat. Under a bistable condition, we deal with the system in a domain of half-lines with a single junction and investigate the condition for the invasion from some of the half-lines beyond the junction or blocking the propagation of the superior species. We first give a sufficient condition for the invasion in the whole domain by a subsolution. Then, making use of sub- and supersolutions, we construct a standing front solution blocking the propagation if the number of half-lines occupied by the inferior species is sufficiently larger than that occupied by the superior species.Unstable dynamics of solitary traveling waves in a lattice with long-range interactionshttps://zbmath.org/1521.350742023-11-13T18:48:18.785376Z"Duran, Henry"https://zbmath.org/authors/?q=ai:duran.henry"Xu, Haitao"https://zbmath.org/authors/?q=ai:xu.haitao.1"Kevrekidis, Panayotis G."https://zbmath.org/authors/?q=ai:kevrekidis.panayotis-g"Vainchtein, Anna"https://zbmath.org/authors/?q=ai:vainchtein.anna(no abstract)Numerous accurate and stable solitary wave solutions to the generalized modified equal-width equationhttps://zbmath.org/1521.350752023-11-13T18:48:18.785376Z"Khater, Mostafa M. A."https://zbmath.org/authors/?q=ai:khater.mostafa-m-aSummary: The generalized modified Equal-Width (\textit{GMEW}) equation is often used to show how a one-dimensional wave moves through a medium that is not linear and has dispersion processes. In this article, we'll use two very precise, cutting-edge analytical and numerical methods to find the exact traveling wave solutions for the model we're looking at. These discoveries are really new, and they could immediately change how people train in engineering and physics. Now that a numerical approach has been described, we can roughly evaluate the replies' accuracy. Analytical and quantitative data were shown using contour plots and two- and three-dimensional graphs. Our method of symbolic computing shows that it has the potential to be a powerful mathematical tool. It can be used to solve a wide range of nonlinear wave problems. Our findings are the outcome of our topic investigation.Nonlocal boundary-value problem for an equation with differentiation operator \(z \partial/\partial z\) in a refined Sobolev scalehttps://zbmath.org/1521.350762023-11-13T18:48:18.785376Z"Ilkiv, V. S."https://zbmath.org/authors/?q=ai:ilkiv.volodymyr-stepanovich"Strap, N. I."https://zbmath.org/authors/?q=ai:strap.nataliya-igorivna"Volyanska, I. I."https://zbmath.org/authors/?q=ai:volyanska.iryna-iSummary: We study a nonlocal boundary-value problem for a differential equation with operator of generalized differentiation \(B = z \partial/\partial z\) acting upon the functions of complex variable \(z\). We establish the conditions of solvability of this problem in the scale of Hörmander spaces, which form the refined Sobolev scale of functions of one complex variable. The analyzed problem is ill posed in Hadamard's sense in the case of many operators of generalized differentiation, and its solvability depends on small denominators appearing in the construction of the solution. It is shown that, in the case of one variable, the corresponding denominators are not small and can be estimated from below by certain constants.A linearized integral equation reconstruction method of admittivity distributions using electrical impedance tomographyhttps://zbmath.org/1521.350772023-11-13T18:48:18.785376Z"Sebu, Cristiana"https://zbmath.org/authors/?q=ai:sebu.cristiana"Amaira, Andrei"https://zbmath.org/authors/?q=ai:amaira.andrei"Curmi, Jeremy"https://zbmath.org/authors/?q=ai:curmi.jeremy(no abstract)New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waveshttps://zbmath.org/1521.350782023-11-13T18:48:18.785376Z"Alam, Md Nur"https://zbmath.org/authors/?q=ai:alam.md-nur"Osman, M. S."https://zbmath.org/authors/?q=ai:osman.mohammed-s|osman.mojahid-saeed|osman.m-sh|osman.mohamed-sayed-aliSummary: This treatise analyzes the coupled nonlinear system of the model for the ion sound and Langmuir waves. The modified \((G^\prime/G)\)-expansion procedure is utilized to raise new closed-form wave solutions. Those solutions are investigated through hyperbolic, trigonometric and rational function. The graphical design makes the dynamics of the equations noticeable. It provides the mathematical foundation in diverse sectors of underwater acoustics, electromagnetic wave propagation, design of specific optoelectronic devices and physics quantum mechanics. Herein, we concluded that the studied approach is advanced, meaningful and significant in implementing many solutions of several nonlinear partial differential equations occurring in applied sciences.On entropy solutions of scalar conservation laws with discontinuous fluxhttps://zbmath.org/1521.350792023-11-13T18:48:18.785376Z"Panov, Evgeny Yu."https://zbmath.org/authors/?q=ai:panov.evgeniy-yuSummary: We introduce the notion of entropy solutions (e.s.) to a conservation law with an arbitrary jump continuous flux vector and prove the existence of the largest and the smallest e.s. to the Cauchy problem. The monotonicity and stability properties of these solutions are also established. In the case of a periodic initial function, we derive the uniqueness of e.s. Generally, the uniqueness property can be violated, which is confirmed by an example. Finally, we prove that in the case of a single space variable a weak limit of a sequence of e.s. is an e.s. as well (under the requirement of the spatial periodicity of the limit Young measure).Boundary control problems for nonlinear reaction-diffusion-convection modelhttps://zbmath.org/1521.350802023-11-13T18:48:18.785376Z"Saritskaya, Zh. Yu."https://zbmath.org/authors/?q=ai:saritskaya.zhanna-yurevna"Brizitskii, R. V."https://zbmath.org/authors/?q=ai:brizitskii.roman-victorovich|brizitskii.roman-viktorovichSummary: The solvability of the boundary control problem for a nonlinear model of mass transfer is proven in the case, when the reaction coefficient depends nonlinearly on concentration of substance and depends on spatial variables. The role of the control is played by the concentration value specified on the entire boundary of the domain.Existence and regularity results for some fully nonlinear singular or degenerate equationhttps://zbmath.org/1521.350812023-11-13T18:48:18.785376Z"Ndaw, C. O."https://zbmath.org/authors/?q=ai:ndaw.cheikhou-oumarSummary: In this article we prove existence, uniqueness and regularity for the singular equation \[\begin{cases}|\nabla u|^\alpha(F(D^2u)+h(x)\cdot\nabla u)+c(x)|u|^\alpha u+p(x)u^{-\gamma}=0 &\text{in }\Omega\\ u>0\text{ in }\Omega,\ u=0 & \text{on }\partial\Omega\end{cases}\] when \(p\) is some continuous and positive function, \(c\) and \(h\) are continuous, \(\alpha>-1\) and \(F\) is fully nonlinear elliptic. Some conditions on the first eigenvalue for the operator \(-|\nabla u|^\alpha(F(D^2u)+h(x)\cdot\nabla u)-c(x)|u|^\alpha u\) are required. The results generalize the well known results of Lazer and McKenna.Large-scale asymptotics of velocity-jump processes and nonlocal Hamilton-Jacobi equationshttps://zbmath.org/1521.350822023-11-13T18:48:18.785376Z"Bouin, Emeric"https://zbmath.org/authors/?q=ai:bouin.emeric"Calvez, Vincent"https://zbmath.org/authors/?q=ai:calvez.vincent"Grenier, Emmanuel"https://zbmath.org/authors/?q=ai:grenier.emmanuel"Nadin, Grégoire"https://zbmath.org/authors/?q=ai:nadin.gregoireSummary: We investigate a simple velocity jump process in the regime of large deviation asymptotics. New velocities are taken randomly at a constant, large, rate from a Gaussian distribution with vanishing variance. The Kolmogorov forward equation associated with this process is the linear BGK kinetic transport equation. We derive a new type of Hamilton-Jacobi equation which is nonlocal with respect to the velocity variable. We introduce a suitable notion of viscosity solution, and we prove well-posedness in the viscosity sense. We also prove convergence of the logarithmic transformation toward this limit problem. Furthermore, we identify the variational formulation of the solution by means of an action functional supported on piecewise linear curves. As an application of this theory, we compute the exact rate of acceleration in a kinetic version of the celebrated F-KPP equation in the one-dimensional case.Nonlinear semigroups for nonlocal conservation lawshttps://zbmath.org/1521.350832023-11-13T18:48:18.785376Z"Kovács, Mihály"https://zbmath.org/authors/?q=ai:kovacs.mihaly"Vághy, Mihály A."https://zbmath.org/authors/?q=ai:vaghy.mihaly-aSummary: We investigate a class of nonlocal conservation laws in several space dimensions, where the continuum average of weighted nonlocal interactions are considered over a finite horizon. We establish well-posedness for a broad class of flux functions and initial data via semigroup theory in Banach spaces and, in particular, via the celebrated Crandall-Liggett Theorem. We also show that the unique mild solution satisfies a Kružkov-type nonlocal entropy inequality. Similarly to the local case, we demonstrate an efficient way of proving various desirable qualitative properties of the unique solution.Solving a system of first-order partial differential equations using decomposition methodshttps://zbmath.org/1521.350842023-11-13T18:48:18.785376Z"Bazylevych, Y. N."https://zbmath.org/authors/?q=ai:bazylevych.y-n"Kostiushko, I. A."https://zbmath.org/authors/?q=ai:kostiushko.i-a"Stanina, O. D."https://zbmath.org/authors/?q=ai:stanina.o-dSummary: The paper describes simplifying a system of equations by decomposing it into independent subsystems or by hierarchical (sequential) decomposition. The authors have developed algebraic methods for transforming coefficient matrices into block-diagonal or block-triangular forms. They allow one to simplify the problem significantly and obtain an analytical solution in many cases.The nonlocal solvability conditions in original coordinates for a system with constant termshttps://zbmath.org/1521.350852023-11-13T18:48:18.785376Z"Dontsova, M. V."https://zbmath.org/authors/?q=ai:dontsova.marina-vladimirovnaSummary: Conditions of a nonlocal solvability of the Cauchy problem in original coordinates are obtained for a system of two quasilinear first-order partial differential equations with constant terms. The investigation is based on the method of an additional argument. The proof of the nonlocal solvability of the Cauchy problem in original coordinates relies on global estimates.On solution of the boundary value problems posed for an equation with the third-order multiple characteristics in semi-bounded domains in three dimensional spacehttps://zbmath.org/1521.350862023-11-13T18:48:18.785376Z"Apakov, Yu. P."https://zbmath.org/authors/?q=ai:apakov.yusupjon-p|apakov.yusufjon-pulatovich|apakov.yusupzhon-pulatovich"Hamitov, A. A."https://zbmath.org/authors/?q=ai:hamitov.a-aSummary: In this work, the boundary value problems posed for an equation with the third-order multiple characteristics in semi-bounded domains in three-dimensional space are considered. The uniqueness of solutions of the posed problems is proved by the method of energy integrals. The existence of the solutions is proved by the method of variables' separation. The solutions were constructed explicitly, in the form infinite series, an opportunity of term-by-term differentiation of the series with respect to all variables is justified.Partial normal form for the semilinear Klein-Gordon equation with quadratic potentials and algebraic non-resonant masseshttps://zbmath.org/1521.350872023-11-13T18:48:18.785376Z"Brun, Pierre"https://zbmath.org/authors/?q=ai:brun.pierreSummary: In this paper, we study small solutions of the non-linear Klein-Gordon equation on \(\mathbb{R}^d\) with quadratic potential. The goal is to understand for which masses, we can apply a normal form procedure. We prove two mains theorems. Our first contribution gives explicit for which the procedure of partial normal form works due to algebraic assumptions. The second one shows that this strategy works for almost all \(m\) in the sense of Lebesgue measure.A highly accurate perfectly-matched-layer boundary integral equation solver for acoustic layered-medium problemshttps://zbmath.org/1521.350882023-11-13T18:48:18.785376Z"Lu, Wangtao"https://zbmath.org/authors/?q=ai:lu.wangtao"Xu, Liwei"https://zbmath.org/authors/?q=ai:xu.liwei"Yin, Tao"https://zbmath.org/authors/?q=ai:yin.tao"Zhang, Lu"https://zbmath.org/authors/?q=ai:zhang.luSummary: Based on the perfectly matched layer (PML) technique, this paper develops a highly accurate boundary integral equation (BIE) solver for acoustic scattering problems in locally defected layered media in both two and three dimensions. The original scattering problem is truncated onto a bounded domain by the PML. Assuming the vanishing of the scattered field on the PML boundary, we derive BIEs on local defects only in terms of using PML-transformed free-space Green's function, and the four standard integral operators: single-layer, double-layer, transpose of double-layer, and hyper-singular boundary integral operators. The hyper-singular integral operator is transformed into a combination of weakly-singular integral operators and tangential derivatives. We develop a high-order Chebyshev-based rectangular-polar singular-integration solver to discretize all weakly singular integrals. Numerical experiments for both two- and three-dimensional problems are carried out to demonstrate the accuracy and efficiency of the proposed solver.Degenerate kernels of polyharmonic and poly-Helmholtz operators in polar and spherical coordinateshttps://zbmath.org/1521.350892023-11-13T18:48:18.785376Z"Tsai, Chia-Cheng"https://zbmath.org/authors/?q=ai:tsai.chia-cheng"Hematiyan, M. R."https://zbmath.org/authors/?q=ai:hematiyan.mohammad-rahim(no abstract)Recovering the Laplacian from centered means on balls and spheres of fixed radiushttps://zbmath.org/1521.350902023-11-13T18:48:18.785376Z"Volchkova, N. P."https://zbmath.org/authors/?q=ai:volchkova.natalia-p|volchkova.natalya-petrovich|volchkova.natalya-petrovna"Volchkov, Vit. V."https://zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: Various issues related to restrictions on radii in mean-value formulas are well-known in the theory of harmonic functions. In particular, using the Brown-Schreiber-Taylor theorem on spectral synthesis for motion-invariant subspaces in \(C(\mathbb{R}^n)\), one can obtain the following strengthening of the classical mean-value theorem for harmonic functions: if a continuous function on \(\mathbb{R}^n\) satisfies the mean-value equations for all balls and spheres of a fixed radius \(r\), then it is harmonic on \(\mathbb{R}^n\). In connection with this result, the following problem arises: recover the Laplacian from the deviation of a function from its average values on balls and spheres of a fixed radius. The aim of this work is to solve this problem. The article uses methods of harmonic analysis, as well as the theory of entire and special functions. The key step in the proof of the main result is expansion of the Dirac delta function in terms of a system of radial distributions supported in a fixed ball, biorthogonal to some system of spherical functions. A similar approach can be used to invert a number of convolution operators with compactly supported radial distributions.Correction to: ``Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions''https://zbmath.org/1521.350912023-11-13T18:48:18.785376Z"Chen, Yongpeng"https://zbmath.org/authors/?q=ai:chen.yongpeng"Jin, Baoxia"https://zbmath.org/authors/?q=ai:jin.baoxiaCorrection to the authors' paper [ibid. 2022, Paper No. 86, 13 p. (2022; Zbl 1518.35263)].BV capacity for the Schrödinger operator with an inverse-square potentialhttps://zbmath.org/1521.350922023-11-13T18:48:18.785376Z"Han, Yang"https://zbmath.org/authors/?q=ai:han.yang|han.yang.1"Liu, Yu"https://zbmath.org/authors/?q=ai:liu.yu"Wang, Haihui"https://zbmath.org/authors/?q=ai:wang.haihuiSummary: For \(a \ge - (\frac{d}{2} - 1)^2\) and \(2\sigma = {{d - 2}}-({{{(d - 2)}^2} + 4a})^{1/2} \), let \(\mathcal{\widetilde{H}}_{\sigma}= 2( { - \Delta + \frac{{{\sigma^2}}}{{{{ | x |}^2}}}})\) be a Schrödinger operator with an inverse-square potential on \({{\mathbb{R}}^d}\backslash \{0\} \). In this paper, we introduce and investigate the \({{\mathcal{\widetilde{H}}_{\sigma}}} \)-BV capacity, whence discovering some capacitary inequalities on \(\Omega \subseteq{{\mathbb{R}}^d}\backslash \{0\} \).Uniform rectifiability, elliptic measure, square functions, and \(\varepsilon\)-approximability via an ACF monotonicity formulahttps://zbmath.org/1521.350932023-11-13T18:48:18.785376Z"Azzam, Jonas"https://zbmath.org/authors/?q=ai:azzam.jonas"Garnett, John"https://zbmath.org/authors/?q=ai:garnett.john-brady"Mourgoglou, Mihalis"https://zbmath.org/authors/?q=ai:mourgoglou.mihalis"Tolsa, Xavier"https://zbmath.org/authors/?q=ai:tolsa.xavierSummary: Let \(\Omega\subset\mathbb{R}^{n+1}\), \(n\geq 2\), be an open set with Ahlfors regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator \(L\) in divergence form associated with a matrix \(A\) with real, merely bounded and possibly nonsymmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if \(L^*\) is the operator in divergence form associated with the transpose matrix of \(A\), then \(\partial \Omega\) is uniformly \(n\)-rectifiable if and only if every bounded solution of \(Lu=0\) and every bounded solution of \(L^*v=0\) in \(\Omega\) is \(\varepsilon\)-approximable if and only if every bounded solution of \(Lu=0\) and every bounded solution of \(L^*v=0\) in \(\Omega\) satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called ``\(S<N\)'' estimates, and another in terms of a suitable corona decomposition involving \(L\)-harmonic and \(L^*\)-harmonic measures. We also prove that if \(L\)-harmonic measure and \(L^*\)-harmonic measure satisfy a weak \(A_\infty\)-type condition, then \(\partial \Omega\) is \(n\)-uniformly rectifiable. In the process we obtain a version of the Alt-Caffarelli-Friedman monotonicity formula for a fairly wide class of elliptic operators which is of independent interest and plays a fundamental role in our arguments.Green function estimates for second order elliptic operators in non-divergence form with Dini continuous coefficientshttps://zbmath.org/1521.350942023-11-13T18:48:18.785376Z"Chen, Zhen-Qing"https://zbmath.org/authors/?q=ai:chen.zhen-qing"Wang, Jie-Ming"https://zbmath.org/authors/?q=ai:wang.jiemingSummary: Two-sided sharp Green function estimates are obtained for second order uniformly elliptic operators in non-divergence form with Dini continuous coefficients in bounded \({C^{1,1}}\) domains, which are shown to be comparable to that of the Dirichlet Laplace operator in the domain. The first and second order derivative estimates of the Green functions are also derived. Moreover, boundary Harnack inequality with an explicit boundary decay rate and interior Schauder's estimates for these differential operators are established, which may be of independent interest.Hyperbolic formulas in elliptic Cauchy problemshttps://zbmath.org/1521.350952023-11-13T18:48:18.785376Z"Fedchenko, Dmitry P."https://zbmath.org/authors/?q=ai:fedchenko.dmitrii-p"Tarkhanov, Nikolai"https://zbmath.org/authors/?q=ai:tarkhanov.nikolai-nSummary: We study the Cauchy problem for the Laplace equation in a cylindrical domain with data on a part of it's boundary which is a cross-section of the cylinder. On reducing the problem to the Cauchy problem for the wave equation in a complex domain and using hyperbolic theory we obtain explicit formulas for the solution, thus developing the classical approach of \textit{H. Lewy} [Math. Ann. 101, 609--619 (1929; JFM 55.0882.03)].Positive continuous solutions for some semilinear elliptic problems in the half spacehttps://zbmath.org/1521.350962023-11-13T18:48:18.785376Z"Alsaedi, Ramzi"https://zbmath.org/authors/?q=ai:alsaedi.ramzi-s-n"Ghanmi, Abdeljabbar"https://zbmath.org/authors/?q=ai:ghanmii.abdeljabbar"Zeddini, Noureddine"https://zbmath.org/authors/?q=ai:zeddini.noureddineSummary: The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions \(K^{\infty}({\mathbb{R}}_+^d)\) that was defined in [\textit{I. Bachar} et al., Electron. J. Differ. Equ. 2002, Paper No. 41 (2002; Zbl 0995.31003)] for \(d=2\) and in [\textit{I. Bachar} and \textit{H. Mâagli}, Positivity 9, No. 2, 153--192 (2005; Zbl 1151.35304)] for \(d\geq 3\) and adapted to study some nonlinear elliptic problems in the half space. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants \(\lambda\) and \(\mu\) to the following system: \(\Delta u=\lambda f(x,u,v)\), \(\Delta v=\mu g(x,u,v)\) in \({\mathbb{R}}_+^d\), \(\lim_{x\to (\xi ,0)} u(x)=a_1\phi_1(\xi)\), \(\lim_{x\to(\xi ,0)} v(x)=a_2 \phi_2(\xi)\) for all \(\xi \in{\mathbb{R}}^{d-1}\), \(\lim_{x_d \to \infty} \frac{u(x)}{x_d}=b_1\), \(\lim_{x_d \to \infty}\frac{v(x)}{x_d}=b_2\), where \(\phi_1\) and \(\phi_2\) are nontrivial nonnegative continuous functions on \(\partial{\mathbb{R}}_+^d= {\mathbb{R}}^{d-1} \times \{0\}\), \(a_1, a_2, b_1, b_2\) are nonnegative constants such that \((a_1+b_1)(a_2+b_2)>0\). The functions \(f\) and \(g\) are nonnegative and belong to a class of functions containing in particular all functions of the type \(f(x,u,v)=p(x) u^{\alpha}g_1(v)\) and \(g(x,u,v)=q(x)g_2(u)v^{\beta}\) with \(\alpha \geq 1\), \(\beta \geq 1\), \(g_1\), \(g_2\) are continuous on \([0,\infty)\), and \(p, q\) are nonnegative functions in \(K^{\infty}({\mathbb{R}}_+^d)\).Generalized integral equation method for an elliptic nonlocal equation in measure spacehttps://zbmath.org/1521.350972023-11-13T18:48:18.785376Z"Zheng, Chunxiong"https://zbmath.org/authors/?q=ai:zheng.chunxiong"Yin, Jia"https://zbmath.org/authors/?q=ai:yin.jiaSummary: A solution strategy, called generalized integral equation method, is proposed to solve a class of elliptic nonlocal equations in measure space, within which both the continuous and discrete nonlocal problems can be taken as specific instances. By extracting the main ingredients of integral equation method, we develop a generalized integral equation method in an abstract operator framework. As a matter of fact, the classic integral equation method for continuous local partial differential equations can be categorized into this framework. The key ingredient of the proposed method is to derive the generalized boundary integral equations, which can be coupled appropriately with the interior operator equation to obtain a reduced problem. We prove that the resulting system is well-posed by showing that it admits an equivalent formulation with strong coercivity, and the solution of the reduced problem is the same as that of the original one. The proposed method is applied to a nonlocal equation in two-dimensional space discretized by an asymptotically compatible scheme. Numerical experiments validate the effectiveness.Gaussian estimates for heat kernels of higher order Schrödinger operators with potentials in generalized Schechter classeshttps://zbmath.org/1521.350982023-11-13T18:48:18.785376Z"Cao, Jun"https://zbmath.org/authors/?q=ai:cao.jun"Liu, Yu"https://zbmath.org/authors/?q=ai:liu.yu"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.12|zhang.chao.16|zhang.chao.10|zhang.chao.2|zhang.chao.11|zhang.chao|zhang.chao.3|zhang.chao.6|zhang.chao.5|zhang.chao.1Summary: Let \(m\in \mathbb{N}\), \(P(D):=\sum_{|\alpha |=2m}(-1)^m a_\alpha D^\alpha\) be a \(2m\)-order homogeneous elliptic operator with real constant coefficients on \(\mathbb{R}^n\), and \(V\) a real-valued measurable function on \(\mathbb{R}^n\). In this article, the authors introduce a new generalized Schechter class concerning \(V\) and show that the higher order Schrödinger operator \(\mathcal{L}:=P(D)+V\) possesses a heat kernel that satisfies the Gaussian upper bound and the Hölder regularity when \(V\) belongs to this new class. The Davies-Gaffney estimates for the associated semigroup and their local versions are also given. These results pave the way for many further studies on the analysis of \(\mathcal{L} \).Heat kernels for a class of hybrid evolution equationshttps://zbmath.org/1521.350992023-11-13T18:48:18.785376Z"Garofalo, Nicola"https://zbmath.org/authors/?q=ai:garofalo.nicola"Tralli, Giulio"https://zbmath.org/authors/?q=ai:tralli.giulioSummary: The aim of this paper is to construct (explicit) heat kernels for some \textit{hybrid} evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form \({\mathscr{L}}_1 + {\mathscr{L}}_2 - \partial_t\), but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators \({\mathscr{L}}_1 - \partial_t\) and \({\mathscr{L}}_2 - \partial_t\). Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander's 1967 groundbreaking paper on hypoellipticity.Regularity properties of passive scalars with rough divergence-free driftshttps://zbmath.org/1521.351002023-11-13T18:48:18.785376Z"Albritton, Dallas"https://zbmath.org/authors/?q=ai:albritton.dallas"Dong, Hongjie"https://zbmath.org/authors/?q=ai:dong.hongjieSummary: We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation
\[
\partial_t \theta - \Delta \theta + b \cdot \nabla \theta = 0,
\]
to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts \(b\) belonging to \(L^q_t L^p_x\), where \(\frac{2}{q} + \frac{n}{p} < 2\), or \(L^p_x L^q_t\), where \(\frac{3}{q} + \frac{n-1}{p} < 2\). For steady drifts, the condition reduces to \(b \in L^{\frac{n-1}{2}+}\). The space \(L^1_t L^\infty_x\) of drifts with `bounded total speed' is a borderline case and plays a special role in the theory. To demonstrate sharpness, we construct counterexamples whose goal is to transport anomalous singularities into the domain `before' they can be dissipated.Cahn-Hilliard-Brinkman model for tumor growth with possibly singular potentialshttps://zbmath.org/1521.351012023-11-13T18:48:18.785376Z"Colli, Pierluigi"https://zbmath.org/authors/?q=ai:colli.pierluigi"Gilardi, Gianni"https://zbmath.org/authors/?q=ai:gilardi.gianni.1"Signori, Andrea"https://zbmath.org/authors/?q=ai:signori.andrea"Sprekels, Jürgen"https://zbmath.org/authors/?q=ai:sprekels.jurgenSummary: We analyze a phase field model for tumor growth consisting of a Cahn-Hilliard-Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn-Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.Spreading dynamics for a predator-prey system with two predators and one prey in a shifting habitathttps://zbmath.org/1521.351022023-11-13T18:48:18.785376Z"Guo, Jong-Shenq"https://zbmath.org/authors/?q=ai:guo.jong-shenq"Shimojo, Masahiko"https://zbmath.org/authors/?q=ai:shimojo.masahiko"Wu, Chin-Chin"https://zbmath.org/authors/?q=ai:wu.chin-chinSummary: We study the spreading dynamics for a three-species predator-prey system with two weak competing predators and one prey in a shifting habitat. First, we derive some extinction results for each species. Then we provide some persistence theorems for each species with moving speeds exceed the shifting speed, but less than some certain quantities. Finally, the convergence to a certain constant state is proven in each persistent regime.Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary modelshttps://zbmath.org/1521.351032023-11-13T18:48:18.785376Z"Khan, Kamruzzaman"https://zbmath.org/authors/?q=ai:khan.kamruzzaman"Schaerf, Timothy M."https://zbmath.org/authors/?q=ai:schaerf.timothy-m"Du, Yihong"https://zbmath.org/authors/?q=ai:du.yihongSummary: This paper investigates the effect of environmental heterogeneity on species spreading via numerical simulation of suitable reaction-diffusion models with free boundaries. We focus on the changes of long-time dynamics (establishment or extinction) and spreading speeds of the species as the parameters describing the heterogeneity of the environment are varied. For the single species model in time-periodic environment and in space-periodic environment theoretically treated in
[\textit{Y. Du} et al., J. Funct. Anal. 265, No. 9, 2089--2142 (2013; Zbl 1282.35419); \textit{Y. Du} and \textit{X. Liang}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 32, No. 2, 279--305 (2015; Zbl 1321.35263)], we obtain more detailed properties here. Among other results, our numerical simulation suggests that, in a time-periodic or space-periodic environment, moderate increase of the oscillation scale enhances the chances of establishment as well as the spreading speed of the species. We also numerically examine a related model with two competing species, which was treated in
[\textit{K. Khan} et al., J. Math. Biol. 83, No. 3, Paper No. 23, 43 p. (2021; Zbl 1477.35285); \textit{J.-S. Guo} and \textit{C.-H. Wu}, Nonlinearity 28, No. 1, 1--27 (2015; Zbl 1316.92066); \textit{Y. Du} and \textit{C.-H. Wu}, Calc. Var. Partial Differ. Equ. 57, No. 2, Paper No. 52, 36 p. (2018; Zbl 1396.35028)] recently and reduces to the single species free boundary model when one of the species is absent. Our numerical results, obtained by varying the parameters in the time-periodic and space-periodic terms of the model, suggest that heterogeneity of the environment enhances the invasion of the two species (as in the single species model), although there are subtle differences of the influences felt by the two. Some intriguing phenomena revealed in our simulations suggest that heterogeneity of the environment decreases the level of predictability of the competition outcome.Weak entire solutions of reaction-interface systemshttps://zbmath.org/1521.351042023-11-13T18:48:18.785376Z"Chen, Yan-Yu"https://zbmath.org/authors/?q=ai:chen.yanyu"Ninomiya, Hirokazu"https://zbmath.org/authors/?q=ai:ninomiya.hirokazu"Wu, Chang-Hong"https://zbmath.org/authors/?q=ai:wu.changhongSummary: In this paper, the singular limit problems arising from FitzHugh-Nagumo-type reaction-diffusion systems are studied, which are called reaction-interface systems. All weak entire solutions originating from finitely many excited intervals are completely characterized. For weak entire solutions originating from infinitely many excited intervals, periodic wave trains and time-periodic solutions are discussed. In particular, we study the dispersion relation of periodic wave trains and investigate the dependence of the propagation speed on the period.Global existence and spatial analyticity for a nonlocal flux with fractional diffusionhttps://zbmath.org/1521.351052023-11-13T18:48:18.785376Z"Gao, Yu"https://zbmath.org/authors/?q=ai:gao.yu"Wang, Cong"https://zbmath.org/authors/?q=ai:wang.cong"Xue, Xiaoping"https://zbmath.org/authors/?q=ai:xue.xiaopingSummary: In this paper, we study a one dimensional nonlinear equation with diffusion \(-\nu(-\partial_{xx})^{\frac{\alpha}{2}}\) for \(0 \leq \alpha \leq 2\) and \(\nu > 0\). We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space \(L^1(\mathbb{R}) \cap H^{1/2}(\mathbb{R})\) when \(0 \leq \alpha \leq 2\). For the subcritical case \(1 < \alpha \leq 2\) and critical case \(\alpha = 1\), we obtain the global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrapping method to improve the regularity of mild solutions in the Bessel potential spaces for the subcritical case \(1 < \alpha \leq 2\). Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case \(\alpha = 1\), if the initial data \(\rho_0\) satisfies \(-\nu < \inf \rho_0 < 0\), we use the method of characteristics for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If \(\rho_0 \geq 0\), the solution exists globally and converges to steady state.
{\copyright 2023 American Institute of Physics}Spatial segregation of multiple species: a singular limit approachhttps://zbmath.org/1521.351062023-11-13T18:48:18.785376Z"Izuhara, Hirofumi"https://zbmath.org/authors/?q=ai:izuhara.hirofumi"Monobe, Harunori"https://zbmath.org/authors/?q=ai:monobe.harunori"Wu, Chang-Hong"https://zbmath.org/authors/?q=ai:wu.changhongSummary: The spatial segregation of the populations occurs commonly in ecology. One typical way to understand this phenomenon is to consider strong competition in some species. In this paper, we shall consider multiple-species competition-diffusion models. Under the condition that some interspecies competition rates are large, we show that the segregation phenomenon occurs. Furthermore, we derive some two-phase Stefan-like problems appearing as the singular limit, which may provide some modeling interpretation for free boundary problems studied in the literature.Continuity of attractors for singularly perturbed semilinear problems with nonlinear boundary conditions and large diffusionhttps://zbmath.org/1521.351072023-11-13T18:48:18.785376Z"Pires, L."https://zbmath.org/authors/?q=ai:pires.leonardo"Samprogna, R. A."https://zbmath.org/authors/?q=ai:samprogna.rodrigo-aSummary: We exhibit singularly perturbed parabolic problems with large diffusion and nonhomogeneous boundary conditions for which the asymptotic behavior can be described by a one-dimensional ordinary differential equation. We estimate the continuity of attractors in Hausdorff's metric by the rate of convergence of resolvent operators. Moreover, we will show explicitly how this estimate of continuity varies exponentially with the fractional power spaces \(X^\alpha\) for \(\alpha\) in an appropriate interval.
{\copyright 2023 American Institute of Physics}Spatial dynamics of a nonlocal reaction-diffusion epidemic model in time-space periodic habitathttps://zbmath.org/1521.351082023-11-13T18:48:18.785376Z"Xin, Ming-Zhen"https://zbmath.org/authors/?q=ai:xin.mingzhen"Wang, Bin-Guo"https://zbmath.org/authors/?q=ai:wang.binguoSummary: This paper is devoted to the study of a partially degenerate reaction-diffusion epidemic model with latent period in time-space periodic habitat. First, we obtain a threshold result on the global stability of either zero or the positive time-space periodic solution in terms of the basic reproduction ratio \(\mathcal{R}_0 \). Second, we establish the existence of the spreading speeds and its coincidence with the minimal speed of time-space periodic traveling waves. At last, we use numerical simulations to investigate the influence of model parameters on spreading speeds.Semilinear parabolic equations in Herz spaceshttps://zbmath.org/1521.351092023-11-13T18:48:18.785376Z"Drihem, Douadi"https://zbmath.org/authors/?q=ai:drihem.douadiSummary: In this paper, we will study local and global Cauchy problems for the semilinear parabolic equations \(\partial_tu-\Delta u=G(u)\) with initial data in Herz spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights. Our results cover the results obtained with initial data in Lebesgue spaces. Moreover, the results in Herz spaces are a little different from the results in Lebesgue spaces.Initial traces and solvability for a semilinear heat equation on a half space of \(\mathbb{R}^N\)https://zbmath.org/1521.351102023-11-13T18:48:18.785376Z"Hisa, Kotaro"https://zbmath.org/authors/?q=ai:hisa.kotaro"Ishige, Kazuhiro"https://zbmath.org/authors/?q=ai:ishige.kazuhiro"Takahashi, Jin"https://zbmath.org/authors/?q=ai:takahashi.jinSummary: We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of \(\mathbb{R}^N\) under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy-Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy-Dirichlet problem.Weak and strong interaction of excitation kinks in scalar parabolic equationshttps://zbmath.org/1521.351112023-11-13T18:48:18.785376Z"Pauthier, Antoine"https://zbmath.org/authors/?q=ai:pauthier.antoine"Rademacher, Jens D. M."https://zbmath.org/authors/?q=ai:rademacher.jens-d-m"Ulbrich, Dennis"https://zbmath.org/authors/?q=ai:ulbrich.dennisSummary: Motivated by studies of the Greenberg-Hastings cellular automata (GHCA) as a caricature of excitable systems, in this paper we study kink-antikink dynamics in the perhaps simplest PDE model of excitable media given by the scalar reaction diffusion-type \(\theta\)-equations for excitable angular phase dynamics. On the one hand, we qualitatively study geometric kink positions using the comparison principle and the theory of terraces. This yields the minimal initial distance as a global lower bound, a well-defined sequence of collision data for kinks- and antikinks, and implies that periodic pure kink sequences are asymptotically equidistant. On the other hand, we study metastable dynamics of finitely many kinks using weak interaction theory for certain analytic kink positions, which admits a rigorous reduction to ODE. By blow-up type singular rescaling we show that distances become ordered in finite time, and eventually diverge. We conclude that diffusion implies a loss of information on kink distances so that the entropic complexity based on positions and collisions in the GHCA does not simply carry over to the PDE model.Velocity averaging for diffusive transport equations with discontinuous fluxhttps://zbmath.org/1521.351122023-11-13T18:48:18.785376Z"Erceg, M."https://zbmath.org/authors/?q=ai:erceg.marko"Mišur, M."https://zbmath.org/authors/?q=ai:misur.marin"Mitrović, D."https://zbmath.org/authors/?q=ai:mitrovic.darkoSummary: We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve the result, we introduce a new variant of micro-local defect functionals which are able to `recognise' changes of the type of the equation. As a corollary, we show the existence of a weak solution for the Cauchy problem for non-linear degenerate parabolic equation with discontinuous flux. We also show existence of strong traces at \(t=0\) for so-called quasi-solutions to degenerate parabolic equations under non-degeneracy conditions on the diffusion term.Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic \(p\)-Laplace equationshttps://zbmath.org/1521.351132023-11-13T18:48:18.785376Z"Banerjee, Agnid"https://zbmath.org/authors/?q=ai:banerjee.agnid"Garain, Prashanta"https://zbmath.org/authors/?q=ai:garain.prashanta"Kinnunen, Juha"https://zbmath.org/authors/?q=ai:kinnunen.juhaSummary: We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional \(p\)-Laplace equations which includes the fractional parabolic \(p\)-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.Exact boundary controllability of the structural acoustic model with variable coefficientshttps://zbmath.org/1521.351142023-11-13T18:48:18.785376Z"Liu, Yu-Xiang"https://zbmath.org/authors/?q=ai:liu.yuxiangSummary: We consider the boundary controllability of the structural acoustic model with variable coefficients. The structural acoustic model is a coupled partial differential equation, which comprises an acoustic wave equation in the interior domain, a Kirchoff plate equation on the boundary portion, with the coupling being accomplished across a boundary interface. In this model, the wave propagation medium and the plate material are all inhomogeneous. By the Riemannian geometry theory and the multiplier technique, our paper derives the exact controllability with two boundary controls under some checkable conditions and the exact-approximate boundary reachability with only one control for the boundary Kirchoff plate equation.Existence of entropy weak solutions for 1D non-local traffic models with space-discontinuous fluxhttps://zbmath.org/1521.351152023-11-13T18:48:18.785376Z"Chiarello, F. A."https://zbmath.org/authors/?q=ai:chiarello.felisia-angela"Contreras, H. D."https://zbmath.org/authors/?q=ai:contreras.harold-deivi"Villada, L. M."https://zbmath.org/authors/?q=ai:villada.luis-miguelSummary: We study a 1D scalar conservation law whose non-local flux has a single spatial discontinuity. This model is intended to describe traffic flow on a road with rough conditions. We approximate the problem through an upwind-type numerical scheme and provide compactness estimates for the sequence of approximate solutions. Then, we prove the existence and the uniqueness of entropy weak solutions. Numerical simulations corroborate the theoretical results and the limit model as the kernel support tends to zero is numerically investigated.Utilizing time-reversibility for shock capturing in nonlinear hyperbolic conservation lawshttps://zbmath.org/1521.351162023-11-13T18:48:18.785376Z"Dzanic, T."https://zbmath.org/authors/?q=ai:dzanic.t"Trojak, W."https://zbmath.org/authors/?q=ai:trojak.will"Witherden, F. D."https://zbmath.org/authors/?q=ai:witherden.freddie-dSummary: In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or \textit{a priori} knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.Lagrangian solutions to the transport-Stokes systemhttps://zbmath.org/1521.351172023-11-13T18:48:18.785376Z"Inversi, Marco"https://zbmath.org/authors/?q=ai:inversi.marcoSummary: In this paper we consider the transport-Stokes system, which describes the sedimentation of a particles in a viscous fluid in inertialess regime. We show existence of Lagrangian solutions to the Cauchy problem with \(L^1\) initial data. We prove uniqueness of solutions as a corollary of a stability estimate with respect to the 1-Wasserstein distance for solutions with initial data in a Yudovich-type refinement of \(L^3\), with finite first moment. Moreover, we describe the evolution starting from axisymmetric initial data. Our approach is purely Lagrangian.Persistence of the steady planar normal shock structure in 3-D unsteady potential flowshttps://zbmath.org/1521.351182023-11-13T18:48:18.785376Z"Fang, Beixiang"https://zbmath.org/authors/?q=ai:fang.beixiang"Huang, Feimin"https://zbmath.org/authors/?q=ai:huang.feimin"Xiang, Wei"https://zbmath.org/authors/?q=ai:xiang.wei"Xiao, Feng"https://zbmath.org/authors/?q=ai:xiao.fengSummary: This paper concerns the dynamic stability of the steady three-dimensional (3-D) wave structure of a planar normal shock front intersecting perpendicularly to a planar solid wall for unsteady potential flows. The stability problem can be formulated as a free boundary problem of a quasi-linear hyperbolic equation of second order in a dihedral-space domain between the shock front and the solid wall. The key difficulty is brought by the edge singularity of the space domain, the intersection curve between the shock front and the solid wall. Different from the two-dimensional (2-D) case, for which the singular part of the boundary is only a point, it is a curve for the 3-D case in this paper. This difference brings new difficulties to the mathematical analysis of the stability problem. A modified partial hodograph transformation is introduced such that the extension technique developed for the 2-D case can be employed to establish the well-posed theory for the initial-boundary value problem of the linearized hyperbolic equation of second order in a dihedral-space domain. Moreover, the extension technique is improved in this paper such that loss of regularity in the \textit{a priori} estimates on the shock front does not occur. Thus, the classical nonlinear iteration scheme can be constructed to prove the existence of the solution to the stability problem, which shows the dynamic stability of the steady planar normal shock without applying the Nash-Moser iteration method.Shock waves and characteristic discontinuities in ideal compressible two-fluid MHDhttps://zbmath.org/1521.351192023-11-13T18:48:18.785376Z"Ruan, Lizhi"https://zbmath.org/authors/?q=ai:ruan.lizhi"Trakhinin, Yuri"https://zbmath.org/authors/?q=ai:trakhinin.yuri-lSummary: We are concerned with a model of ideal compressible isentropic two-fluid magnetohydrodynamics (MHD). Introducing an entropy-like function, we reduce the equations of two-fluid MHD to a symmetric form which looks like the classical MHD system written in the nonconservative form in terms of the pressure, the velocity, the magnetic field and the entropy. This gives a number of instant results. In particular, we conclude that all compressive extreme shock waves exist locally in time in the limit of weak magnetic field. We write down a condition sufficient for the local-in-time existence of current-vortex sheets in two-fluid flows. For the 2D case and a particular equation of state, we make the conclusion that contact discontinuities in two-fluid MHD flows exist locally in time provided that the Rayleigh-Taylor sign condition on the jump of the normal derivative of the pressure is satisfied at the first moment.Semilinear evolution models with scale-invariant friction and visco-elastic dampinghttps://zbmath.org/1521.351202023-11-13T18:48:18.785376Z"Mezadek, Abdelatif Kainane"https://zbmath.org/authors/?q=ai:mezadek.abdelatif-kainane"Reissig, Michael"https://zbmath.org/authors/?q=ai:reissig.michaelSummary: In this paper we study the global (in time) existence of small data Sobolev solutions and blow-up of Sobolev solutions to the Cauchy problem for semilinear evolution models with scale-invariant friction, visco-elastic damping and power nonlinearity. We are interested in critical exponents and the question how higher regularity in the data influences the admissible range of exponents \(p\) in the power nonlinearity to get global (in time) small data Sobolev solutions.Longer lifespan for many solutions of the Kirchhoff equationhttps://zbmath.org/1521.351212023-11-13T18:48:18.785376Z"Baldi, Pietro"https://zbmath.org/authors/?q=ai:baldi.pietro"Haus, Emanuele"https://zbmath.org/authors/?q=ai:haus.emanueleSummary: We consider the Kirchhoff equation \(\partial_{tt} u - \Delta u \big( 1 + \int_{\mathbb{T}^d} |\nabla u|^2 \big) = 0\) on the \(d\)-dimensional torus \(\mathbb{T}^d\), and its Cauchy problem with initial data \(u(0,x), \partial_t u(0,x)\) of size \(\varepsilon\) in the Sobolev class. The effective equation for the dynamics at the quintic order, obtained in previous papers by quasilinear normal form, contains resonances corresponding to nontrivial terms in the energy estimates. Such resonances cannot be avoided by tuning external parameters (simply because the Kirchhoff equation does not contain parameters). In this paper we introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound \(\varepsilon^{-6}\) for the lifespan of the corresponding solutions (the standard local theory gives \(\varepsilon^{-2}\), and the normal form for the cubic terms gives \(\varepsilon^{-4})\). The proof relies on the fact that, under these nonresonance conditions, the growth rate of the ``superactions'' of the effective equations on large time intervals is smaller (by a factor \(\varepsilon^2)\) than its a priori estimate based on the normal form for the cubic terms. The set of initial data satisfying such nonresonance conditions contains several nontrivial examples that are discussed in the paper.Asymptotic behavior for a viscoelastic problem with acoustic boundary conditions and variable-exponent nonlinearitieshttps://zbmath.org/1521.351222023-11-13T18:48:18.785376Z"Rahmoune, Abita"https://zbmath.org/authors/?q=ai:rahmoune.abita"Benyattou, Benabderrahmane"https://zbmath.org/authors/?q=ai:benyattou.benabderrahmaneSummary: The main goal of this work is to investigate the existence and uniqueness of the global solution of a mixed problem associated with a nonlinear viscoelastic Kirchho equation with Balakrishnan-Taylor damping and nonlinear boundary interior sources with variable exponents and acoustic boundary conditions in a bounded domain. We establish a general stability result for the equation without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function.Boundary problems for Helmholtz equation and the Cauchy problem for Dirac operatorshttps://zbmath.org/1521.351232023-11-13T18:48:18.785376Z"Shlapunov, Alexander A."https://zbmath.org/authors/?q=ai:shlapunov.alexander-aSummary: Studying an operator equation \(Au=f\) in Hilbert spaces one usually needs the adjoint operator \(A^\star\) for \(A\). Solving the ill-posed Cauchy problem for Dirac type systems in the Lebesgue spaces by an iteration method we propose to construct the corresponding adjoint operator with the use of normally solvable mixed problem for Helmholtz Equation. This leads to the description of necessary and sufficient solvability conditions for the Cauchy Problem and formulae for its exact and approximate solutions.Weyl's law for the Steklov problem on surfaces with rough boundaryhttps://zbmath.org/1521.351242023-11-13T18:48:18.785376Z"Karpukhin, Mikhail"https://zbmath.org/authors/?q=ai:karpukhin.mikhail-a"Lagacé, Jean"https://zbmath.org/authors/?q=ai:lagace.jean"Polterovich, Iosif"https://zbmath.org/authors/?q=ai:polterovich.iosifSummary: The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundary is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl's law holds for an even larger class of surfaces with rough boundaries. This class includes domains with interior cusps as well as ``slow'' exterior cusps. Moreover, the condition on the speed of exterior cusps cannot be improved, which makes our result, in a sense optimal. The proof is based on the methods of Suslina and Agranovich combined with some observations about the boundary behaviour of conformal mappings.Reverse isoperimetric inequality for the lowest Robin eigenvalue of a trianglehttps://zbmath.org/1521.351252023-11-13T18:48:18.785376Z"Krejčiřík, David"https://zbmath.org/authors/?q=ai:krejcirik.david"Lotoreichik, Vladimir"https://zbmath.org/authors/?q=ai:lotoreichik.vladimir"Vu, Tuyen"https://zbmath.org/authors/?q=ai:vu.tuyenSummary: We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.Dirac points for the honeycomb lattice with impenetrable obstacleshttps://zbmath.org/1521.351262023-11-13T18:48:18.785376Z"Li, Wei"https://zbmath.org/authors/?q=ai:li.wei.22"Lin, Junshan"https://zbmath.org/authors/?q=ai:lin.junshan"Zhang, Hai"https://zbmath.org/authors/?q=ai:zhang.hai|zhang.hai.2|zhang.hai.1|zhang.hai.3|zhang.hai.4Summary: This work is concerned with the Dirac points for the honeycomb lattice with impenetrable obstacles arranged periodically in a homogeneous medium. We consider both the Dirichlet and Neumann eigenvalue problems and prove the existence of Dirac points for both eigenvalue problems at crossing of the lower band surfaces as well as higher band surfaces. Furthermore, we perform quantitative analyses for the eigenvalues and the slopes of two conical dispersion surfaces near each Dirac point based on a combination of the layer potential technique and asymptotic analysis. It is shown that the eigenvalues are in the neighborhood of the singular frequencies associated with the Green's function for the honeycomb lattice, and the slopes of the dispersion surfaces are reciprocal to the eigenvalues.Fourier coefficients of restrictions of eigenfunctionshttps://zbmath.org/1521.351272023-11-13T18:48:18.785376Z"Wyman, Emmett L."https://zbmath.org/authors/?q=ai:wyman.emmett-l"Xi, Yakun"https://zbmath.org/authors/?q=ai:xi.yakun"Zelditch, Steve"https://zbmath.org/authors/?q=ai:zelditch.steveSummary: Let \(\{e_j\}\) be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold \((M, g)\). Let \(H \subset M\) be a submanifold and \(\{\psi_k\}\) be an orthonormal basis of Laplace eigenfunctions of \(H\) with the induced metric. We obtain joint asymptotics for the Fourier coefficients
\[
\langle\gamma_H e_j, \psi_k \rangle_{L^2 (H)} = \int_H e_j \overline{\psi}_k dV_H
\]
of restrictions \(\gamma_H e_j\) of \(e_j\) to \(H\). In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum \(\left\{ (\mu_k, \lambda_j) \right\}_{j,k-0}^{\infty}\) of the (square roots of the) Laplacian \(\Delta_M\) on \(M\) and the Laplacian \(\Delta_H\) on \(H\) in a family of suitably `thick' regions in \(\mathbb{R}^2\). Thick regions include (1) the truncated cone \(\mu_k / \lambda_j \in [a, b] \subset (0, 1)\) and \(\lambda_j \leqslant \lambda\), and (2) the slowly thickening strip \(|\mu_k - c \lambda_j |\leqslant w (\lambda)\) and \(\lambda_j \leqslant \lambda\), where \(w (\lambda)\) is monotonic and \(1 \ll w (\lambda) \precsim \lambda^{1/2}\). Key tools for obtaining the asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.The method of fundamental solutions for scattering of electromagnetic waves by a chiral objecthttps://zbmath.org/1521.351282023-11-13T18:48:18.785376Z"Athanasiadou, E. S."https://zbmath.org/authors/?q=ai:athanasiadou.evagelia-s"Arkoudis, I."https://zbmath.org/authors/?q=ai:arkoudis.iSummary: The scattering of a time-harmonic electromagnetic wave by a penetrable chiral obstacle in an achiral environment is considered. The method of fundamental solutions is employed in order to obtain numerically the solution of the problem using fundamental solutions in dyadic form. Surface vector potentials in terms of dyadic fundamental solutions together with the associated boundary integral operators are defined and their regularity properties are presented. Based on the dependence of the solution to the boundary data, appropriate systems of functions containing elements of dyadic fundamental solutions on the surface of the scatterer are constructed. Completeness and linear independence for these systems are proved with the usage of surface vector potentials. Using the transmission conditions, the scattering problem is transformed into a linear algebraic system with a coefficient matrix which consists of chiral and achiral blocks.Erratum to: ``Convexity in \(x\) of the level sets of the first Dirichlet eigenfunction''https://zbmath.org/1521.351292023-11-13T18:48:18.785376Z"Chu, Chie-Ping"https://zbmath.org/authors/?q=ai:chu.chiepingCorrects Lemma 3.2 in the author's paper [ibid. 280, No. 13--14, 1467--1474 (2007; Zbl 1136.35065)].On the Riemann-Hilbert problem of a generalized derivative nonlinear Schrödinger equationhttps://zbmath.org/1521.351302023-11-13T18:48:18.785376Z"Hu, Bei-Bei"https://zbmath.org/authors/?q=ai:hu.beibei"Zhang, Ling"https://zbmath.org/authors/?q=ai:zhang.ling.2"Xia, Tie-Cheng"https://zbmath.org/authors/?q=ai:xia.tie-chengSummary: In this work, we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schrödinger (DNLS) equation. By establishing a matrix Riemann-Hilbert problem and reconstructing potential function \(q(x, t)\) from eigenfunctions \(\{G_j(x, t, \eta)\}_1^3\) in the inverse problem, the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed. Moreover, we also obtain that the spectral functions \(f(\eta)\), \(s(\eta)\), \(F(\eta)\), \(S(\eta)\) are not independent of each other, but meet an important global relation. As applications, the generalized DNLS equation can be reduced to the Kaup-Newell equation and Chen-Lee-Liu equation on the half-line.A scalar Riemann-Hilbert problem on the torus: applications to the KdV equationhttps://zbmath.org/1521.351312023-11-13T18:48:18.785376Z"Piorkowski, Mateusz"https://zbmath.org/authors/?q=ai:piorkowski.mateusz"Teschl, Gerald"https://zbmath.org/authors/?q=ai:teschl.geraldThis article revisits the Riemann-Hilbert problem representing one-gap solutions of the Korteweg-de Vries equation. The main point of the article is to formulate a \textit{scalar-valued} Riemann-Hilbert problem on an elliptic Riemann surface, whose solution can be written in terms of Jacobi theta functions. From this, the authors derive the solution of the vector-valued Riemann-Hilbert problem representation of the one-gap solutions of the Korteweg-de Vries equation and also revisit the (singular) matrix-valued Riemann-Hilbert problem in this setting.
Reviewer: Deniz Bilman (Cincinnati)Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary conditionhttps://zbmath.org/1521.351322023-11-13T18:48:18.785376Z"Bernou, Armand"https://zbmath.org/authors/?q=ai:bernou.armand"Carrapatoso, Kleber"https://zbmath.org/authors/?q=ai:carrapatoso.kleber"Mischler, Stéphane"https://zbmath.org/authors/?q=ai:mischler.stephane"Tristani, Isabelle"https://zbmath.org/authors/?q=ai:tristani.isabelleThe subject of this work consists of the consideration of kinetic equations describing a variation of the density function corresponding to particles moving within a sufficiently smooth boundary in such a way that either the Boltzmann or the Landau collision operator is applicable. The specific feature of the considered problem is the position-dependent Maxwell boundary conditions varying from the pure specular reflection to the pure diffusive boundary condition and allowing the consideration of a weakly non-equilibrium system interacting with the equilibrated thermostat. The main result consists of establishing the convergence to the equilibrium in such a general situation in a uniform way including the transition to the hydrodynamic limit.
Reviewer: Eugene Postnikov (Kursk)High-order modeling of multiphase flows: based on discrete Boltzmann methodhttps://zbmath.org/1521.351332023-11-13T18:48:18.785376Z"Wang, Shuange"https://zbmath.org/authors/?q=ai:wang.shuange"Lin, Chuandong"https://zbmath.org/authors/?q=ai:lin.chuandong"Yan, Weiwei"https://zbmath.org/authors/?q=ai:yan.weiwei"Su, Xianli"https://zbmath.org/authors/?q=ai:su.xianli"Yang, Lichen"https://zbmath.org/authors/?q=ai:yang.lichenSummary: The high-order kinetic model for compressible multiphase flow is presented within the framework of the discrete Boltzmann method (DBM). Based on the Carnahan-Starling state equation, this model can describe the phase transition by introducing a source term of molecular interaction on the right-hand side of the Boltzmann equation. Meanwhile, the force term is incorporated to describe the external force. Through Hermite polynomial expansion, the equilibrium distribution function is expressed. Compared to the Navier-Stokes equations, the DBM provides more detailed and accurate information on both hydrodynamic and thermodynamic non-equilibrium effects. Finally, the model is verified through several typical benchmarks, including the liquid-vapor coexistence curve, free-falling process, shock wave, sound wave, thermal phase separation, and two-bubble oblique collision.\(\Gamma\)-convergence for nearly incompressible fluidshttps://zbmath.org/1521.351342023-11-13T18:48:18.785376Z"Bella, Peter"https://zbmath.org/authors/?q=ai:bella.peter"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Oschmann, Florian"https://zbmath.org/authors/?q=ai:oschmann.florianSummary: We consider the time-dependent compressible Navier-Stokes equations in the low Mach number regime in a family of domains \(\Omega_\varepsilon\subset \mathbb{R}^d\) converging in the sense of Mosco to a domain \(\Omega\subset \mathbb{R}^d\), \(d\in\{2, 3\}\). We show the limit is the incompressible Navier-Stokes system in \(\Omega\).
{\copyright 2023 American Institute of Physics}Modification of optimal homotopy asymptotic method for multi-dimensional time-fractional model of Navier-Stokes equationhttps://zbmath.org/1521.351352023-11-13T18:48:18.785376Z"Jan, Himayat Ullah"https://zbmath.org/authors/?q=ai:jan.himayat-ullah"Ullah, Hakeem"https://zbmath.org/authors/?q=ai:ullah.hakeem"Fiza, Mehreen"https://zbmath.org/authors/?q=ai:fiza.mehreen"Khan, Ilyas"https://zbmath.org/authors/?q=ai:khan.ilyas"Mohamed, Abdullah"https://zbmath.org/authors/?q=ai:mohamed.abdullah"Mousa, Abd Allah A."https://zbmath.org/authors/?q=ai:mousa.abd-allah-a(no abstract)Stability and asymptotic analysis for instationary gas transport via relative energy estimateshttps://zbmath.org/1521.351362023-11-13T18:48:18.785376Z"Egger, H."https://zbmath.org/authors/?q=ai:egger.herbert"Giesselmann, J."https://zbmath.org/authors/?q=ai:giesselmann.janThis paper first studies (an equivalent reformulation of) the one-dimensional compressible barotropic Euler equations with gravitation and friction on a bounded interval, which models the gas flows in a pipe. Under the assumptions of non-degenerate pipe cross-section, no-vacuum, and no-concentration, the high-friction (\textit{i.e.}, parabolic) limit and the stability results with respect to coefficients and initial data are established.
The key ingredient of the proof is the estimates for the relative energy/entropy functional of an abstract evolutionary system (Equations~(11) and (12)), which encompasses the above PDE model for pipe flow. It, in particular, only requires minimal regularity of pipe flow solutions.
Notably, such analyses can be naturally extended to study the gas flows in networks of pipes. See Section 5 of the paper.
Reviewer: Siran Li (Shanghai)Which measure-valued solutions of the monoatomic gas equations are generated by weak solutions?https://zbmath.org/1521.351372023-11-13T18:48:18.785376Z"Gallenmüller, Dennis"https://zbmath.org/authors/?q=ai:gallenmuller.dennis"Wiedemann, Emil"https://zbmath.org/authors/?q=ai:wiedemann.emilSummary: Contrary to the incompressible case, not every measure-valued solution of the compressible Euler equations can be generated by weak solutions or a vanishing viscosity sequence. In the present paper we give sufficient conditions on an admissible measure-valued solution of the isentropic Euler system to be generated by weak solutions. As one of the crucial steps we prove a characterization result for generating \({\mathcal{A}} \)-free Young measures in terms of potential operators including uniform \(L^{\infty } \)-bounds. More concrete versions of our results are presented in the case of a solution consisting of two Dirac measures. We conclude by discussing that are also necessary conditions for generating a measure-valued solution by weak solutions or a vanishing viscosity sequence and will point out that the resulting gap mainly results from obtaining only uniform \(L^p\)-bounds for \(1<p<\infty\) instead of \(p=\infty \).On the motion of a large number of small rigid bodies in a viscous incompressible fluidhttps://zbmath.org/1521.351382023-11-13T18:48:18.785376Z"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Roy, Arnab"https://zbmath.org/authors/?q=ai:roy.arnab"Zarnescu, Arghir"https://zbmath.org/authors/?q=ai:zarnescu.arghir-daniConsider a viscous incompressible fluid within a domain (in \(\mathbb{R}^2\) or \(\mathbb{R}^3\)). Now, let a finite number of rigid bodies be immersed in the fluid, and allow then eventually collide. The purpose of this paper is to better understand and clarify this phenomena under the conditions stated in its introduction.
Reviewer: Igor Leite Freire (São Carlos)Unstable Stokes waveshttps://zbmath.org/1521.351392023-11-13T18:48:18.785376Z"Hur, Vera Mikyoung"https://zbmath.org/authors/?q=ai:hur.vera-mikyoung"Yang, Zhao"https://zbmath.org/authors/?q=ai:yang.zhao.1Summary: We investigate the spectral instability of a \(2\pi /\kappa\) periodic Stokes wave of sufficiently small amplitude, traveling in water of unit depth, under gravity. Numerical evidence suggests instability whenever the unperturbed wave is resonant with its infinitesimal perturbations. This has not been analytically studied except for the Benjamin-Feir instability in the vicinity of the origin of the complex plane. Here we develop a periodic Evans function approach to give an alternative proof of the Benjamin-Feir instability and, also, a first proof of spectral instability away from the origin. Specifically, we prove instability near the origin for \(\kappa >\kappa_1:=1.3627827\dots\), and instability due to resonance of order two so long as an index function is positive. Validated numerics establishes that the index function is indeed positive for some \(\kappa <\kappa_1\), whereby there exists a Stokes wave that is spectrally unstable even though it is insusceptible to the Benjamin-Feir instability. The proofs involve center manifold reduction, Floquet theory, and methods of ordinary and partial differential equations. Numerical evaluation reveals that the index function remains positive unless \(\kappa =1.8494040\dots \). Therefore we conjecture that all Stokes waves of sufficiently small amplitude are spectrally unstable. For the proof of the conjecture, one has to verify that the index function is positive for \(\kappa\) sufficiently small.A new regularity criterion for the 3D incompressible Boussinesq equations in terms of the middle eigenvalue of the strain tensor in the homogeneous Besov spaces with negative indiceshttps://zbmath.org/1521.351402023-11-13T18:48:18.785376Z"Ines, Ben Omrane"https://zbmath.org/authors/?q=ai:ines.ben-omrane"Sadek, Gala"https://zbmath.org/authors/?q=ai:sadek.gala"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraSummary: This paper is concerned with the logarithmically improved regularity criterion in terms of the middle eigenvalue of the strain tensor to the 3D Boussinesq equations in Besov spaces with negative indices. It is shown that a weak solution is regular on \((0, T]\) provided that
\[
\int_0^T \frac{\| \lambda_2^+ (\cdot, t) \|^{\frac{2}{2-\delta}}_{\dot{B}_{\infty, \infty}^{-\delta}}}{\ln (e + \| u(\cdot, t) \|_{\dot{B}_{\infty,\infty}^{-\delta}}} dt < \infty.
\]
for some \(0< \delta <1\). As a consequence, this result is some improvements of recent works by \textit{J. Neustupa} and \textit{P. Penel} [Adv. Math. Fluid Mech. 237--268, 237--268 (2001; Zbl 1027.35094)] and \textit{E. Miller} [Arch. Ration. Mech. Anal. 235, No. 1, 99--139 (2020; Zbl 1434.35060)].Wave diffraction and radiation by a fully-submerged body in front of a vertical wall by using an exact DtN artificial boundary conditionhttps://zbmath.org/1521.351412023-11-13T18:48:18.785376Z"Rim, Un-Ryong"https://zbmath.org/authors/?q=ai:rim.un-ryong"Ri, Yong-Gwang"https://zbmath.org/authors/?q=ai:ri.yong-gwang"Do, Won-Chol"https://zbmath.org/authors/?q=ai:do.won-chol"Dong, Pil-Sung"https://zbmath.org/authors/?q=ai:dong.pil-sung"Kim, Chol-Won"https://zbmath.org/authors/?q=ai:kim.chol-won"Kim, Jin-Sim"https://zbmath.org/authors/?q=ai:kim.jin-sim(no abstract)\(n\)-soliton, breather, lump solutions and diverse traveling wave solutions of the fractional \((2+1)\)-Dimensional Boussinesq equationhttps://zbmath.org/1521.351422023-11-13T18:48:18.785376Z"Wang, Kang-Jia"https://zbmath.org/authors/?q=ai:wang.kang-jia"Liu, Jing-Hua"https://zbmath.org/authors/?q=ai:liu.jinghua"Si, Jing"https://zbmath.org/authors/?q=ai:si.jing"Shi, Feng"https://zbmath.org/authors/?q=ai:shi.feng"Wang, Guo-Dong"https://zbmath.org/authors/?q=ai:wang.guodong(no abstract)Asymptotic stability of a nonlinear wave for an outflow problem of the bipolar Navier-Stokes-Poisson system under large initial perturbationhttps://zbmath.org/1521.351432023-11-13T18:48:18.785376Z"Wu, Qiwei"https://zbmath.org/authors/?q=ai:wu.qiwei"Hou, Xiaofeng"https://zbmath.org/authors/?q=ai:hou.xiaofeng"Zhu, Peicheng"https://zbmath.org/authors/?q=ai:zhu.peichengSummary: In this paper, we study the time-asymptotic behavior of solutions to an outflow problem for the one-dimensional bipolar Navier-Stokes-Poisson system in the half space. First, we make some suitable assumptions on the boundary data and space-asymptotic states such that the time-asymptotic state of the solution is a nonlinear wave, which is the superposition of the transonic stationary solution and the 2-rarefaction wave. Next, we show the stability of this nonlinear wave under a class of large initial perturbation, provided that the strength of the transonic stationary solution is small enough, while the amplitude of the 2-rarefaction wave can be arbitrarily large. The proof is completed by a delicate energy method and a continuation argument. The key point is to derive the positive upper and lower bounds of the particle densities.Application of Megrabov's differential identities to the two-velocity hydrodynamics equations with one pressurehttps://zbmath.org/1521.351442023-11-13T18:48:18.785376Z"Zhabborov, Nasriddin M."https://zbmath.org/authors/?q=ai:zhabborov.nasriddin-m"Korobov, Pëtr V."https://zbmath.org/authors/?q=ai:korobov.petr-v"Imomnazarov, Kholmatzhon Kh."https://zbmath.org/authors/?q=ai:imomnazarov.kholmatzhon-khudainazarovichSummary: A series of the differential identities connecting velocities, pressure and body force in the two-velocity hydrodynamics equations with equilibrium of pressure phases are found. Some of these identities have a divergent form and can be considered as some conservation laws. It is detected that the flow functions for plane motion satisfy the Monge-Ampere system of equations.Padé approximations of quantized-vortex solutions of the Gross-Pitaevskii equationhttps://zbmath.org/1521.351452023-11-13T18:48:18.785376Z"Chen, Weiru"https://zbmath.org/authors/?q=ai:chen.weiru"Lan, Shanquan"https://zbmath.org/authors/?q=ai:lan.shanquan"Liu, Xiyi"https://zbmath.org/authors/?q=ai:liu.xiyi"Mo, Jiexiong"https://zbmath.org/authors/?q=ai:mo.jiexiong"Xu, Xiaobao"https://zbmath.org/authors/?q=ai:xu.xiaobao"Li, Guqiang"https://zbmath.org/authors/?q=ai:li.guqiangSummary: Quantized vortices are important topological excitations in Bose-Einstein condensates. The Gross-Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padé approximate solutions for quantized vortices with winding numbers \(\omega = 1, 2, 3, 4, 5, 6\) in the context of the Gross-Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that (1) they approximate the entire solutions quite well from the core to infinity; (2) higher-order Padé approximate solutions have higher accuracy; (3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with \(^{87}\mathrm{Rb}\) cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross-Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.Vector kink-dark complex solitons in a three-component Bose-Einstein condensatehttps://zbmath.org/1521.351462023-11-13T18:48:18.785376Z"Li, Yan"https://zbmath.org/authors/?q=ai:li.yan.12|li.yan.24|li.yan.7|li.yan.19|li.yan.43|li.yan.15|li.yan.28|li.yan|li.yan.11|li.yan.41|li.yan.54|li.yan.25|li.yan.2|li.yan.10|li.yan.9|li.yan.14|li.yan.16|li.yan.21|li.yan.5"Qin, Yan-Hong"https://zbmath.org/authors/?q=ai:qin.yanhong"Zhao, Li-Chen"https://zbmath.org/authors/?q=ai:zhao.li-chen"Yang, Zhan-Ying"https://zbmath.org/authors/?q=ai:yang.zhanying"Yang, Wen-Li"https://zbmath.org/authors/?q=ai:yang.wenliSummary: We investigate kink-dark complex solitons (KDCSs) in a three-component Bose-Einstein condensate (BEC) with repulsive interactions and pair-transition (PT) effects. Soliton profiles critically depend on the phase differences between dark solitons excitation elements. We report a type of kink-dark soliton profile which shows a droplet-bubble-droplet with a density dip, in sharp contrast to previously studied bubble-droplets. The interaction between two KDCSs is further investigated. It demonstrates some striking particle transition behaviours during their collision processes, while soliton profiles survive after the collision. Additionally, we exhibit the state transition dynamics between a kink soliton and a dark soliton. These results suggest that PT effects can induce more abundant complex solitons dynamics in multi-component BEC.Damping-like effects in Heisenberg spin chain caused by the site-dependent bilinear interactionhttps://zbmath.org/1521.351472023-11-13T18:48:18.785376Z"Zhang, Yu-Juan"https://zbmath.org/authors/?q=ai:zhang.yujuan"Zhao, Dun"https://zbmath.org/authors/?q=ai:zhao.dun"Li, Zai-Dong"https://zbmath.org/authors/?q=ai:li.zaidongSummary: We investigate a continuous Heisenberg spin chain equation which models the local magnetization in ferromagnet with time- and site-dependent inhomogeneous bilinear interaction and time-dependent spin-transfer torque. By establishing the gauge equivalence between the spin chain equation and an integrable generalized nonlinear Schrödinger equation, we present explicitly a novel nonautonomous magnetic soliton solution for the spin chain equation. The results display how the dynamics of the magnetic soliton can be controlled by the bilinear interaction and spin-polarized current. Especially, we find that the site-dependent bilinear interaction may break some conserved quantity, and give rise to damping-like effect in the spin evolution.Schrödinger-Lohe type models of quantum synchronization with nonidentical oscillatorshttps://zbmath.org/1521.351482023-11-13T18:48:18.785376Z"Antonelli, Paolo"https://zbmath.org/authors/?q=ai:antonelli.paolo"Reynolds, David N."https://zbmath.org/authors/?q=ai:reynolds.david-nAuthors' abstract: We study the asymptotic emergent dynamics of two models that can be thought of as extensions of the well known Schrödinger-Lohe model for quantum synchronization. More precisely, the interaction strength between different oscillators is determined by intrinsic parameters, following Cucker-Smale communication protocol. Unlike the original Schrödinger-Lohe system, where the interaction strength was assumed to be uniform, in the cases under our consideration the total mass of each quantum oscillator is allowed to vary in time. A striking consequence of this property is that these extended models yield configurations exhibiting phase, but not space, synchronization. The results are mainly based on the analysis of the ODE systems arising from the correlations, control over the well known Cucker-Smale dynamics, and the dynamics satisfied by the quantum order parameter.
Reviewer: Abderrazek Benhassine (Monastir)Stabilization of an interconnected system of Schrödinger and wave equations with boundary couplinghttps://zbmath.org/1521.351492023-11-13T18:48:18.785376Z"Moumen, Latifa"https://zbmath.org/authors/?q=ai:moumen.latifa"Sidiali, Fatima Zohra"https://zbmath.org/authors/?q=ai:sidiali.fatima-zohra"Rebiai, Salah-Eddine"https://zbmath.org/authors/?q=ai:rebiai.salah-eddineSummary: In this paper, we consider the Schrödinger equation coupled by the interface with a wave equation and with a boundary damping. The dissipation is acting on the wave equation through the Neumann boundary condition. We formulate the coupled system as an abstract evolution equation in an appropriate Hilbert space and use linear semigroup theory to show the well-posedness of the system. Then under some assumptions on the geometry of the spatial domain, we prove exponential stability of the solution. The proof of this result is based on a frequency domain approach which consists in verifying that the imaginary axis is included in the resolvent set of the system and analyzing the behavior of the resolvent operator of the system on the imaginary axis. The analysis of the resolvent is carried out by combining contradiction argument with the multipliers technique. This result extends Theorem 3.2 in [textit{J.-M. Wang} and \textit{J. Wang}, ``Energy decay rates for the coupled wave and Schrödinger system with boundary control'', in: Proceedings of the 15th international conference on control and automation, ICCA 2019. Los Alamitos, CA: IEEE Computer Society. 732--737 (2019; \url{doi:10.1109/ICCA.2019.8899735})] to multimensional spatial domains.Blowup of solutions for a transport equation with nonlocal velocity and dampinghttps://zbmath.org/1521.351502023-11-13T18:48:18.785376Z"Li, You"https://zbmath.org/authors/?q=ai:li.you"Liu, Yannan"https://zbmath.org/authors/?q=ai:liu.yannan|liu.yannan.1"Zhang, Wanwan"https://zbmath.org/authors/?q=ai:zhang.wanwanSummary: We study a transport equation with nonlocal velocity and power law type damping. We show that a certain class of initial data smooth solution blows up in finite time.
{\copyright 2023 American Institute of Physics}Dynamics of exact soliton solutions to the coupled nonlinear system using reliable analytical mathematical approacheshttps://zbmath.org/1521.351512023-11-13T18:48:18.785376Z"Bilal, Muhammad"https://zbmath.org/authors/?q=ai:bilal.muhammad"Younas, Usman"https://zbmath.org/authors/?q=ai:younas.usman"Ren, Jingli"https://zbmath.org/authors/?q=ai:ren.jingliSummary: Nonlinear Schrödinger-type equations are important models that have emerged from a wide variety of fields, such as fluids, nonlinear optics, the theory of deep-water waves, plasma physics, and so on. In this work, we obtain different soliton solutions to coupled nonlinear Schrödinger-type (CNLST) equations by applying three integration tools known as the \(\left(\frac{G^\prime}{G^2}\right)\)-expansion function method, the modified direct algebraic method (MDAM), and the generalized Kudryashov method. The soliton and other solutions obtained by these methods can be categorized as single (dark, singular), complex, and combined soliton solutions, as well as hyperbolic, plane wave, and trigonometric solutions with arbitrary parameters. The spectrum of the solitons is enumerated along with their existence criteria. Moreover, 2D, 3D, and contour profiles of the reported results are also plotted by choosing suitable values of the parameters involved, which makes it easier for researchers to comprehend the physical phenomena of the governing equation. The solutions acquired demonstrate that the proposed techniques are efficient, valuable, and straightforward when constructing new solutions for various types of nonlinear partial differential equation that have important applications in applied sciences and engineering. All the reported solutions are verified by substitution back into the original equation through the software package Mathematica.High-order breather, \(M\)-kink lump and semi-rational solutions of potential Kadomtsev-Petviashvili equationhttps://zbmath.org/1521.351522023-11-13T18:48:18.785376Z"Cao, Yulei"https://zbmath.org/authors/?q=ai:cao.yulei"Cheng, Yi"https://zbmath.org/authors/?q=ai:cheng.yi"He, Jingsong"https://zbmath.org/authors/?q=ai:he.jingsong"Chen, Yiren"https://zbmath.org/authors/?q=ai:chen.yirenSummary: \(N\)-kink soliton and high-order synchronized breather solutions for potential Kadomtsev-Petviashvili equation are derived by means of the Hirota bilinear method, and the limit process of high-order synchronized breathers are shown. Furthermore, \(M\)-lump solutions are also presented by taking the long wave limit. Additionally, a family of semi-rational solutions with elastic collision are generated by taking a long-wave limit of only a part of exponential functions, their interaction behaviors are shown by three-dimensional plots and contour plots.Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero-Bogoyavlenskii-Schiff equationhttps://zbmath.org/1521.351532023-11-13T18:48:18.785376Z"Khater, Mostafa M. A."https://zbmath.org/authors/?q=ai:khater.mostafa-m-a"Elagan, S. K."https://zbmath.org/authors/?q=ai:elagan.sayed-khalil|elagan.sayed-k-m"El-Shorbagy, M. A."https://zbmath.org/authors/?q=ai:el-shorbagy.mohammed-a"Alfalqi, S. H."https://zbmath.org/authors/?q=ai:alfalqi.s-h"Alzaidi, J. F."https://zbmath.org/authors/?q=ai:alzaidi.j-f"Alshehri, Nawal A."https://zbmath.org/authors/?q=ai:alshehri.nawal-aSummary: This paper studies the analytical and semi-analytic solutions of the generalized Calogero-Bogoyavlenskii-Schiff (CBS) equation. This model describes the \((2 + 1)\)-dimensional interaction between Riemann-wave propagation along the \(y\)-axis and the \(x\)-axis wave. The extended simplest equation (ESE) method is applied to the model, and a variety of novel solitary-wave solutions is given. These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma. The accuracy of the obtained solution is verified using a variational iteration (VI) semi-analytical scheme. The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained. The adopted scheme's performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.Resonance Y-type soliton solutions and some new types of hybrid solutions in the \((2+1)\)-dimensional Sawada-Kotera equationhttps://zbmath.org/1521.351542023-11-13T18:48:18.785376Z"Li, Jiaheng"https://zbmath.org/authors/?q=ai:li.jiaheng"Chen, Qingqing"https://zbmath.org/authors/?q=ai:chen.qingqing"Li, Biao"https://zbmath.org/authors/?q=ai:li.biaoSummary: In this paper, based on \(N\)-soliton solutions, we introduce a new constraint among parameters to find the resonance Y-type soliton solutions in \((2+1)\)-dimensional integrable systems. Then, we take the \((2+1)\)-dimensional Sawada-Kotera equation as an example to illustrate how to generate these resonance Y-type soliton solutions with this new constraint. Next, by the long wave limit method, velocity resonance and module resonance, we can obtain some new types of hybrid solutions of resonance Y-type solitons with line waves, breather waves, high-order lump waves respectively. Finally, we also study the dynamics of these interaction solutions and indicate mathematically that these interactions are elastic.Dynamics of a D'Alembert wave and a soliton molecule for an extended BLMP equationhttps://zbmath.org/1521.351552023-11-13T18:48:18.785376Z"Ren, Bo"https://zbmath.org/authors/?q=ai:ren.boSummary: The D'Alembert solution of the wave motion equation is an important basic formula in linear partial differential theory. The study of the D'Alembert wave is worthy of deep consideration in nonlinear partial differential systems. In this paper, we construct a \((2+1)\)-dimensional extended Boiti-Leon-Manna-Pempinelli (eBLMP) equation which fails to pass the Painlevé property. The D'Alembert-type wave of the eBLMP equation is still obtained by introducing one arbitrary function of the traveling-wave variable. The multi-solitary wave which should satisfy the velocity resonance condition is obtained by solving the Hirota bilinear form of the eBLMP equation. The dynamics of the three-soliton molecule, the three-kink soliton molecule, the soliton molecule bound by an asymmetry soliton and a one-soliton, and the interaction between the half periodic wave and a kink soliton molecule from the eBLMP equation are investigated by selecting appropriate parameters.Localization of nonlocal symmetries and interaction solutions of the Sawada-Kotera equationhttps://zbmath.org/1521.351562023-11-13T18:48:18.785376Z"Wu, Jian-wen"https://zbmath.org/authors/?q=ai:wu.jian-wen"Cai, Yue-jin"https://zbmath.org/authors/?q=ai:cai.yuejin"Lin, Ji"https://zbmath.org/authors/?q=ai:lin.jiSummary: The nonlocal symmetry of the Sawada-Kotera (SK) equation is constructed with the known Lax pair. By introducing suitable and simple auxiliary variables, the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further. On the other hand, the group invariant solutions of the SK equation are constructed with the classic Lie group method. In particular, by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.Shape-changed propagations and interactions for the \((3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation in fluidshttps://zbmath.org/1521.351572023-11-13T18:48:18.785376Z"Zhang, Dan-Dan"https://zbmath.org/authors/?q=ai:zhang.dandan"Wang, Lei"https://zbmath.org/authors/?q=ai:wang.lei"Liu, Lei"https://zbmath.org/authors/?q=ai:liu.lei.2"Liu, Tai-Xing"https://zbmath.org/authors/?q=ai:liu.tai-xing"Sun, Wen-Rong"https://zbmath.org/authors/?q=ai:sun.wen-rongSummary: In this article, we consider the \((3+1)\)-dimensional generalized Kadomtsev-Petviashvili (GKP) equation in fluids. We show that a variety of nonlinear localized waves can be produced by the breath wave of the GKP model, such as the (oscillating-) W- and M-shaped waves, rational W-shaped waves, multi-peak solitary waves, (quasi-) Bell-shaped and W-shaped waves and (quasi-) periodic waves. Based on the characteristic line analysis and nonlinear superposition principle, we give the transition conditions analytically. We find the interesting dynamic behavior of the converted nonlinear waves, which is known as the time-varying feature. We further offer explanations for such phenomenon. We then discuss the classification of the converted solutions. We finally investigate the interactions of the converted waves including the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. And the mechanisms of the collisions are analyzed in detail. The results could enrich the dynamic features of the high-dimensional nonlinear waves in fluids.Novel travelling wave structures: few-cycle-pulse solitons and soliton moleculeshttps://zbmath.org/1521.351582023-11-13T18:48:18.785376Z"Chen, Zitong"https://zbmath.org/authors/?q=ai:chen.zitong"Jia, Man"https://zbmath.org/authors/?q=ai:jia.man.1|jia.manSummary: We discuss a fifth order KdV (FOKdV) equation via a novel travelling wave method by introducing a background term. Results show that the background term plays an essential role in finding new abundant travelling wave structures, such as the soliton induced by negative background, the periodic travelling wave excited by the positive background, the few-cycle-pulse (FCP) solitons with and without background, the soliton molecules excited by the background. The FCP solitons are first obtained for the FOKdV equation.Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg-de Vries equationhttps://zbmath.org/1521.351592023-11-13T18:48:18.785376Z"Hashemi, Mir Sajjad"https://zbmath.org/authors/?q=ai:hashemi.mir-sajjad"Haji-Badali, Ali"https://zbmath.org/authors/?q=ai:haji-badali.ali"Alizadeh, Farzaneh"https://zbmath.org/authors/?q=ai:alizadeh.farzaneh"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafaSummary: In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann-Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries (LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg-de Vries (KdV) equation, and second, we find the related exact solutions for the derived generators. Finally, according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field KdV equation.Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equationhttps://zbmath.org/1521.351602023-11-13T18:48:18.785376Z"Liang, Zu-feng"https://zbmath.org/authors/?q=ai:liang.zufeng"Tang, Xiao-yan"https://zbmath.org/authors/?q=ai:tang.xiaoyan"Ding, Wei"https://zbmath.org/authors/?q=ai:ding.weiSummary: Nonlocal symmetries related to the Bäcklund transformation (BT) for the modified KdV-sine-Gordon (mKdV-SG) equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations. Then through the parameter expansion procedure, an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived. Finally, several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.Linear superposition of Wronskian rational solutions to the KdV equationhttps://zbmath.org/1521.351612023-11-13T18:48:18.785376Z"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiuSummary: A linear superposition is studied for Wronskian rational solutions to the KdV equation, which include rogue wave solutions. It is proved that it is equivalent to a polynomial identity that an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders is again a solution to the bilinear KdV equation. It is also conjectured that there is no other rational solutions among general linear superpositions of Wronskian rational solutions.Construction of fractal soliton solutions for the fractional evolution equations with conformable derivativehttps://zbmath.org/1521.351622023-11-13T18:48:18.785376Z"Wang, Kangle"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)New solitary wave solutions of the fractional modified KdV-Kadomtsev-Petviashvili equationhttps://zbmath.org/1521.351632023-11-13T18:48:18.785376Z"Wang, Kang-Le"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)Totally new soliton phenomena in the fractional Zoomeron model for shallow waterhttps://zbmath.org/1521.351642023-11-13T18:48:18.785376Z"Wang, Kang-Le"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)Lie symmetry analysis, optimal system and conservation laws of a new \((2+1)\)-dimensional KdV systemhttps://zbmath.org/1521.351652023-11-13T18:48:18.785376Z"Wang, Mengmeng"https://zbmath.org/authors/?q=ai:wang.mengmeng"Shen, Shoufeng"https://zbmath.org/authors/?q=ai:shen.shoufeng"Wang, Lizhen"https://zbmath.org/authors/?q=ai:wang.lizhenSummary: In this paper, Lie point symmetries of a new \((2+1)\)-dimensional KdV system are constructed by using the symbolic computation software Maple. Then, the one-dimensional optimal system, associated with corresponding Lie algebra, is obtained. Moreover, the reduction equations and some explicit solutions based on the optimal system are presented. Finally, the nonlinear self-adjointness is provided and conservation laws of this KdV system are constructed.The well-posedness for the Camassa-Holm type equations in critical Besov spaces \(B_{p , 1}^{1 + \frac{ 1}{ p}}\) with \(1 \leq p < + \infty \)https://zbmath.org/1521.351662023-11-13T18:48:18.785376Z"Ye, Weikui"https://zbmath.org/authors/?q=ai:ye.weikui"Yin, Zhaoyang"https://zbmath.org/authors/?q=ai:yin.zhaoyang"Guo, Yingying"https://zbmath.org/authors/?q=ai:guo.yingyingIn this paper, the authors prove local well-posedness for the Camassa-Holm (CH) equation
\[
u_t - u_{xxt} + 3uu_x = 2u_xu_{xx} + uu_{xxx}\tag{1}
\]
and the Novikov equation
\[
u_t - u_{xxt} = 3uu_xu_{xx} + u^2u_{xxx} - 4u^2u_x\tag{2}
\]
in the critical Besov spaces \(\mathcal C([0,T]:B_{p,1}^{1+\frac 1p}(\mathbb R))\) with \(1 \leq p < \infty\). In addition, they prove local well-posedness for the two-component Camassa-Holm system
\[
\begin{split}
& u_t + uu_x = -\partial_x(1-\partial_{xx})^{-1}(u^2 + \frac 12 u_x^2 + \frac 12 \rho^2) \\
& \rho_t + u \rho_x = - u_x\rho
\end{split}\tag{3}
\]
in \(\mathcal C([0,T]:B_{p,1}^{1+\frac 1p}) \times \mathcal C([0,T]:B_{p,1}^{\frac 1p})\).
Their results follow by first establishing well-posedness results for the abstract equation
\[
\begin{split}
& u_t + A(u) u_x = F(u) \qquad t >0, x \in \mathbb R \\
& u(0,x) = u_0(x)
\end{split}\tag{4}
\]
where \(A(u)\) is a polynomial and \(F\) is a 'good operator' such that for any \(\phi \in C_0^\infty\) and any \(\epsilon > 0\), the following fact holds.
If \(u_n \phi \rightarrow u\phi\) in \(B_{p,1}^{1+\frac 1p-\epsilon}\), then \(\langle F(u_n),\phi \rangle \rightarrow \langle F(u),\phi \rangle\). In addition, they establish similar well-posedness results for an abstract system related to the two-component Camassa-Holm system. These results enable them to prove local well-posedness for (1), (2), and (3) in the Besov spaces discussed above.
The local well-posedness results for the abstract equation (and abstract system) are obtained by using a Lagrangian coordinate transformation. Specifically, the associated Lagrangian scale of (4) is
\[
\begin{split}
& \frac d{dt}y(t,\xi)=A(u)(t,y(t,\xi)) \qquad t >0, \xi \in \mathbb R \\
& y(0,\xi)=\xi.
\end{split}
\]
Then introducing the new variable \(U(t,\xi) = u(t,y(t,\xi))\), (4) becomes
\[
\begin{split}
& U_t = (F(u))(t,y(t,\xi)) := \widetilde{F}(U,y) \qquad t > 0, \xi \in \mathbb R \\
& U(0,\xi) = U_0(\xi) = u_0(\xi).
\end{split}\tag{5}
\]
Establishing properties of (5) allow the authors to prove well-posedness results for (4) and thus (1) and (5). A similar technique is used to prove local well-posedness for the system (3).
Reviewer: Julie L. Levandosky (Framingham)Modulation instability, rogue waves and conservation laws in higher-order nonlinear Schrödinger equationhttps://zbmath.org/1521.351672023-11-13T18:48:18.785376Z"Dong, Min-Jie"https://zbmath.org/authors/?q=ai:dong.minjie"Tian, Li-Xin"https://zbmath.org/authors/?q=ai:tian.lixinSummary: In this paper, the modulation instability (MI), rogue waves (RWs) and conservation laws of the coupled higher-order nonlinear Schrödinger equation are investigated. According to MI and the \(2 \times 2\) Lax pair, Darboux-dressing transformation with an asymptotic expansion method, the existence and properties of the one-, second-, and third-order RWs for the higher-order nonlinear Schrödinger equation are constructed. In addition, the main characteristics of these solutions are discussed through some graphics, which are draw widespread attention in a variety of complex systems such as optics, Bose-Einstein condensates, capillary flow, superfluidity, fluid dynamics, and finance. In addition, infinitely-many conservation laws are established.Vector NLS solitons interacting with a boundaryhttps://zbmath.org/1521.351682023-11-13T18:48:18.785376Z"Zhang, Cheng"https://zbmath.org/authors/?q=ai:zhang.cheng"Zhang, Da-jun"https://zbmath.org/authors/?q=ai:zhang.dajunSummary: We construct multi-soliton solutions of the \(n\)-component vector nonlinear Schrödinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the so-called \textit{dressing the boundary}, which generates soliton solutions by preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes different components of the vector solitons. Connection of our construction to the inverse scattering transform is also provided.On the stability of solitons for the Maxwell-Lorentz equations with rotating particlehttps://zbmath.org/1521.351692023-11-13T18:48:18.785376Z"Komech, A. I."https://zbmath.org/authors/?q=ai:komech.alexander-ilich"Kopylova, E. A."https://zbmath.org/authors/?q=ai:kopylova.elena-aThe authors show the stability of rotating solitons of the Maxwell-Lorentz system with an extended charged rotating particle, assuming the effective moments of inertia are sufficiently large. The solitons are solutions which correspond to the uniform rotation of the particle. The Hamilton-Poisson representation of the Maxwell-Lorentz system is constructed in order to show stability. This allows to construct a Lyapunov function corresponding to a soliton. One key part of the proof is a lower bound for the Lyapunov function, which implies that the soliton is a strict local minimizer of the Lyapunov function.
Reviewer: Eric Stachura (Marietta)The perfect conductivity problem with arbitrary vanishing orders and non-trivial topologyhttps://zbmath.org/1521.351702023-11-13T18:48:18.785376Z"Sherman, Morgan"https://zbmath.org/authors/?q=ai:sherman.morgan"Weinkove, Ben"https://zbmath.org/authors/?q=ai:weinkove.benThe authors consider mathematical issues related to the electromagnetic problem of the behaviour of the electric potential in the gap between two perfect conductors when the latter are approaching to each other. The key feature is a complicated mutual topology of two adjacent conducting subdomains and the main result consists of obtaining optimal smooth bounds in terms of exploring the potential's gradient.
Reviewer: Eugene Postnikov (Kursk)A porous-media model for reactive fluid-rock interaction in a dehydrating rockhttps://zbmath.org/1521.351712023-11-13T18:48:18.785376Z"Zafferi, Andrea"https://zbmath.org/authors/?q=ai:zafferi.andrea"Huber, Konstantin"https://zbmath.org/authors/?q=ai:huber.konstantin"Peschka, Dirk"https://zbmath.org/authors/?q=ai:peschka.dirk"Vrijmoed, Johannes"https://zbmath.org/authors/?q=ai:vrijmoed.johannes"John, Timm"https://zbmath.org/authors/?q=ai:john.timm"Thomas, Marita"https://zbmath.org/authors/?q=ai:thomas.maritaSummary: We study the General Equations of Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) structure of models for reactive two-phase flows and their connection to a porous-media model for a reactive fluid-rock interaction used in geosciences. For this, we discuss the equilibration of fast dissipative processes in the GENERIC framework. Mathematical properties of the porous-media model and first results on its mathematical analysis are provided. The mathematical assumptions imposed for the analysis are critically validated with the thermodynamical rock datasets.
{\copyright 2023 American Institute of Physics}Generalized heat diffusion equations with variable coefficients and their fractalization from the Black-Scholes equationhttps://zbmath.org/1521.351722023-11-13T18:48:18.785376Z"El-Nabulsi, Rami Ahmad"https://zbmath.org/authors/?q=ai:el-nabulsi.rami-ahmad"Golmankhaneh, Alireza Khalili"https://zbmath.org/authors/?q=ai:golmankhaneh.alireza-khaliliSummary: In this study, we prove that modified diffusion equations, including the generalized Burgers' equation with variable coefficients, can be derived from the Black-Scholes equation with a time-dependent parameter based on the propagator method known in quantum and statistical physics. The extension for the case of a local fractal derivative is also addressed and analyzed.Optimal control of lake eutrophicationhttps://zbmath.org/1521.351732023-11-13T18:48:18.785376Z"Choquet, Catherine"https://zbmath.org/authors/?q=ai:choquet.catherine"Comte, Eloïse"https://zbmath.org/authors/?q=ai:comte.eloiseThe authors consider a domain \(\Omega \) in \(\mathbb{R}^{3}\), which represents a lake, with a \(C^{1}\) boundary \(\partial \Omega \) which is the union of three disjoint sets \(\partial \Omega =\Gamma _{\mathrm{in}}\cup \Gamma _{\mathrm{out}}\cup \Gamma \), where \(\Gamma _{\mathrm{in}}\) is the lake entrance and \(\Gamma _{\mathrm{out}}\) is the lake exit (\(\Gamma _{\mathrm{in}}\) and \(\Gamma _{\mathrm{out}}\) being not necessarily connected). They consider the space-time dynamics of the phosphorus stock \(\overline{S}\) and of the cyanobacteria concentration \(c\) in the lake written in \(\Omega \times (0,T)\) as: \(\partial _{t}\overline{S} +\operatorname{div}(v\overline{S})-d_{s}\Delta \overline{S}+b(\overline{S})\overline{S}-h( \overline{S})+f(\overline{S},c)c=0\), \(\partial _{t}c+\operatorname{div}(vc)-d_{c}\Delta c+m(c)c-f(\overline{S},c)c=0\), where \(v\) is the velocity which governs the convection, \(d_{s}\) and \(d_{c}\) diffusion coefficients, \(b\) and \(m\) nonlinear functions which represent the rates of loss per unit stock and per cyanobacteria concentration, \(h(\overline{S})=\overline{S}^{2}/(k+\overline{S }^{2})\) models the internal discharge of the phosphorus trapped in the sediments, and \(f\) the Monod term, often chosen in the form \(f(\overline{S} )=u_{max}\overline{S}/(k_{s}+\overline{S})\), which relates the phosphorus stock with the cyanobacterial dynamics, \(u_{\max}\) representing the maximal increasing rate and \(k_{s}\) the half-saturation value.
The initial conditions \(\overline{S}\mid _{t=0}=\overline{S}_{0}\), \(c\mid _{t=0}=c_{0}\) are imposed in \(\Omega \), together with the boundary conditions: \((\overline{ S}v-d_{s}\nabla \overline{S})\cdot n=(cv-d_{c}\nabla c)\cdot n=0\), on \( (0,T)\times \Gamma \), \(r_{S}\overline{S}+(\overline{S}v-d_{s}\nabla \overline{S})\cdot n=R_{S}(\overline{S})\), with \(r_{S}\leq 0\), \( r_{c}c+(cv-d_{c}\nabla c)\cdot n=R_{c}(c)\), with \(r_{c}\leq 0\), on \( (0,T)\times \Gamma _{\mathrm{out}}\), \(\overline{S}=\overline{P}\), \((cv-d_{c}\nabla c)\cdot n=0\), on \((0,T)\times \Gamma _{\mathrm{in}}\). Assuming further hypotheses on the different terms of the parabolic problem, the authors prove the existence of a unique weak solution \((\overline{S},c)\in (L^{2}(0,T;H^{1}(\Omega ))\cap L^{\infty }(0,T;L^{2}(\Omega ))^{2}\) to this problem.
For the proof, the authors propose a variational formulation of the above parabolic problem and they use the Faedo-Galerkin method, maximum principles and a uniqueness result. They then propose an optimal control associated to the above problem written as: Find \((\overline{P}^{\ast }, \overline{S}^{\ast },c^{\ast })\) such that \(J(\overline{P}^{\ast },c^{\ast })=\max\{J(\overline{P},c)\); \(\overline{P}\in E\) with \((\overline{S},c)\) satisfying the above parabolic problem\(\}\). Here \(E\) is the set of admissible controls defined as \(E=\{\overline{P}\in L^{\infty }((0,T)\times \Gamma _{in})\); \(0\leq \overline{P}(t,x)\leq \overline{P}_{\max}\) a.e. in \( (0,T)\times \Gamma _{in}\}\), where \(\overline{P}_{\max}\) is any given real number, and \(J\) the objective function defined as: \(J(\overline{P} ,c)=\int_{0}^{T}\left( \int_{\Gamma _{\mathrm{in}}}B(t,\sigma ,\overline{P}(t,\sigma ))e^{-\rho t}d\sigma -\int_{\Omega }D(t,x,c(t,x))e^{-\rho t}dx\right) dt\), where \(\rho \in ]0,1[\) is the social discount rate and \((\overline{S},c)\) satisfies the above parabolic problem. Setting \(S=\overline{S}-P\), where \(P\) is the unique solution to: \(\partial _{t}P-d_{s}\Delta P=0\) in \(\Omega \times (0,T)\), \(P\mid _{t=0}=P_{0}\) in \(\Omega \), \(P\mid _{\Gamma _{in}}= \overline{P}\), \(P\mid _{\partial \Omega \setminus \Gamma _{\mathrm{in}}}=0\) on \( \partial \Omega \times (0,T)\), \(P_{0}\) being the solution to: \(-\Delta P_{0}=0\) in \(\Omega \), \(P\mid _{\Gamma _{in}}=\overline{P}\mid _{t=0}\), \( P_{0}\mid _{\partial \Omega \setminus \Gamma _{in}}=0\) on \(\partial \Omega \times (0,T)\), the authors rewrite the above problem and the optimal control problem. Under the same hypotheses as in the existence and uniqueness result for a weak solution to the parabolic problem, they prove the existence of a global solution \((\overline{P}^{\ast },\overline{S}^{\ast },c^{\ast })\) to the optimal control problem. The proof is based on some boundedness property of a maximizing sequence for the objective function \(J\) and on the analysis of the convergence of this sequence.
Reviewer: Alain Brillard (Riedisheim)The dynamics of pulse solutions for reaction diffusion systems on a star shaped metric graph with the Kirchhoff's boundary conditionhttps://zbmath.org/1521.351742023-11-13T18:48:18.785376Z"Ei, Shin-Ichiro"https://zbmath.org/authors/?q=ai:ei.shin-ichiro"Mitsuzono, Ken"https://zbmath.org/authors/?q=ai:mitsuzono.ken"Shimatani, Haruki"https://zbmath.org/authors/?q=ai:shimatani.harukiSummary: In this paper, we consider motions of localized patterns for reaction-diffusion systems of general types on a metric star graph which consists of several half-lines with a common end point called ``the junction point'', where the Kirchhoff boundary condition is imposed. Assuming the existence and the stability of pulse and front like patterns for corresponding 1dimensional problems of reaction-diffusion systems, we rigorously derive ordinary differential equations describing the motions of them on a metric star graph. As the application, the attractive motion of a single pulse solution for the Gray-Scott model toward the junction point is shown. It is also shown that a single front solution of Allen-Cahn equation is repulsive against the junction point. The motion of multi pulse solutions and front solutions are also treated.Existence theorems for a generalized Chern-Simons equation on finite graphshttps://zbmath.org/1521.351752023-11-13T18:48:18.785376Z"Gao, Jia"https://zbmath.org/authors/?q=ai:gao.jia"Hou, Songbo"https://zbmath.org/authors/?q=ai:hou.songboSummary: Consider \(G = (V, E)\) as a finite graph, where \(V\) and \(E\) correspond to the vertices and edges, respectively. We study a generalized Chern-Simons equation \(\Delta u = \lambda\mathrm{e}^u(\mathrm{e}^{bu} - 1) + 4\pi\sum_{j = 1}^N \delta_{p_j}\) on \(G\), where \(\lambda\) and \(b\) are positive constants; \(N\) is a positive integer; \(p_1, p_2, \dots, p_N\) are distinct vertices of \(V\); and \(\delta_{p_j}\) is the Dirac delta mass at \(p_j\). We prove that there exists a critical value \(\lambda_c\) such that the equation has a solution if \(\lambda \geq \lambda_c\) and the equation has no solution if \(\lambda < \lambda_c\). We also prove that if \(\lambda > \lambda_c\), the equation has at least two solutions that include a local minimizer for the corresponding functional and a mountain-pass type solution. Our results extend and complete those of \textit{A. Huang} et al. [Commun. Math. Phys. 377, No. 1, 613--621 (2020; Zbl 1447.35338)] and \textit{S. Hou} and \textit{J. Sun} [Calc. Var. Partial Differ. Equ. 61, No. 4, Paper No. 139, 13 p. (2022; Zbl 1491.35238)].
{\copyright 2023 American Institute of Physics}Optimal Hardy weights on the Euclidean latticehttps://zbmath.org/1521.351762023-11-13T18:48:18.785376Z"Keller, Matthias"https://zbmath.org/authors/?q=ai:keller.matthias"Lemm, Marius"https://zbmath.org/authors/?q=ai:lemm.mariusSummary: We investigate the large-distance asymptotics of optimal Hardy weights on \(\mathbb{Z}^d\), \(d\geq 3\), via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar \(\frac{(d-2)^2}{4}|x|^{-2}\) as \(|x|\to \infty \). We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on \(\mathbb{Z}^d\): (1) averages over large sectors have inverse-square scaling, (2) for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3) for i.i.d. coefficients, there is a matching inverse-square lower bound on moments. The results imply \(|x|^{-4}\)-scaling for Rellich weights on \(\mathbb{Z}^d\). Analogous results are also new in the continuum setting. The proofs leverage Green's function estimates rooted in homogenization theory.Blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphshttps://zbmath.org/1521.351772023-11-13T18:48:18.785376Z"Wu, Yiting"https://zbmath.org/authors/?q=ai:wu.yiting(no abstract)Liouville-type results for elliptic equations with advection and potential terms on the Heisenberg grouphttps://zbmath.org/1521.351782023-11-13T18:48:18.785376Z"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed-boussairi"Kirane, Mokhtar"https://zbmath.org/authors/?q=ai:kirane.mokhtar"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: We investigate nonlinear elliptic equations of the form \[-\Delta_Hu(\xi)+A(\xi)\cdot\nabla_Hu(\xi)=V(\xi)f(u),\quad\xi\in\mathbb{H}^n,\] where \(\mathbb{H}^n=(\mathbb{R}^{2n+1},\circ)\) is the \((2n+1)\)-dimensional Heisenberg group, \(\Delta_H\) is the Kohn-Laplacian operator, \(\nabla_H\) is the Heisenberg gradient, \(\cdot\) is the inner product in \(\mathbb{R}^{2n}\), the advection term \(A:\mathbb{H}^n\to\mathbb{R}^{2n}\) is a \(C^1\) vector field satisfying a certain decay condition, the potential function \(V:\mathbb{H}^n\to(0,\infty)\) is continuous, and the nonlinearity \(f(u)\) has the form \(-u^{-p}\), \(p>0\), \(u>0\), or \(e^u\). Namely, we establish Liouville-type results for the class of stable solutions to the considered problems. Next, some special cases of the potential function \(V\) are discussed.Noncoercive diffusion equations with Radon measures as initial datahttps://zbmath.org/1521.351792023-11-13T18:48:18.785376Z"Porzio, Maria Michaela"https://zbmath.org/authors/?q=ai:porzio.maria-michaela"Smarrazzo, Flavia"https://zbmath.org/authors/?q=ai:smarrazzo.flavia"Tesei, Alberto"https://zbmath.org/authors/?q=ai:tesei.albertoSummary: We study Radon measure-valued solutions of the Cauchy-Dirichlet problem for \(\partial_t u = \Delta \phi (u)\) for a continuous, nondecreasing, at most powerlike \(\phi \). We prove well-posedness and regularity results, which depend on whether or not the initial data charge sets of suitable capacity (determined both by the Laplacian and by the growth order of \(\phi \)), and on suitable \textit{compatibility conditions}.Diffusive spatial movement with memory in an advective environmenthttps://zbmath.org/1521.351802023-11-13T18:48:18.785376Z"Zhang, Hua"https://zbmath.org/authors/?q=ai:zhang.hua.5"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.4"Song, Yongli"https://zbmath.org/authors/?q=ai:song.yongli"Wei, Junjie"https://zbmath.org/authors/?q=ai:wei.junjieSummary: The movements of species in a river are driven by random diffusion, unidirectional water flow, and cognitive judgement with spatial memory. In this paper, we formulate a reaction-diffusion-advection model with memory-based diffusion and homogeneous Dirichlet boundary conditions. The existence of a nonconstant positive steady state is proven. We obtain the linear stability of the steady state by analysing the eigenvalues of the associated linear operator: the nonconstant steady state can always be linearly stable regardless of the memory delay, while the model can also possess Hopf bifurcation as the memory delay varies. Moreover, theoretical and numerical results show that large advection annihilates oscillation patterns and drives the species to concentrate downstream.Singular boundary behaviour and large solutions for fractional elliptic equationshttps://zbmath.org/1521.351812023-11-13T18:48:18.785376Z"Abatangelo, Nicola"https://zbmath.org/authors/?q=ai:abatangelo.nicola"Gómez-Castro, David"https://zbmath.org/authors/?q=ai:gomez-castro.david"Vázquez, Juan Luis"https://zbmath.org/authors/?q=ai:vazquez.juan-luisSummary: We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data.
In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary.
Here, we show that, for equations driven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function.Correction to: ``Solvability and Volterra property of nonlocal problems for mixed fractional-order diffusion-wave equation''https://zbmath.org/1521.351822023-11-13T18:48:18.785376Z"Adil, Nauryzbay"https://zbmath.org/authors/?q=ai:adil.nauryzbay"Berdyshev, Abdumauvlen S."https://zbmath.org/authors/?q=ai:berdyshev.abdumauvlen-suleymanovich"Eshmatov, B. E."https://zbmath.org/authors/?q=ai:eshmatov.bakhodir-e"Baishemirov, Zharasbek D."https://zbmath.org/authors/?q=ai:baishemirov.zharasbek-dFrom the text: Following publication of the original article [ibid. 2023, Paper No. 47, 29 p. (2023; Zbl 1518.35613)], the authors identified an error in the author name Abdumauvlen S. Berdyshev.
The incorrect author name is: Abdumauvlen S. Bersyhev
The correct author name is: Abdumauvlen S. Berdyshev
The author group has been updated above and the original article [loc. cit.] has been corrected.On the critical behavior for time-fractional reaction diffusion problemshttps://zbmath.org/1521.351832023-11-13T18:48:18.785376Z"Aldawish, Ibtisam"https://zbmath.org/authors/?q=ai:aldawish.ibtisam"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: We first investigate the existence and nonexistence of weak solutions to the time-fractional reaction diffusion problem
\[
\frac{\partial^\alpha u}{\partial t^\alpha}-\frac{\partial^2 u}{\partial x^2}+u\ge x^{-a}|u|^p, \ t>0, \ x\in(0,1],\quad u(0,x)=u_0(x), \ x\in(0,1]
\]
under the inhomogeneous Dirichlet boundary condition
\[
u(t,1)=\delta,\quad t>0,
\]
where \(u=u(t,x)\), \(0<\alpha<1\), \(\frac{\partial^\alpha}{\partial t^\alpha}\) is the time-Caputo fractional derivative of order \(\alpha\), \(a\ge 0\), \(p>1\) and \(\delta>0\). We show that, if \(a\le 2\), the existence holds for all \(p>1\) while if \(a>2\), then the dividing line with respect to existence or nonexistence is given by the critical exponent \(p^*=a-1\). The proof of the nonexistence result is based on nonlinear capacity estimates specifically adapted to the nonlocal nature of the problem, the modified Helmholtz operator \(-\frac{\partial^2}{\partial x^2}+I\), and the considered boundary condition. The existence part is proved by the construction of explicit solutions. We next extend our study to the case of systems.Spike solutions for a fractional elliptic equation in a compact Riemannian manifoldhttps://zbmath.org/1521.351842023-11-13T18:48:18.785376Z"Bendahou, Imene"https://zbmath.org/authors/?q=ai:bendahou.imene"Khemiri, Zied"https://zbmath.org/authors/?q=ai:khemiri.zied"Mahmoudi, Fethi"https://zbmath.org/authors/?q=ai:mahmoudi.fethiSummary: Given an \(n\)-dimensional compact Riemannian manifold \((M,g)\) without boundary, we consider the nonlocal equation
\[\varepsilon^{2s} P_g^s u + u = u^p \quad \hbox{in }\, (M,g),\]
where \(P_g^s\) stands for the fractional Paneitz operator with principal symbol \((-\Delta_g)^s\), \(s \in (0,1)\), \( p \in (1,2_s^*-1)\) with \(2_s^* := \frac{2n}{n-2s} \), \(n>2s\), represents the critical Sobolev exponent and \(\varepsilon > 0\) is a small real parameter. We construct a family of positive solutions \(u_\varepsilon\) that concentrate, as \(\varepsilon \to 0\) goes to zero, near critical points of the mean curvature \(H\) for \(0 <s< \frac{1}{2}\) and near critical points of a reduced function involving the scalar curvature of the manifold~ \(M\) for \( \frac{1}{2} \leq s < 1\).On the equivalence of classical Helmholtz equation and fractional Helmholtz equation with arbitrary orderhttps://zbmath.org/1521.351852023-11-13T18:48:18.785376Z"Cheng, Xinyu"https://zbmath.org/authors/?q=ai:cheng.xinyu"Li, Dong"https://zbmath.org/authors/?q=ai:li.dong"Yang, Wen"https://zbmath.org/authors/?q=ai:yang.wen.1|yang.wenSummary: We show the equivalence of the classical Helmholtz equation and the fractional Helmholtz equation with arbitrary order. This improves a recent result of \textit{V. Guan} et al. [Commun. Contemp. Math. 25, No. 2, Article ID 2250016, 18 p. (2023; Zbl 1509.35344)].Analysis of fractional differential equations with the help of different operatorshttps://zbmath.org/1521.351862023-11-13T18:48:18.785376Z"Iqbal, Naveed"https://zbmath.org/authors/?q=ai:iqbal.naveed-h"Al Harbi, Moteb Fheed Saad"https://zbmath.org/authors/?q=ai:al-harbi.moteb-fheed-saad"Alshammari, Saleh"https://zbmath.org/authors/?q=ai:alshammari.saleh"Zaland, Shamsullah"https://zbmath.org/authors/?q=ai:zaland.shamsullah(no abstract)Application of Hosoya polynomial to solve a class of time-fractional diffusion equationshttps://zbmath.org/1521.351872023-11-13T18:48:18.785376Z"Jafari, Hossein"https://zbmath.org/authors/?q=ai:jafari.hossein"Ganji, Roghayeh Moallem"https://zbmath.org/authors/?q=ai:ganji.roghayeh-moallem"Narsale, Sonali Mandar"https://zbmath.org/authors/?q=ai:narsale.sonali-mandar"Kgarose, Maluti"https://zbmath.org/authors/?q=ai:kgarose.maluti"Nguyen, Van Thinh"https://zbmath.org/authors/?q=ai:nguyen.van-thinh(no abstract)Uniqueness of the potential in a time-fractional diffusion equationhttps://zbmath.org/1521.351882023-11-13T18:48:18.785376Z"Jing, Xiaohua"https://zbmath.org/authors/?q=ai:jing.xiaohua"Peng, Jigen"https://zbmath.org/authors/?q=ai:peng.jigenSummary: This article concerns the uniqueness of an inverse coefficient problem of identifying a spatially varying potential in a one-dimensional time-fractional diffusion equation. The input sources are given by a complete system in \(L^2 (0,1)\), and measurements are observed at the end point of the spatial interval. Firstly, we provide the positive lower bound of the Green function for the differential operator with different boundary conditions. Then, based on the positive lower bound estimation of the Green function, the relationship between the Green function, the solution of the forward problem, and the potential, such measurements uniquely determine the potential on the entire interval under different boundary conditions.Global solutions of a fractional semilinear pseudo-parabolic equation with nonlocal sourcehttps://zbmath.org/1521.351892023-11-13T18:48:18.785376Z"Li, Na"https://zbmath.org/authors/?q=ai:li.na.4|li.na"Fang, Shaomei"https://zbmath.org/authors/?q=ai:fang.shaomeiSummary: In this paper, the initial boundary value problem for a fractional nonlocal semilinear pseudo-parabolic equation is established. Firstly, we get the local solution by the standard Galerkin method and the priori estimates. Next, by applying potential well argument, the existence and uniqueness of the global solution are proved for initial energy \(J(u_0)\le d\).Global well-posedness of a Cauchy problem for a nonlinear parabolic equation with memoryhttps://zbmath.org/1521.351902023-11-13T18:48:18.785376Z"Nguyen, Anh Tuan"https://zbmath.org/authors/?q=ai:nguyen.anh-tuan"Nghia, Bui Dai"https://zbmath.org/authors/?q=ai:nghia.bui-dai"Nguyen, Van Thinh"https://zbmath.org/authors/?q=ai:nguyen.van-thinh(no abstract)Classification of solutions to mixed order elliptic system with general nonlinearityhttps://zbmath.org/1521.351912023-11-13T18:48:18.785376Z"Peng, Shaolong"https://zbmath.org/authors/?q=ai:peng.shaolongSummary: In this paper, we consider the mixed order elliptic system
\[
\begin{cases} (-\Delta)^{\frac{\alpha}{2}}u(x)=f(u,v), \\
(-\Delta) v(x)=g(u,v), \end{cases}
\]
where \(u\geq 0\), \(\alpha \in (0,2)\), \(v\) may change signs. We aim to study the classification results of solutions to the above semilinear elliptic system in \(\mathbb{R}^2\). We first derive the equivalence between the above PDE system and the corresponding IE (integral equation) system. Then, applying the method of moving spheres in integral form combined with integral inequalities, under certain assumptions, we give a complete classification of the classical solutions to the above system in \(\mathbb{R}^2\) .Terminal value problem for stochastic fractional equation within an operator with exponential kernelhttps://zbmath.org/1521.351922023-11-13T18:48:18.785376Z"Phuong, Nguyen Duc"https://zbmath.org/authors/?q=ai:phuong.nguyen-duc"Hoan, Luu Vu Cam"https://zbmath.org/authors/?q=ai:hoan.luu-vu-cam"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-i"Nguyen, Anh Tuan"https://zbmath.org/authors/?q=ai:nguyen.anh-tuan(no abstract)New results on continuity by order of derivative for conformable parabolic equationshttps://zbmath.org/1521.351932023-11-13T18:48:18.785376Z"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Nguyen, Van Tien"https://zbmath.org/authors/?q=ai:nguyen.van-tien"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donal"Can, Nguyen Huu"https://zbmath.org/authors/?q=ai:can.nguyen-huu"Nguyen, Van Thinh"https://zbmath.org/authors/?q=ai:nguyen.van-thinh(no abstract)Space-dependent variable-order time-fractional wave equation: existence and uniqueness of its weak solutionhttps://zbmath.org/1521.351942023-11-13T18:48:18.785376Z"van Bockstal, K."https://zbmath.org/authors/?q=ai:van-bockstal.karel"Hendy, A. S."https://zbmath.org/authors/?q=ai:hendy.ahmed-s"Zaky, M. A."https://zbmath.org/authors/?q=ai:zaky.mahmoud-aSummary: The investigation of an initial-boundary value problem for a fractional wave equation with space-dependent variable-order wherein the coefficients have a dependency on the spatial and time variables is the concern of this work. This type of variable-order fractional differential operator originates in the modelling of viscoelastic materials. The global in time existence of a unique weak solution to the model problem has been proved under appropriate conditions on the data. Rothe's time discretization method is applied to achieve that purpose.On the new exact traveling wave solutions of the time-space fractional strain wave equation in microstructured solids via the variational methodhttps://zbmath.org/1521.351952023-11-13T18:48:18.785376Z"Wang, Kang-Jia"https://zbmath.org/authors/?q=ai:wang.kang-jiaSummary: In this paper, we mainly study the time-space fractional strain wave equation in microstructured solids. He's variational method, combined with the two-scale transform are implemented to seek the solitary and periodic wave solutions of the time-space strain wave equation. The main advantage of the variational method is that it can reduce the order of the differential equation, thus simplifying the equation, making the solving process more intuitive and avoiding the tedious solving process. Finally, the numerical results are shown in the form of 3D and 2D graphs to prove the applicability and effectiveness of the method. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations in physics.Solitary wave dynamics of the local fractional Bogoyavlensky-Konopelchenko modelhttps://zbmath.org/1521.351962023-11-13T18:48:18.785376Z"Wang, Kangle"https://zbmath.org/authors/?q=ai:wang.kangle(no abstract)Normalized ground states and multiple solutions for nonautonomous fractional Schrödinger equationshttps://zbmath.org/1521.351972023-11-13T18:48:18.785376Z"Yang, Chen"https://zbmath.org/authors/?q=ai:yang.chen.2"Yu, Shu-Bin"https://zbmath.org/authors/?q=ai:yu.shubin"Tang, Chun-Lei"https://zbmath.org/authors/?q=ai:tang.chun-leiSummary: In this paper, we consider the following fractional Schrödinger equations with prescribed \(L^2\)-norm constraint:
\[
\begin{cases}
(-\Delta)^s u = \lambda u + h(\varepsilon x)f(u)\text{ in }\mathbb{R}^N,\\
\int_{\mathbb{R}^N}|u|^2dx = a^2,
\end{cases}
\]
where \(0 < s < 1\), \(N \geq 3\), \(a, \varepsilon > 0\), \(h\in C(\mathbb{R}^N, \mathbb{R^+})\) and \(f\in C(\mathbb{R}, \mathbb{R})\). In the mass subcritical case but under general assumptions on \(f\), we prove the multiplicity of normalized solutions to this problem. Specifically, we show that the number of normalized solutions is at least the number of global maximum points of \(h\) when \(\varepsilon\) is small enough. Before that, without any restrictions on \(\varepsilon\) and the number of global maximum points, the existence of normalized ground states can be determined. In this sense, by studying the relationship between \(h_0 := \inf_{x\in\mathbb{R}^N}h(x)\) and \(h_\infty := \lim_{|x|\rightarrow\infty}h(x)\), we establish new results on the existence of normalized ground states for nonautonomous elliptic equations.Almost periodic solutions of the wave equation with damping and impulsive actionhttps://zbmath.org/1521.351982023-11-13T18:48:18.785376Z"Dvornyk, A. V."https://zbmath.org/authors/?q=ai:dvornyk.a-v"Tkachenko, V. I."https://zbmath.org/authors/?q=ai:tkachenko.victor-iSummary: We obtain sufficient conditions for the existence of piecewise continuous almost periodic solutions to the damped wave equation with impulsive action.An inverse problem of recovering the variable order of the derivative in a fractional diffusion equationhttps://zbmath.org/1521.351992023-11-13T18:48:18.785376Z"Artyushin, A. N."https://zbmath.org/authors/?q=ai:artyushin.aleksandr-nikolaevichSummary: We consider a fractional diffusion equation with variable space-dependent order of the derivative in a bounded multidimensional domain. The initial data are homogeneous and the right-hand side and its time derivative satisfy some monotonicity conditions. Addressing the inverse problem with final overdetermination, we establish the uniqueness of a solution as well as some necessary and sufficient solvability conditions in terms of a certain constructive operator \(A \). Moreover, we give a simple sufficient solvability condition for the inverse problem. The arguments rely on the Birkhoff-Tarski theorem.An identification problem of source function in the system of composite typehttps://zbmath.org/1521.352002023-11-13T18:48:18.785376Z"Belov, Yuriĭ Ya."https://zbmath.org/authors/?q=ai:belov.yurii-yaSummary: An identification problem of source function for the semievolutionary system of two partial differential equations, one of which is parabolic, and the second -- elliptic are investigated. The Cauchy problem and the first boundary-value problem are considered. Initial problems are approximated by problems in which the elliptic equation is replaced with the parabolic equation containing the small parameter \(\varepsilon>0\) at a derivative with respect to time.On the problem of identification of two lower coefficients and the coefficient by the derivative with respect to time in the parabolic equationhttps://zbmath.org/1521.352012023-11-13T18:48:18.785376Z"Datsenko, Anzhelika V."https://zbmath.org/authors/?q=ai:datsenko.anzhelika-v"Polyntseva, Svetlana V."https://zbmath.org/authors/?q=ai:polyntseva.svetlana-vSummary: The theorem of existence and uniqueness of classical solution of identification problem of two lower coefficients and the coefficient by the derivative with respect to time in the class of smooth bounded functions is proved.
In the proof of the existence and uniqueness of the inverse problem solution using the overdetermination conditions, the original inverse problem is reduced to the direct problem for the loaded (containing traces of unknown functions and their derivatives) equation. The investigation of the correctness of the direct problem is obtained by the method of weak approximation.Identification of the potential coefficient in the wave equation with incomplete data: a sentinel methodhttps://zbmath.org/1521.352022023-11-13T18:48:18.785376Z"Elhamza, Billal"https://zbmath.org/authors/?q=ai:elhamza.billal"Hafdallah, Abdelhak"https://zbmath.org/authors/?q=ai:hafdallah.abdelhakSummary: In this paper, we consider a wave equation with incomplete data, where we do not know the potential coefficient and the initial conditions. From observing the system in the boundary, we want to get information on the potential coefficient independently of the initial conditions. This can be obtained using the sentinel method of Lions, which is a functional insensitive to certain parameters. Shows us through the adjoint system that the existence of the sentinel is equivalent to an optimal control problem. We solve this optimal control problem by using the Hilbert uniqueness method (HUM).An identification problem of the source function of the special form in two-dimensional parabolic equationhttps://zbmath.org/1521.352032023-11-13T18:48:18.785376Z"Frolenkov, Igor' V."https://zbmath.org/authors/?q=ai:frolenkov.igor-v"Kriger, Ekaterina N."https://zbmath.org/authors/?q=ai:kriger.ekaterina-nSummary: The existence, uniqueness and stability of solution by input data of the identification problem for parabolic equation with source function of the special form in the case of Cauchy's data has been proved in this article.An representation of the solution of the inverse problem for a multidimensional parabolic equation with initial data in the form of a producthttps://zbmath.org/1521.352042023-11-13T18:48:18.785376Z"Frolenkov, Igor' V."https://zbmath.org/authors/?q=ai:frolenkov.igor-v"Romanenko, Galina V."https://zbmath.org/authors/?q=ai:romanenko.galina-vSummary: An identification problem of the coefficient at differential operator of second order in multidimensional parabolic equation with Cauchy data was studied in this article. The theorems of existence and uniqueness of the solution for direct and inverse problems has been proved.On reconstruction of small sources from Cauchy data at a fixed frequencyhttps://zbmath.org/1521.352052023-11-13T18:48:18.785376Z"Harris, Isaac"https://zbmath.org/authors/?q=ai:harris.isaac"Le, Thu"https://zbmath.org/authors/?q=ai:le.thu-minh"Nguyen, Dinh-Liem"https://zbmath.org/authors/?q=ai:nguyen.dinh-liemSummary: This short paper is concerned with the numerical reconstruction of small sources from boundary Cauchy data for a single frequency. We study a sampling method to determine the location of small sources in a very fast and robust way. Furthermore, the method can also compute the intensity of point sources provided that the sources are well separated. A simple justification of the method is done using the Green representation formula and an asymptotic expansion of the radiated field for small volume sources. The implementation of the method is non-iterative, computationally cheap, fast, and very simple. Numerical examples are presented to illustrate the performance of the method.A Tikhonov regularization method for Cauchy problem based on a new relaxation modelhttps://zbmath.org/1521.352062023-11-13T18:48:18.785376Z"Huang, Qin"https://zbmath.org/authors/?q=ai:huang.qin"Gong, Rongfang"https://zbmath.org/authors/?q=ai:gong.rongfang"Jin, Qinian"https://zbmath.org/authors/?q=ai:jin.qinian"Zhang, Ye"https://zbmath.org/authors/?q=ai:zhang.yeSummary: In this paper, we consider a Cauchy problem of recovering both missing value and flux on inaccessible boundary from Dirichlet and Neumann data measured on the remaining accessible boundary. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. With an introduction of a relaxation parameter, the Dirichlet boundary conditions are approximated by two Robin ones. Compared to the existing work, weaker regularity is required on the Dirichlet data. This makes the proposed model simpler and more efficient in computation. A series of theoretical results are established for the new reconstruction model. Several numerical examples are provided to show feasibility and effectiveness of the proposed method. For simplicity of the statements, we take Poisson equation as the governed equation. However, the proposed method can be applied directly to Cauchy problems governed by more general equations, even other linear or nonlinear inverse problems.Extracting discontinuity using the probe and enclosure methodshttps://zbmath.org/1521.352072023-11-13T18:48:18.785376Z"Ikehata, Masaru"https://zbmath.org/authors/?q=ai:ikehata.masaruSummary: This is a review article on the development of the probe and enclosure methods from past to present, focused on their central ideas together with various applications.Convergence analysis of an alternating direction method of multipliers for the identification of nonsmooth diffusion parameters with total variationhttps://zbmath.org/1521.352082023-11-13T18:48:18.785376Z"Ouakrim, Y."https://zbmath.org/authors/?q=ai:ouakrim.youssef"Boutaayamou, I."https://zbmath.org/authors/?q=ai:boutaayamou.idriss"El Yazidi, Y."https://zbmath.org/authors/?q=ai:el-yazidi.youness"Zafrar, A."https://zbmath.org/authors/?q=ai:zafrar.abderrahimSummary: The paper presents a numerical method for identifying discontinuous conductivities in elliptic equations from boundary observations. The solutions to this inverse problem are obtained through a constrained optimization problem, where the cost functional is a combination of the Kohn-Vogelius and Total Variation functionals. Instead of regularizing the Total Variation stabilization functional, which is commonly used in the literature, we introduce an Alternating Direction Method of Multipliers to preserve the favorable properties of non-smoothness and convexity. The discretization is carried out using a mixed finite element/volume method, while the numerical solutions are iteratively computed using a variant of the Uzawa algorithm. We show the surjectivity of the derivatives of the constraints related to the discrete optimization problem and derive a source condition for the discrete inverse problem. We then investigate the convergence analysis and establish the convergence rate. Finally, we conclude with some numerical experiments to illustrate the efficiency of the proposed method.Lipschitz stability of recovering the conductivity from internal current densitieshttps://zbmath.org/1521.352092023-11-13T18:48:18.785376Z"Qiu, Lingyun"https://zbmath.org/authors/?q=ai:qiu.lingyun"Zheng, Siqin"https://zbmath.org/authors/?q=ai:zheng.siqinSummary: In recent years, coupled physics imaging techniques have been developed to produce clearer images than those produced by electrical impedance tomography. This paper focuses on the inverse problem arising in current density impedance imaging and magneto-acousto-electric tomography. We consider the electrostatic equation \(\nabla\cdot(\sigma\nabla w_b)=0\) in a bounded domain \(\Omega\subset\mathbb{R}^3\) with either the Dirichlet or Neumann boundary condition \(b\), where \(\sigma\) is a scalar conductivity function. The inverse problem is formulated as recovering \(\sigma\) from vector fields \(J_b=\sigma\nabla w_b\) with different boundary conditions \(b\). We provide a local Lipschitz stability, stating that near some known \(\sigma_0\) and under some regularity assumptions, we can find \(b_1\) and \(b_2\) by constructing complex geometrical optics (CGO) solutions such that \(\Vert\ln\sigma^{(1)}-\ln\sigma^{(2)}\Vert_{C^{m,\alpha}(\overline{\Omega})}\leqslant C\sum^2_{j=1}\Vert J^{(1)}_{b_j}-J^{(2)}_{b_j}\Vert_{C^{m,\alpha}(\overline{\Omega})}\). Furthermore, we modify the CGO solutions using the reflection method to make \(b_1\) and \(b_2\) vanish on a portion of a plane, and prove a local Lipschitz stability with partial data.Dynamics of nonlocal and local SIR diffusive epidemic model with free boundarieshttps://zbmath.org/1521.352102023-11-13T18:48:18.785376Z"Li, Chenglin"https://zbmath.org/authors/?q=ai:li.chenglinSummary: This paper is concerned with nonlocal and local diffusive SIR epidemic model with free boundaries including convolution, which is natural extension of reaction diffusion systems with free boundary problems and local diffusions. The existence of unique global solution for this model is considered. Dichotomy of the spreading and vanishing is established. A spreading barrier line is found to determine whether the spreading of disease will fail finally. The spreading of disease will fail when it cannot spread across the spreading barrier line \(l^\ast\), while it will be successful when it transcends over this barrier line. The results show that if the basic reproduction number \(\mathcal{R}_0 < 1\), the spreading of disease will fail eventually, and if \(\mathcal{R}_0 > 1 + \frac{d_2}{\mu_2 + \alpha}\), the spreading of disease will get success finally. We also find that the spreading coefficients play important role in the spreading achievement. When \(1 + \frac{\beta}{\theta\mu_1} < \mathcal{R}_0 < 1 + \frac{d_2}{\mu_2 + \alpha}\), the spreading coefficient decides whether the spreading of disease will be successful. It is shown that the spreading will be successful when the spreading coefficient is relatively big, while the spreading will fail if the spreading coefficient is relatively small.Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficientshttps://zbmath.org/1521.352112023-11-13T18:48:18.785376Z"Nauryz, Targyn"https://zbmath.org/authors/?q=ai:nauryz.targyn-atanbekovich"Briozzo, Adriana C."https://zbmath.org/authors/?q=ai:briozzo.adriana-cSummary: In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.Fourth-order nonlinear degenerate problem for image decompositionhttps://zbmath.org/1521.352122023-11-13T18:48:18.785376Z"Nokrane, Ahmed"https://zbmath.org/authors/?q=ai:nokrane.ahmed"Alaa, Nour Eddine"https://zbmath.org/authors/?q=ai:alaa.noureddine"Aqel, Fatima"https://zbmath.org/authors/?q=ai:aqel.fatima-al-zahraSummary: The aim of this work is to study a new coupled fourth-order reaction-diffusion system, applied to image decomposition into cartoons and textures. The existence and uniqueness of an entropy solution to the system with initial data \(BH\) are established using Galerkin's method. Then, numerical experiments and comparisons with other models have been performed to show the efficiency of the proposed model in image decomposition.Reaction-diffusion on a time-dependent interval: refining the notion of `critical length'https://zbmath.org/1521.352132023-11-13T18:48:18.785376Z"Allwright, Jane"https://zbmath.org/authors/?q=ai:allwright.janeSummary: A reaction-diffusion equation is studied in a time-dependent interval whose length varies with time. The reaction term is either linear or of KPP type. On a fixed interval, it is well known that if the length is less than a certain critical value then the solution tends to zero. When the domain length may vary with time, we prove conditions under which the solution does and does not converge to zero in long time. We show that, even with the length always strictly less than the `critical length', either outcome may occur. Examples are given. The proof is based on upper and lower estimates for the solution, which are derived in this paper for a general time-dependent interval.Sharp interface limit of stochastic Cahn-Hilliard equation with singular noisehttps://zbmath.org/1521.352142023-11-13T18:48:18.785376Z"Baňas, Ľubomír"https://zbmath.org/authors/?q=ai:banas.lubomir"Yang, Huanyu"https://zbmath.org/authors/?q=ai:yang.huanyu"Zhu, Rongchan"https://zbmath.org/authors/?q=ai:zhu.rongchanSummary: We study the sharp interface limit of the two dimensional stochastic Cahn-Hilliard equation driven by two types of singular noise: a space-time white noise and a space-time singular divergence-type noise. We show that with appropriate scaling of the noise the solutions of the stochastic problems converge to the solutions of the determinisitic Mullins-Sekerka/Hele-Shaw problem.Infinite pinninghttps://zbmath.org/1521.352152023-11-13T18:48:18.785376Z"Dondl, Patrick"https://zbmath.org/authors/?q=ai:dondl.patrick-w"Jesenko, Martin"https://zbmath.org/authors/?q=ai:jesenko.martin"Scheutzow, Michael"https://zbmath.org/authors/?q=ai:scheutzow.michaelSummary: In this work, we address the occurrence of infinite pinning in a random medium. We suppose that an initially flat interface starts to move through the medium due to some constant driving force. The medium is assumed to contain random obstacles. We model their positions by a Poisson point process and their strengths are not bounded. We determine a necessary condition on its distribution so that regardless of the driving force the interface gets pinned.Fractional soliton dynamics of electrical microtubule transmission line model with local \(M\)-derivativehttps://zbmath.org/1521.352162023-11-13T18:48:18.785376Z"Raza, Nauman"https://zbmath.org/authors/?q=ai:raza.nauman"Arshed, Saima"https://zbmath.org/authors/?q=ai:arshed.saima"Khan, Kashif Ali"https://zbmath.org/authors/?q=ai:khan.kashif-ali"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafaSummary: In this paper, two integrating strategies namely \(\exp[-\phi(\chi)]\) and \(\frac{G^\prime}{G^2}\)-expansion methods together with the attributes of local-\(M\) derivatives have been acknowledged on the electrical microtubule (MT) model to retrieve soliton solutions. The said model performs a significant role in illustrating the waves propagation in nonlinear systems. MTs are also highly productive in signaling, cell motility, and intracellular transport. The proposed algorithms yielded solutions of bright, dark, singular, and combo fractional soliton type. The significance of the fractional parameters of the fetched results is explained and presented vividly.Special Fourier integral operators of types I and II with function-variable symbols: definition, relation to metaplectic transform, and Heisenberg's uncertainty principleshttps://zbmath.org/1521.352172023-11-13T18:48:18.785376Z"Wang, Ga"https://zbmath.org/authors/?q=ai:wang.ga"Zhang, Zhichao"https://zbmath.org/authors/?q=ai:zhang.zhichaoSummary: This study devotes to Heisenberg's uncertainty principles for Fourier integral operators of types I and II with function-variable symbols, i.e., the symbol \(\sigma \in S_{1 \otimes v_s}^{\infty} (\mathbb{R}^N)\) of the type I Fourier integral operator is only \(\mathbf{w}\)-dependent and the symbol \(\tau \in S_{1 \otimes v_s}^{\infty} (\mathbb{R}^N)\) of the type II Fourier integral operator is only \(\mathbf{y}\)-dependent. These two special Fourier integral operators are abbreviated as the FIO-I-FV and FIO-II-FV, respectively. We disclose an equivalence relation between the FIO-I-FV and the classical metaplectic transform, as well as the FIO-II-FV and the metaplectic transform, based on which we employ various versions of Heisenberg's uncertainty principles for the metaplectic transform, ranging from the general metaplectic transform of real-valued functions to some specific (e.g., the orthogonal, the orthonormal, the minimum eigenvalue commutative, the maximum eigenvalue commutative) metaplectic transforms of complex-valued functions, in the establishment of the corresponding results for the FIO-I-FV and FIO-II-FV.Catching all geodesics of a manifold with moving balls and application to controllability of the wave equationhttps://zbmath.org/1521.370342023-11-13T18:48:18.785376Z"Letrouit, Cyril"https://zbmath.org/authors/?q=ai:letrouit.cyrilSummary: We address the problem of catching all speed-1 geodesics of a Riemannian manifold with a moving ball: given a compact Riemannian manifold \((M, g)\) and small parameters \(\varepsilon>0\) and \(v>0\), is it possible to find \(T > 0\) and an absolutely continuous map \(x:[0,T] \to M, t \mapsto x(t)\) satisfying \(\|\dot{x}\|_\infty \leqslant v\) and such that any geodesic of \((M, g)\) traveled at speed 1 meets the open ball \(B_g (x(t), \varepsilon) \subset M\) within time \(T\)? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation can sometimes be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold \((M, g)\) satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any \(\varepsilon>0\) and \(v>0\), and we give examples of Riemannian manifolds \((M, g)\) for which (GRC) is satisfied. Then, we build an explicit example of a domain \(X \subset \mathbb{R}^2\) (with flat metric) containing convex obstacles, not satisfying (GRC), for which our problem has a negative answer if \(\varepsilon\) and \(v\) are small enough, \textit{i.e.}, no sufficiently small ball moving sufficiently slowly can catch all geodesics of \(X\).Steady Euler flows on \({\mathbb{R}}^3\) with wild and universal dynamicshttps://zbmath.org/1521.370362023-11-13T18:48:18.785376Z"Berger, Pierre"https://zbmath.org/authors/?q=ai:berger.pierre"Florio, Anna"https://zbmath.org/authors/?q=ai:florio.anna"Peralta-Salas, Daniel"https://zbmath.org/authors/?q=ai:peralta-salas.danielSummary: Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set \({\mathscr{G}}\) of stationary solutions to the Euler equations in \({\mathbb{R}}^3\) such that each vector field \(X\in{\mathscr{G}}\) is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincaré map of \(X\) at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension 3 or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.Generalized conditional symmetries and pre-Hamiltonian operatorshttps://zbmath.org/1521.370702023-11-13T18:48:18.785376Z"Wang, Bao"https://zbmath.org/authors/?q=ai:wang.baoSummary: In this paper, we consider the connection between generalized conditional symmetries (GCSs) and pre-Hamiltonian operators. The set of GCSs of an evolutionary partial differential equations system is divided into a union of many linear subspaces by different characteristic operators, and we consider the mappings between two of them, which generalize the recursion operators of symmetries and the pre-Hamiltonian operators. Finally, we give a systematic method to construct infinitely many GCSs for integrable systems, including the Gelfand-Dickey hierarchy and the AKNS-D hierarchy. All time flows in one integrable hierarchy, admitting infinitely many common GCSs.
{\copyright 2023 American Institute of Physics}Cauchy problems related to integrable matrix hierarchieshttps://zbmath.org/1521.370712023-11-13T18:48:18.785376Z"Helminck, G. F."https://zbmath.org/authors/?q=ai:helminck.gerard-frantsisk|helminck.gerardus-franciscusSummary: We discuss the solvability of two Cauchy problems in matrix pseudodifferential operators. The first is associated with a set of matrix pseudodifferential operators of negative order, a prominent example being the set of strict integral operator parts of products of a solution \((L,\{U_\alpha\})\) of the \(\mathbf{h}[\partial]\)-hierarchy, where \(\mathbf{h}\) is a maximal commutative subalgebra of \(gl_n(\mathbb{C})\). We show that it can be solved in the case of compatibility completeness of the adopted setting. The second Cauchy problem is slightly more general and relates to a set of matrix pseudodifferential operators of order zero or less. The key example here is the collection of integral operator parts of the different products of a solution \(\{V_\alpha\}\) of the strict \(\mathbf{h}[\partial]\)-hierarchy. This system is solvable if two properties hold: the Cauchy solvability in dimension \(n\) and the compatibility completeness. Both conditions are shown to hold in the formal power series setting.Four-component integrable hierarchies of Hamiltonian equations with \((m+n+2)\)th-order Lax pairshttps://zbmath.org/1521.370742023-11-13T18:48:18.785376Z"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiuSummary: A class of higher-order matrix spectral problems is formulated and the associated integrable hierarchies are generated via the zero-curvature formulation. The trace identity is used to furnish Hamiltonian structures and thus explore the Liouville integrability of the obtained hierarchies. Illuminating examples are given in terms of coupled nonlinear Schrödinger equations and coupled modified Korteweg-de Vries equations with four components.Solving the modified Camassa-Holm equation via the inverse scattering transformhttps://zbmath.org/1521.370792023-11-13T18:48:18.785376Z"Mao, Hui"https://zbmath.org/authors/?q=ai:mao.hui"Qian, Yu"https://zbmath.org/authors/?q=ai:qian.yu"Miao, Yuanyuan"https://zbmath.org/authors/?q=ai:miao.yuanyuanSummary: With the aid of the reciprocal transformation and the associated equation, we study the inverse scattering transform with a matrix Riemann-Hilbert problem for the modified Camassa-Holm (mCH) equation with nonzero boundary conditions (NZBC) at infinity. In terms of a suitable uniformization variable, the direct and inverse scattering problems are presented for the associated modified Camassa-Holm (amCH) equation. By means of the reciprocal transformation and the reconstruction formula for the potential of the amCH equation, we present the \(N\)-soliton solution for the mCH equation with NZBC. As applications, various solutions including both bright and dark types, smooth soliton solutions, singular soliton solutions, and multi-valued singular soliton solutions of the mCH equation and their interactions are exhibited.Equilibria of vortex type Hamiltonians on closed surfaceshttps://zbmath.org/1521.370802023-11-13T18:48:18.785376Z"Ahmedou, Mohameden"https://zbmath.org/authors/?q=ai:ould-ahmedou.mohameden"Bartsch, Thomas"https://zbmath.org/authors/?q=ai:bartsch.thomas.1|bartsch.thomas.2"Fiernkranz, Tim"https://zbmath.org/authors/?q=ai:fiernkranz.timThis paper is concerned with the proof of existence of critical points of a certain type of functionals on a closed Riemannian surface. These manifolds are assumed to be not homeomorphic to the sphere or the projection plane.
The functionals under consideration (vortex-type Hamiltonians) involve the Green function of the Laplace-Beltrami operator and a certain \(C^1\) function. In particular, under a suitable choice of this function one gets the Kirchhoff-Routh Hamiltonian of fluid dynamics.
The main results are stated in Theorems 1.1 and 1.3.
Reviewer: Igor Leite Freire (São Carlos)Bäcklund transformation, infinite number of conservation laws and fission properties of an integro-differential model for ocean internal solitary waveshttps://zbmath.org/1521.370822023-11-13T18:48:18.785376Z"Yu, Di"https://zbmath.org/authors/?q=ai:yu.di"Zhang, Zong-Guo"https://zbmath.org/authors/?q=ai:zhang.zongguo"Dong, Huan-He"https://zbmath.org/authors/?q=ai:dong.huanhe"Yang, Hong-Wei"https://zbmath.org/authors/?q=ai:yang.hongweiSummary: This paper presents an analytical investigation of the propagation of internal solitary waves in the ocean of finite depth. Using the multi-scale analysis and reduced perturbation methods, the integro-differential equation is derived, which is called the intermediate long wave (ILW) equation and can describe the amplitude of internal solitary waves. It can reduce to the Benjamin-Ono equation in the deep-water limit, and to the KdV equation in the shallow-water limit. Little attention has been paid to the features of integro-differential equations, especially for their conservation laws. Here, based on Hirota bilinear method, Bäcklund transformations in bilinear form of ILW equation are derived and infinite number of conservation laws are given. Finally, we analyze the fission phenomenon of internal solitary waves theoretically and verify it through numerical simulation. All of these have potential value for the further research on ocean internal solitary waves.Limit speed of traveling wave solutions for the perturbed generalized KdV equationhttps://zbmath.org/1521.370832023-11-13T18:48:18.785376Z"Chen, Aiyong"https://zbmath.org/authors/?q=ai:chen.aiyong"Zhang, Chi"https://zbmath.org/authors/?q=ai:zhang.chi.4"Huang, Wentao"https://zbmath.org/authors/?q=ai:huang.wentao.1|huang.wentaoSummary: The existence of solitary waves and periodic waves for a perturbed generalized KdV equation is established by using geometric singular perturbation theory. It is proven that the limit wave speed \(c_0(h)\) is decreasing by analyzing the ratio of abelian integrals for \(n=2\) and \(n=3\). The upper and lower bounds of the limit wave speed are given. Moreover, the relation between the wave speed and the wavelength of traveling waves is obtained. Our results answer partially an open question proposed by
\textit{W. Yan} et al. [Math. Model. Anal. 19, No. 4, 537--555 (2014; Zbl 1488.34329)].Quasi-periodic solutions for quintic completely resonant derivative beam equations on \(\mathbb{T}^2\)https://zbmath.org/1521.370842023-11-13T18:48:18.785376Z"Ge, Chuanfang"https://zbmath.org/authors/?q=ai:ge.chuanfang"Geng, Jiansheng"https://zbmath.org/authors/?q=ai:geng.jianshengSummary: In the present paper, we consider two dimensional completely resonant, derivative, quintic nonlinear beam equations with reversible structure. Because of this reversible system without external parameters or potentials, Birkhoff normal form reduction is necessary before applying Kolmogorov-Arnold-Moser (KAM) theorem. As application of KAM theorem, the existence of partially hyperbolic, small amplitude, quasi-periodic solutions of the reversible system is proved in this paper.
{\copyright 2023 American Institute of Physics}On the analytic Birkhoff normal form of the Benjamin-Ono equation and applicationshttps://zbmath.org/1521.370852023-11-13T18:48:18.785376Z"Gérard, Patrick"https://zbmath.org/authors/?q=ai:gerard.patrick|gerard.patrick-d"Kappeler, Thomas"https://zbmath.org/authors/?q=ai:kappeler.thomas"Topalov, Petar"https://zbmath.org/authors/?q=ai:topalov.peter-jSummary: In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in \(H_0^s ( \mathbb{T} , \mathbb{R} )\) for any \(s > - 1 / 2\) where \(H_0^s ( \mathbb{T} , \mathbb{R} )\) denotes the subspace of the Sobolev space \(H^s ( \mathbb{T} , \mathbb{R} )\) of elements with mean 0. As an application we show that for any \(- 1 / 2 < s < 0\), the flow map of the Benjamin-Ono equation \(\mathcal{S}_0^t : H_0^s ( \mathbb{T} , \mathbb{R} ) \to H_0^s ( \mathbb{T} , \mathbb{R} )\) is \textit{nowhere locally uniformly continuous} in a neighborhood of zero in \(H_0^s ( \mathbb{T} , \mathbb{R} )\).Long time stability of plane wave solutions to Schrödinger equation on torushttps://zbmath.org/1521.370862023-11-13T18:48:18.785376Z"Mi, Lufang"https://zbmath.org/authors/?q=ai:mi.lufang"Sun, Yingte"https://zbmath.org/authors/?q=ai:sun.yingte"Wang, Peizhen"https://zbmath.org/authors/?q=ai:wang.peizhenSummary: We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrödinger equation (NLS) in the defocusing \((\lambda=1)\) or focusing \((\lambda=-1)\) case,
\[
\mathrm{i}u_t= -\Delta u+ \lambda |u|^2 u, \quad x \in \mathbb{T}^d, \quad t \in \mathbb{R}.
\] More precisely, in a Gevrey space
\[
G_\sigma := \left\{ u: \|u\|_\sigma^2 = \sum_{a \in \mathbb{Z}^d} e^{2\sigma \sqrt{|a|}} |u_a|^2 < \infty \right\}
\] for some positive constant \(\sigma \), we show that solution with the initial datum in the \(4 \epsilon\)-neighborhood of the plane wave solution still stays in the \(C \epsilon\)-neighborhood \((C>4)\) of the plane wave solution for a subexponential long time \(|t| \leq \epsilon^{-\zeta |\ln \epsilon|^\varrho}\), where \(\zeta = \min \{\frac{1}{4}, \sigma-\sigma^\prime\}\), \(\sigma > \sigma^\prime > 0\) and \(0 < \varrho < 1/6\).Well-posedness of fractional stochastic complex Ginzburg-Landau equations driven by regular additive noisehttps://zbmath.org/1521.370922023-11-13T18:48:18.785376Z"Liu, Aili"https://zbmath.org/authors/?q=ai:liu.aili"Zou, Yanyan"https://zbmath.org/authors/?q=ai:zou.yanyan"Ren, Die"https://zbmath.org/authors/?q=ai:ren.die"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.jiSummary: This paper deals with the well-posedness of the solutions of the fractional complex Ginzburg-Landau equation driven by locally Lipschitz nonlinear diffusion terms defined on \(R^n\). We first give the pathwise uniform estimates and uniform estimates on average. Then we prove the existence, uniqueness and measurability of solutions for the equation.A physics-constrained deep residual network for solving the sine-Gordon equationhttps://zbmath.org/1521.370992023-11-13T18:48:18.785376Z"Li, Jun"https://zbmath.org/authors/?q=ai:li.jun.8|li.jun.18|li.jun.15|li.jun.23|li.jun.7|li.jun.39|li.jun.1|li.jun.16|li.jun.6|li.jun|li.jun.9|li.jun.19|li.jun.3|li.jun.21|li.jun.12|li.jun.24|li.jun.11|li.jun.36"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.2|chen.yong|chen.yong.1|chen.yong.6|chen.yong.4|chen.yong.5|chen.yong.9|chen.yong.8|chen.yong.10|chen.yong.3Summary: Despite some empirical successes for solving nonlinear evolution equations using deep learning, there are several unresolved issues. First, it could not uncover the dynamical behaviors of some equations where highly nonlinear source terms are included very well. Second, the gradient exploding and vanishing problems often occur for the traditional feedforward neural networks. In this paper, we propose a new architecture that combines the deep residual neural network with some underlying physical laws. Using the sine-Gordon equation as an example, we show that the numerical result is in good agreement with the exact soliton solution. In addition, a lot of numerical experiments show that the model is robust under small perturbations to a certain extent.Geometric harmonic analysis II. Function spaces measuring size and smoothness on rough setshttps://zbmath.org/1521.420032023-11-13T18:48:18.785376Z"Mitrea, Dorina"https://zbmath.org/authors/?q=ai:mitrea.dorina"Mitrea, Irina"https://zbmath.org/authors/?q=ai:mitrea.irina"Mitrea, Marius"https://zbmath.org/authors/?q=ai:mitrea.mariusThe present book is the second in a series of five volumes, at the confluence of Harmonic Analysis, Geometric Measure Theory, Function Space Theory, and Partial Differential Equations. The series is generically branded as Geometric Harmonic Analysis, with the individual volumes carrying the following subtitles: Volume I: A Sharp Divergence Theorem with Nontangential Pointwise Traces; Volume II: Function Spaces Measuring Size and Smoothness on Rough Sets; Volume III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering; Volume IV: Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis; Volume V: Fredholm Theory and Finer Estimates for Integral Operators, with Applications to Boundary Problems.
As a whole, the series develops tools capable of dealing with boundary value problems for elliptic systems in general geometric settings, going beyond the class of Lipschitz domains. In this second volume, the focus is on the functional analytic aspects of this extensive program. The authors assemble an expansive library of function spaces, measuring size and smoothness, in very inclusive geometric environments. This is accomplished both by developing a large body of novel results and by upgrading the existing theory to ensure its compatibility with the Calderón-Zygmund theory for singular integral operators of boundary layer type and, ultimately, the kind of boundary value problems considered in subsequent volumes.
Chapter 1 sets the stage by elaborating on basic terminology and results pertaining to vector spaces and operator theory, quasi-normed spaces, real interpolation of quasilinear operators, complex interpolation of quasi-Banach spaces, and some useful categories of topological vector spaces.
The main topic in Chapter 2 is Fredholm's theory. The starting point is the classical setting on Banach spaces. Since a variety of function spaces naturally employed in the formulation of boundary value problems, such as certain Hardy, Besov, and Triebel-Lizorkin spaces, fail to be Banach (as their ``norms'' only satisfy a quasi-triangle inequality), the authors provide a refinement of the ``standard'' Fredholm theory on Banach spaces which is applicable to more general topological vector spaces that are not necessarily locally convex.
Chapter 3 is devoted to the study of functions whose oscillations, measured in a mean integral sense, vanish at infinitesimal scales. Here the authors succeed in producing characterizations of the space VMO as the closure in BMO of classes of smooth functions contained in BMO within which uniform continuity may be suitably quantified, such as the class of smooth functions satisfying a Hölder condition. The proof of this result is of purely real variable nature, which works in the general context of spaces of homogeneous type. The authors make use of this technology to study a related scale, namely a new brand of Hölder spaces, obtained by imposing the condition that the local Hölder semi-norm vanishes as the scale at which this is considered approaches zero.
The pioneering work of R. Coifman and G. Weiss, which has fundamentally reconfigured the classical theory of Hardy spaces, has brought to prominence a brand of Hardy spaces placing minimal regularity and structural demands on the underlying geometric ambient. The present Chapter 4 is concerned with Hardy spaces on Ahlfors regular subsets of the Euclidean ambient, considering topics such as the Fefferman-Stein grand maximal function, Lebesgue-based and Lorentz-based Hardy spaces, real interpolation, atoms and molecules, duality, weak-star convergence, the compatibility of various duality pairings, and a novel filtering operator extracting functions out of Hardy distributions.
Chapter 5 debuts by providing an inclusive notion of Banach function space, which the authors call Generalized Banach Function Space. Its relevance is apparent from the fact that a variety of scales of spaces of interest, such as Morrey spaces, block spaces, as well as Beurling algebras and their pre-duals, now fit naturally into this more accommodating label. One of the main results in this chapter is a versatile extrapolation result that allows passing from estimates on Muckenhoupt weighted Lebesgue spaces to estimates on this brand of Generalized Banach Function Spaces. This chapter also contains a treatment of Orlicz spaces.
In Chapter 6 the authors develop a theory for the scales of Morrey and Morrey-Campanato spaces, which is comparable in scope and power to its Euclidean counterpart, in more general geometric settings. Subsequently, Chapters 7--9 are concerned with adaptations of the scales of Besov, Triebel-Lizorkin, and weighted Sobolev spaces to geometric ambients, which only enjoy but a small fraction of the structural richness of the Euclidean space. There is a wealth of topics of interest covered here, including atomic and molecular theory, real and complex interpolation, duality, embeddings, envelopes, boundary traces, and extension operators.
In Chapter 10 the authors study both the strong and weak versions of the normal traces of vector fields defined in a given domain, seeking conditions that guarantee membership to Hardy spaces on the boundary of said domain. The technology developed here makes essential use of the bullet product (introduced in Volume I: A Sharp Divergence Theorem with Nontangential Pointwise Traces) and is at the core of the Fatou-type results established in subsequent volumes.
While there is ample literature on Sobolev spaces on open sets, the goal of Chapter 11 is to introduce a scale of Sobolev spaces on the geometric measure theoretic boundaries of sets of locally finite perimeter in the Euclidean setting and on Riemannian manifolds. This brand of ``boundary'' Sobolev spaces are analytic and geometric, in the sense that they are defined using ``weak derivatives'' and integration by parts along the boundary which, in turn, are manufactured using the scalar components of the geometric measure theoretic outward unit typical to the set of locally finite perimeter in question. In contrast to other types of Sobolev spaces on generic measure metric spaces introduced and studied elsewhere in the literature, this brand of Sobolev spaces allows for integration by parts on the boundary. Such a feature is critical in light of the applications to singular integral operators and boundary value problems.
Reviewer: Mohammed El Aïdi (Bogotá)\(L^{\infty}\)-BMO boundedness of some pseudo-differential operatorshttps://zbmath.org/1521.420162023-11-13T18:48:18.785376Z"Ruan, Jianmiao"https://zbmath.org/authors/?q=ai:ruan.jianmiao"Zhu, Xiangrong"https://zbmath.org/authors/?q=ai:zhu.xiangrongThe authors introduce a new Hörmander class of symbols \(\mathcal{S}^{m}_{\rho}(\ln^{6} L)\), which satisfies \(\mathcal{S}^{m}_{\rho, \delta} \subsetneqq \mathcal{S}^{m}_{\rho}(\ln^{6} L) \subsetneqq \mathcal{S}^{m}_{\rho, 1}\) for \(0 \leq \delta < 1\), where \(\mathcal{S}^{m}_{\rho, \delta}\) is the classical Hörmander class. A symbol \(a( \cdot )\) belongs to \(\mathcal{S}^{m}_{\rho}(\ln^{6} L)\) (\(m \in \mathbb{R}\),\ \(0 \leq \rho \leq 1\)) if
\[
\sup_{x, \xi \in \mathbb{R}^{n}} (1+|\xi|)^{-m+\rho N-M} \ln^{6M}(2+|\xi|) \left|\nabla_{\xi}^{N} \nabla_{x}^{M} a(x, \xi)\right| < \infty,
\]
for every \(N, M \in \mathbb{N} = \{ 0, 1, 2,\dots\}\).
The main result in this article is the obtaining of the \((L^{\infty}, \mathrm{BMO})\) estimate for pseudo-differential operators with symbols belonging to the class \(\mathcal{S}^{\frac{n(\rho -1)}{2}}_{\rho}(\ln^{6} L)\), \(0 \leq \rho < 1\).
This result is an improvement of Theorem 3.2-(c) in [\textit{J. Alvarez} and \textit{J. Hounie}, Ark. Mat. 28, No. 1, 1--22 (1990; Zbl 0713.35106)].
Reviewer: Pablo Alejandro Rocha (Buenos Aires)Correction to: ``Hardy spaces associated with generalized degenerate Schrödinger operators with applications to Carleson measure''https://zbmath.org/1521.420202023-11-13T18:48:18.785376Z"Liu, Xiong"https://zbmath.org/authors/?q=ai:liu.xiong|liu.xiong.1"He, Jianxun"https://zbmath.org/authors/?q=ai:he.jianxun"Li, Jinxia"https://zbmath.org/authors/?q=ai:li.jinxiaCorrects the first equation first equation on page 28 in the authors' paper [ibid. 46, No. 4, Paper No. 132, 33 p. (2023; Zbl 1518.42034)].A pair of Barut-Girardello type transforms and allied pseudo-differential operatorshttps://zbmath.org/1521.440052023-11-13T18:48:18.785376Z"Mandal, U. K."https://zbmath.org/authors/?q=ai:mandal.upain-kumar"Prasad, Akhilesh"https://zbmath.org/authors/?q=ai:prasad.akhileshThe Barut-Girardello transform was introduced in 1971 by \textit{A. O. Barut} and \textit{L. Girardello} [Comm. Math. Phys. 21, 41--55 (1971; Zbl 0214.38203)] with the modified Bessel function of first kind \(I_\nu\) as its kernel. Later \textit{A. Torre} [Int. Trans. Spec. Funct. 19, No. 4, 277--292 (2008; Zbl 1145.44002)] gave two new versions of the same and studied some of their properties. The authors in the present paper advance these works and introduce the modified versions of the conventional BG-transform and the first and second BG-transforms and investigate a number of their interesting properties. They define the modified BG-transform of an integrable function \(f\) as
\[
({\mathcal{G}{_{\nu ,n}}f})(y)= {2^{\frac{1}{n}}}\int_0^\infty {\sqrt {xy} {I_\nu }({{2^{\frac{1}{n}}}xy}){e^{-\frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx, \tag{1}
\]
where \(\nu \ge \frac{-1}{2}\) and \(n \in \mathbb{N}\). The modified first and second BG-transforms of an integrable function \(f\) are respectively defined as
\[
({\mathcal{G}{_{1,\nu ,\mu ,n}}f})(y)= {2^{\frac{1}{n}}}{y^{2\mu + 1}}\int_0^\infty {{{({xy})}^{ - \mu }}{I_\nu }({{2^{\frac{1}{n}}}xy}){e^{ - \frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx, \tag{2}
\]
\[
({\mathcal{G}{_{2,\nu ,\mu ,n}}f})(y)= {2^{\frac{1}{n}}}\int_0^\infty {{x^{2\mu + 1}}{{({xy})}^{ - \mu }}{I_\nu }({{2^{\frac{1}{n}}}xy}){e^{ - \frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx, \tag{3}
\]
where \(\nu\) and \(n\) are as in (1) and \(\mu\) is any real number. The inversion formulas for (2) and (3) are respectively given by
\[
({\mathcal{G}_{1,\nu ,\mu ,n}^{ - 1}f})(y)= {({ - 1})^{\nu + 1}}{2^{\frac{1}{n}}}{y^{2\mu + 1}}\int_0^\infty {{{({xy})}^{ - \mu }}{I_\nu }({{2^{\frac{1}{n}}}xy}){e^{\frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx \tag{4}
\]
and
\[
({\mathcal{G}_{2,\nu ,\mu ,n}^{ - 1}f})(y)= {({ - 1})^{\nu + 1}}{2^{\frac{1}{n}}}\int_0^\infty {{x^{2\mu + 1}}{{({xy})}^{ - \mu }}{I_\nu }({{2^{\frac{1}{n}}}xy}){e^{\frac{1}{n}({{x^n} + {y^n}})}}} f(x)\,dx. \tag{5}
\]
The relations between (2) and (3) are given by
\[
({\mathcal{G}{_{1,\nu ,\mu ,n}}f})(y)= {y^{2\mu + 1}}({\mathcal{G}{_{2,\nu ,\mu ,n}}{x^{ - ({2\mu + 1})}}f})(y)
\]
and
\[
({\mathcal{G}{_{2,\nu ,\mu ,n}}f})(y)= {y^{ - ({2\mu + 1})}}({\mathcal{G}{_{1,\nu ,\mu ,n}}{x^{2\mu + 1}}f})(y),
\]
while those between (4) and (5) are given by
\[
({\mathcal{G}_{1,\nu ,\mu ,n}^{ - 1}f})(y)= ({{{\left\{ {\mathcal{G}_{2,\nu ,\mu ,n}^{ - 1}} \right\}}^\prime }f})(y)= {y^{2\mu + 1}}({\mathcal{G}_{2,\nu ,\mu ,n}^{ - 1}{x^{ - ({2\mu + 1})}}f})(y)
\]
and
\[
({\mathcal{G}_{2,\nu ,\mu ,n}^{ - 1}f})(y)= ({{{\left\{ {\mathcal{G}_{1,\nu ,\mu ,n}^{ - 1}} \right\}}^\prime }f})(y)= {y^{ - ({2\mu + 1})}}({\mathcal{G}_{1,\nu ,\mu ,n}^{ - 1}{x^{2\mu + 1}}f})(y).
\]
The authors also give Parseval's and mixed type of Parseval's relations for these two types of BG-transforms. By writing the differential operator \(M_{1,\nu,\mu,n}\) associated to the BG-transforms as
\[
{M_{1,\nu ,\mu ,n}} = \frac{{{d^2}}}{{d{x^2}}} + \left({\frac{{1 + 2\mu }}{x} + 2{x^{n - 1}}}\right)\frac{d}{{dx}} + \left({\frac{{{\mu ^2} - {\nu ^2}}}{{{x^2}}} + {x^{2(n - 1)}} + ({2\mu + n}){x^{n - 2}}}\right)
\]
and its adjoint \(M_{2,\nu,\mu,n}\) as
\[
{M_{2,\nu ,\mu ,n}} = \frac{{{d^2}}}{{d{x^2}}} - \left({\frac{{1 + 2\mu }}{x} + 2{x^{n - 1}}}\right)\frac{d}{{dx}} + \left({\frac{{{{({\mu + 1})}^2} - {\nu ^2}}}{{{x^2}}} + {x^{2n - 2}} + ({2\mu + 2 - n}){x^{n - 2}}}\right)
\]
the authors deduce the relations
\[
({\mathcal{G}{_{1,\nu ,\mu ,n}}{M_{2,\nu ,\mu ,n}}f})(y)= {2^{\frac{2}{n}}}{y^2}({\mathcal{G}{_{1,\nu ,\mu }}f})(y)\quad \text{and}\quad (({\mathcal{G}{_{2,\nu ,\mu ,n}}{M_{1,\nu ,\mu ,n}}f})(y)= {2^{\frac{2}{n}}}{y^2}({\mathcal{G}{_{2,\nu ,\mu }}f})(y).
\]
In Section 2 of the paper the authors introduce some further differential operators \(A_{1,\nu,\mu,n}\), $B_{1,\nu,\mu,n}$, $A_{2,\nu,\mu,n}$, \(B_{2,\nu,\mu,n}\) which are closely related to the operators \(M_{1,\nu,\mu,n}\) and \(M_{2,\nu,\mu,n}\) in the sense \(A_{1,\nu,\mu,n}B_{1,\nu,\mu,n}=M_{1,\nu,\mu,n}\) and \(B_{2,\nu,\mu,n}A_{2,\nu,\mu,n}=M_{2,\nu,\mu,n}\) and generalize these two latter operators up to the \(r^{\mathrm{th}}\) order in Lemma~2.1. In Section~3 the authors introduce Zemanian type function spaces like those defined in the works of \textit{M.~Linares Linares} and \textit{J.~M.~R. Mendez Pérez} [Bull. Calcutta Math. Soc. 83, 447--546 (1991; Zbl 0759.46039)] and \textit{A.~Prasad} and \textit{K.~Mahato} [Rend. Circ. Mat. Palermo 65, No.~2, 209--241 (2016; Zbl 1366.46025)] and discuss the continuity of differential operators and the BG-type transforms on these spaces. The translation and convolution operators associated with the modified BG-type transforms are studied in Section 4 and a number of stimulating results are proven about them. Two pseudo-differential operators involving the BG-type transforms are defined in Section~5 and their continuity theorems are proved. Lastly the authors discuss the applicability of the modified BG-transform to solve a differential equation and a pseudo-differential equation.
Reviewer: Lalit Mohan Upadhyaya (Dehradun)On some extensions of the A-modelhttps://zbmath.org/1521.470282023-11-13T18:48:18.785376Z"Juršėnas, Rytis"https://zbmath.org/authors/?q=ai:jursenas.rytisGiven a lower semibounded selfadjoint operator \(L\) in a Hilbert space \({\mathfrak H}_0\), let \({\mathfrak H}_{n+1}\subseteq{\mathfrak H}_n\) (\(n\in\mathbb{Z}\)) be the scale of Hilbert spaces associated with \(L\), where \(L\) has domain \(\mathrm{dom}L={\mathfrak H}_2\), and let \(\{\varphi_\sigma\}\) be a family of linearly independent functionals \(\varphi_\sigma \in{\mathfrak H}_{-m-2}\setminus{\mathfrak H}_{-m-1}\) (\(m\in\mathbb{N}\)), where \(\sigma\) ranges over the index set \({\mathcal S}\) of dimension \(d\in\mathbb{N}\). Then the symmetric restriction \(L_{\min}\subseteq L\) to the domain of \(f\in{\mathfrak H}_{m+2}\), such that \(\langle\varphi_\sigma,f\rangle=0\) for all \(\sigma\), is an essentially selfadjoint operator on \({\mathfrak H}_0\).
The paper deals with describing nontrivial extensions of \(L_{\min}\) (perturbations of \(L\)) in \({\mathfrak H}_0\). The cascade A-model for rank-\(d\) higher order singular perturbations is studied on the basis of ordinary boundary triples. Assuming the existence of orthogonal decompositions \({\mathfrak H}_n^+\oplus{\mathfrak H}_n^-\) of the Hilbert spaces \({\mathfrak H}_n\) (\(n\in\mathbb{N}\)) with orthogonal projections \(P_n^\pm:{\mathfrak H}_n\to{\mathfrak H}_n^\pm\) such that \(P_{n+1}^\pm\subseteq P_n^\pm\), nontrivial extensions in the \(A\)-model are constructed for the symmetric restrictions of \(L_{\min}\) in the subspaces. Nontrivial realizations of \(L_{\min}\) in the spaces \({\mathfrak H}_m^\pm\) are obtained with the use of the Krein-Naimark resolvent formula, the Weyl functions, the Krein \(Q\)-functions, and the generalized Nevanlinna functions.
Reviewer: Yuri I. Karlovich (Cuernavaca)Nonradiality of second fractional eigenfunctions of thin annulihttps://zbmath.org/1521.470352023-11-13T18:48:18.785376Z"Djitte, Sidy M."https://zbmath.org/authors/?q=ai:djitte.sidy-moctar"Jarohs, Sven"https://zbmath.org/authors/?q=ai:jarohs.svenSummary: In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annuli-like domains and the corresponding eigenfunctions. In the first part, we consider an annulus with inner radius \(R\) and outer radius \(R+1\). We show that for \(R\) sufficiently large any corresponding second eigenfunction of this annulus is nonradial. In the second part, we investigate the second eigenvalue in domains of the form \(B_1 (0)\setminus \overline{B_{\tau}(a)}\), where \(a\) is in the unitary ball and \(0<\tau <1-|a|\). We show that this value is maximized for \(a = 0\), if the set \(B_1 (0)\setminus \overline{B_{\tau}(0)}\) has no radial second eigenfunction. We emphasize that the first part of our paper implies that this assumption is indeed nonempty.Random Hamiltonians with arbitrary point interactions in one dimensionhttps://zbmath.org/1521.470712023-11-13T18:48:18.785376Z"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Fillman, Jake"https://zbmath.org/authors/?q=ai:fillman.jake"Helman, Mark"https://zbmath.org/authors/?q=ai:helman.mark"Kesten, Jacob"https://zbmath.org/authors/?q=ai:kesten.jacob"Sukhtaiev, Selim"https://zbmath.org/authors/?q=ai:sukhtaiev.selimSummary: We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. Under minimal assumptions on the type of disorder, we prove the following dichotomy: Either every realization of the random operator has purely absolutely continuous spectrum or spectral and exponential dynamical localization hold. In particular, we establish Anderson localization for Schrödinger operators with Bernoulli-type random singular potential and singular density.Self-similarity in homogeneous stationary and evolution problemshttps://zbmath.org/1521.470732023-11-13T18:48:18.785376Z"Cholewa, Jan W."https://zbmath.org/authors/?q=ai:cholewa.jan-w"Rodriguez-Bernal, Anibal"https://zbmath.org/authors/?q=ai:rodriguez-bernal.anibalThe authors of this paper investigate the self-similarity properties related to linear elliptic and evolutionary problems involving homogeneous operators in several spaces including measures. They employ these techniques to analyse in particular \(2m\)th-order diffusion equations and the associated fractional problems. Homogenous operators are those that interact in a special form with the dilations of functions in \(\mathbb{R}^N\), defined by
\[
\phi_R(x)=\phi(Rx),\quad x\in\mathbb{R}^N,\ R>0.
\]
An operator \(L\) with domain \(\mathrm{D}(L)\) in a Banach space \(X\) of functions or distributions which is invariant by rescaling is homogeneous of degree \(\sigma \in \mathbb{R}\) if
\[
L(\phi_R)=R^\sigma(L\phi)_R,\quad \phi\in\mathrm{D}(L),\ R>0.
\]
In fact, the operator semigroup \((S(t))_{t\geq0}\) serving as the solution of the equation \(u_t+Lu=0\), \(t\geq0\), is homogenous of degree \(\sigma\), i.e., for all \(t > 0\), \(R > 0\) and \(\phi \in X\), one has
\[
S(t)\phi_R=(S(R^\sigma t)\phi)_R.
\]
The authors analyse these types of semigroups and resolvent operators. Among others, the authors investigate \(2m\)-order parabolic equations as well as Ornstein-Uhlenbeck-type semigroups.
Reviewer: Christian Budde (Bloemfontein)Compensated compactness: continuity in optimal weak topologieshttps://zbmath.org/1521.470762023-11-13T18:48:18.785376Z"Guerra, André"https://zbmath.org/authors/?q=ai:guerra.andre"Raiţă, Bogdan"https://zbmath.org/authors/?q=ai:raita.bogdan"Schrecker, Matthew R. I."https://zbmath.org/authors/?q=ai:schrecker.matthew-r-iAuthors' abstract: For \(l\)-homogeneous linear differential operators \(\mathcal{A}\) of constant rank, we study the implication
\begin{align*}
\left.
\begin{matrix}
v_j \rightharpoonup v \text{ in } X \\
\mathcal{A} v_j \to \mathcal{A} v \text{ in } W^{- l} Y
\end{matrix}
\right\} \Rightarrow F(v_j) \rightsquigarrow F(v) \text{ in } Z,
\end{align*}
where \(F\) is an \(\mathcal{A}\)-quasiaffine function and \(\rightsquigarrow\) denotes an appropriate type of weak convergence. Here, \(Z\) is a local \(L^1\)-type space, either the space \(\mathscr{M}\) of measures, or \(L^1\), or the Hardy space \(\mathscr{H}^1\); \(X, Y\) are \(L^p\)-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of \(X, Y, Z\) are sharp. Analogous statements are also given in the case when \(F(v)\) is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove \(\mathscr{H}^p\)-bounds for the sequence \((F (v_j))_j\), for appropriate \(p < 1\), and new convergence results in the dual of Hölder spaces when \((v_j)\) is \(\mathcal{A}\)-free and lies in a suitable negative order Sobolev space \(W^{- \beta, s}\). The choice of these Hölder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.
Reviewer: Mohammed El Aïdi (Bogotá)Domains of elliptic operators on sets in Wiener spacehttps://zbmath.org/1521.470772023-11-13T18:48:18.785376Z"Addona, Davide"https://zbmath.org/authors/?q=ai:addona.davide"Cappa, Gianluca"https://zbmath.org/authors/?q=ai:cappa.gianluca"Ferrari, Simone"https://zbmath.org/authors/?q=ai:ferrari.simoneSummary: Let \(X\) be a separable Banach space endowed with a non-degenerate centered Gaussian measure \(\mu\). The associated Cameron-Martin space is denoted by \(H\). Consider two sufficiently regular convex functions \(U:X\to\mathbb{R}\) and \(G:X\to\mathbb{R} \). We let \(\nu=e^{- U}\mu\) and \(\Omega=G^{-1}(-\infty,0]\). In this paper, we study the domain of the self-adjoint operator associated with the quadratic form
\[
(\psi,\varphi)\mapsto\int_\Omega\left\langle\nabla_H\psi,\nabla_H\varphi\right\rangle_H\,d\nu, \quad\psi,\varphi\in W^{1,2}(\Omega,\nu),\tag{(0.1)}
\]
and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein-Uhlenbeck operator in Hilbert space with \(\Omega=X\) and on half-spaces, namely, if \(U\equiv 0\) and \(G\) is an affine function, then the domain of the operator defined via (0.1) is the space
\[
\left\{u\in W^{2,2}(\Omega,\mu)|\left\langle\nabla_Hu(x),\nabla_HG(x)\right\rangle_H=0 \text{ for }\rho\text{-a.e. }x \in G^{- 1}(0)\right\},
\]
where \(\rho\) is the Feyel-de La Pradelle Hausdorff-Gauss surface measure.Applications of the Bielecki renorming techniquehttps://zbmath.org/1521.470912023-11-13T18:48:18.785376Z"Bessenyei, Mihály"https://zbmath.org/authors/?q=ai:bessenyei.mihaly"Páles, Zsolt"https://zbmath.org/authors/?q=ai:pales.zsoltIn this interesting and well-written paper, the authors discuss variants of the renorming technique which makes a non-contractive operator contractive after a suitable change of the metric and seems to go back to \textit{A. Bielecki} [Bull. Acad. Pol. Sci., Cl. III 4, 261--264 (1956; Zbl 0070.08103)]. In the second half, the authors illustrate the abstract results by means of Uryson-Fredholm integral equations, Uryson-Volterra integral equations, and Presić-type functional equations involving restrictable operators (also called operators ``with memory'').
Reviewer: Jürgen Appell (Würzburg)Stepanov ergodic perturbations for nonautonomous evolution equations in Banach spaceshttps://zbmath.org/1521.471142023-11-13T18:48:18.785376Z"Dianda, Abdoul Aziz Kalifa"https://zbmath.org/authors/?q=ai:dianda.abdoul-aziz-kalifa"Ezzinbi, Khalil"https://zbmath.org/authors/?q=ai:ezzinbi.khalil"Khalil, Kamal"https://zbmath.org/authors/?q=ai:khalil.kamalSummary: We prove the existence and uniqueness of \(\mu\)-pseudo almost automorphic solutions for a class of semilinear nonautonomous evolution equations of the form: \(u'(t)=A(t)u(t)+f(t,u(t))\), \( t\in \mathbb{R}\) where \((A(t))_{t\in \mathbb{R}}\) is a family of closed densely defined linear operators acting on a Banach space \(X\), generating a strongly continuous evolution family that have an exponential dichotomy on \(\mathbb{R}\). The nonlinear term \(f : \mathbb{R} \times X \longrightarrow X\) is assumed to be only \(\mu\)-pseudo almost automorphic in Stepanov's sense in \(t\) and Lipschitz continuous with respect to the second variable. To illustrate our theoretical results, we provide an application to a reaction-diffusion equation on \(\mathbb{R}\) with time-dependent parameters.A Dixmier trace formula for the density of stateshttps://zbmath.org/1521.471172023-11-13T18:48:18.785376Z"Azamov, N."https://zbmath.org/authors/?q=ai:azamov.nurulla-a"McDonald, E."https://zbmath.org/authors/?q=ai:mcdonald.edward-a"Sukochev, F."https://zbmath.org/authors/?q=ai:sukochev.fedor-a"Zanin, D."https://zbmath.org/authors/?q=ai:zanin.dmitriy-vSummary: A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrödinger operator with bounded potential. In solid state physics, there is another celebrated measure associated with such operators -- the density of states. In this paper, we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.Hamilton-Jacobi equation for state constrained Bolza problems with discontinuous time dependence: a characterization of the value functionhttps://zbmath.org/1521.490022023-11-13T18:48:18.785376Z"Bernis, Julien"https://zbmath.org/authors/?q=ai:bernis.julien"Bettiol, Piernicola"https://zbmath.org/authors/?q=ai:bettiol.piernicolaSummary: We consider a class of state constrained Bolza problems in which the integral cost is merely continuous w.r.t. the state variable, and the dynamics and the integral cost are allowed to have a discontinuous behaviour w.r.t. the time variable \(t\) in the following sense: although they have everywhere one-sided limits in t, they are required to be continuous only for a.e. t. For this class of problems we establish conditions under which the Value Function is characterized as the unique viscosity solution in the class of lower semicontinuous functions to the associated Hamilton-Jacobi equation. We provide some illustrative examples including a ``growth versus consumption'' problem in neo-classical macro-economics, one peculiarity of which is the presence of a fractional singularity w.r.t. the state variable.Nonuniqueness of minimizers for semilinear optimal control problemshttps://zbmath.org/1521.490042023-11-13T18:48:18.785376Z"Pighin, Dario"https://zbmath.org/authors/?q=ai:pighin.darioSummary: A counterexample to uniqueness of global minimizers of semilinear optimal control problems is given. The lack of uniqueness occurs for a special choice of the state-target in the cost functional. Our arguments also show that, for some state-targets, there exist local minimizers which are not global. When this occurs, gradient-type algorithms may be trapped by local minimizers, thus missing global ones. Furthermore, the issue of convexity of a quadratic functional in optimal control is analyzed in an abstract setting.
As a corollary of nonuniqueness of minimizers, a nonuniqueness result for a coupled elliptic system is deduced.
Numerical simulations have been performed illustrating the theoretical results.
We also discuss the possible impact of the multiplicity of minimizers on the turnpike property in long time horizons.\(\Gamma\)-convergence and stochastic homogenization of degenerate integral functionals in weighted Sobolev spaceshttps://zbmath.org/1521.490102023-11-13T18:48:18.785376Z"D'Onofrio, Chiara"https://zbmath.org/authors/?q=ai:donofrio.chiara"Zeppieri, Caterina Ida"https://zbmath.org/authors/?q=ai:zeppieri.caterina-idaThe authors prove a \(\Gamma\)-convergence result for nonconvex bulk integral fucntionals in the vectorial case, whose densities satisfy degenerate growth conditions from above and below. After an exhaustive presentation of the state of the art, they introduce a new assumption in the form of uniform integrability conditions of the functions modeling the growth, i.e. they require that the integrands belong to parameters dependent weighted Sobolev space, whose parameters are in the same Muckenhoupt space \(A_p(K)\).
In this setting the weights converge (up to a subsequence) to a limit Muckenhoupt weight and the corresponding integrals \(\Gamma\)-converge (in the \(L^1\)-topology), again up to a subsequence, to a degenerate integral defined on the corresponding limit weighted Sobolev space.
The results are also extended to cover the random stationary setting, thus providing a stochastic homogenization result in the corresponding case, which, in turn, generalizes to the degenerate setting, the results contained in [\textit{G. Dal Maso} and \textit{L. Modica}, Ann. Mat. Pura Appl. (4) 144, 347--389 (1986; Zbl 0607.49010); \textit{G. Dal Maso} and \textit{L. Modica}, J. Reine Angew. Math. 368, 28--42 (1986; Zbl 0582.60034); \textit{K. Messaoudi} and \textit{G. Michaille}, RAIRO, Modélisation Math. Anal. Numér. 28, No. 3, 329--356 (1994; Zbl 0818.60029)].
Reviewer: Elvira Zappale (Roma)A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functionalhttps://zbmath.org/1521.490112023-11-13T18:48:18.785376Z"Gobbino, Massimo"https://zbmath.org/authors/?q=ai:gobbino.massimo"Picenni, Nicola"https://zbmath.org/authors/?q=ai:picenni.nicolaThe authors consider extensions of the 1D Perona-Malik functional of the type \(\mathbb{PMF}_{\varepsilon }(\beta ,f,\Omega ,u)=\int_{\Omega }\{\varepsilon ^{6}\omega (\varepsilon )^{4}u^{\prime \prime }(x)^{2}+\log (1+u^{\prime }(x)^{2})+\beta (u(x)-f(x))^{2}\}dx\), where \(\beta >0\) is a real number, \(\Omega \subseteq \mathbb{R}\) an open set, \(\varepsilon \in (0,1)\), and \(f\in L^{2}(\Omega )\) a given function called forcing term. The authors first prove the existence of minimizers to the problem \(\min\{\mathbb{ PMF}_{\varepsilon }(\beta ,f,(0,1),u):u\in H^{2}(0,1)\}=m(\varepsilon ,\beta ,f)\). Every minimizer to this problem is proved to belong to \(H^{4}(0,1)\) and in particular to \(C^{2}([0,1])\). When \(\varepsilon \rightarrow 0^{+}\), the minimum \(m(\varepsilon ,\beta ,f)\) converges to \(0\). The proof is a standard application of the direct method in the calculus of variations and of the fact that the convex envelope of the function \(p\rightarrow \log(1+p^{2})\) is identically equal to \(0\).
The first main result proves that if \(\omega (\varepsilon )=\varepsilon \left\vert \log \varepsilon \right\vert ^{1/2}\) and \(f\in C^{1}([0,1])\), the minimum value \( m(\varepsilon ,\beta ,f)\) satisfies \(\lim_{\varepsilon \rightarrow 0^{+}} \frac{m(\varepsilon ,\beta ,f)}{\omega (\varepsilon )^{2}}=10(\frac{2\beta }{ 27})^{1/5}\int_{0}^{1}\left\vert f^{\prime }(x)\right\vert ^{4/5}dx\). The second main result proves blow-up properties of minimizers \(u_{\varepsilon }\) at standard resolution, the sequences \(w_{\varepsilon }(y)=\frac{ u_{\varepsilon }(x_{\varepsilon }+\omega (\varepsilon )y)-f(x_{\varepsilon }) }{\omega (\varepsilon )}\) and \(v_{\varepsilon }(y)=\frac{u_{\varepsilon }(x_{\varepsilon }+\omega (\varepsilon )y)-u_{\varepsilon }(x_{\varepsilon }) }{\omega (\varepsilon )}\) being relatively compact in the sense of locally of locally strict convergence and all their limit points being suitable staircases. These \(w_{\varepsilon }\) and \(v_{\varepsilon }\) correspond to zooms of the graph of a minimizer \(u_{\varepsilon }\) in a neighborhood of \( (x_{\varepsilon },f(x_{\varepsilon })\) and \((x_{\varepsilon },u_{\varepsilon }(x_{\varepsilon }))\) at scale \(\omega (\varepsilon )\). A sequence of functions \(\{u_{n}\}\subseteq BV_{loc}(\mathbb{R})\) converges locally strictly to some \(u_{\infty }\in BV_{loc}(\mathbb{R})\) if \( u_{n}\) converges to \(u_{\infty }\) in \(BV(a,b)\) for every interval \( (a,b)\subseteq \mathbb{R}\) whose endpoints are not jump points of the limit \( u_{\infty }\). The authors also prove that the family \(\{u_{\varepsilon }\}\) of minimizers converges to \(f\) in the strict sense and also as varifolds.
In a further part of their paper, the authors consider the functional defined on \(H^{2}(\Omega )\) by \(\mathbb{RMF}_{\varepsilon }(a,b,u)=\int_{a}^{b}\{\varepsilon ^{6}u^{\prime \prime }(x)^{2}+\frac{1}{ \omega (\varepsilon )^{2}}\log (1+u^{\prime }(x)^{2})\}dx\) and the functional \(\mathbb{J}^{1/2}(a,b,u)=\sum_{s\in S_{u}\cap (a,b)}\left\vert \lim_{x\rightarrow s^{+}}u(x)-\lim_{x\rightarrow s^{-}}u(x)\right\vert ^{1/2} \), defined for every pure jump function that is having a finite or countable jump set \(S_{u}\).\ Both functionals are extended by \(+\infty \) to \(L^{2}(a,b) \). They prove that the \(\Gamma\)-\(\lim_{\varepsilon \rightarrow 0^{+}}\mathbb{RMF}_{\varepsilon }(a,b,u)\) is equal to \(\frac{16}{\sqrt{3}}\mathbb{J} ^{1/2}(a,b,u)\) for every \(u\in L^{2}(a,b)\), with respect to the metric of \( L^{2}(a,b)\). They also prove a compactness result for minimizers and the strict convergence of recovery sequences. The paper ends with some perspectives and open problems.
Reviewer: Alain Brillard (Riedisheim)The optimal Hölder exponent in Massari's regularity theoremhttps://zbmath.org/1521.490342023-11-13T18:48:18.785376Z"Schmidt, Thomas"https://zbmath.org/authors/?q=ai:schmidt.thomas"Schütt, Jule Helena"https://zbmath.org/authors/?q=ai:schutt.jule-helenaIn this instructively written paper, the optimal Hölder exponent in Massari's regularity theorem for sets with variational mean curvarture on \(L^p\), is determined. More precisely, let \(p\in(n,\infty]\), \(U\subseteq\mathbb{R}^n\) be open and \(E\subseteq\mathbb{R}^n\) be a set of finite perimeter in \(U\); denoted by \(P(E,U)<\infty\). If there exists \(H\in L^p(U)\) such that \(H\) is a local variational mean curvature of \(E\) in \(U\); that is, if \(E\) minimizes the functional
\[
\mathcal{F}_H^U(F):=P(F,U)-\int_{F\cap U}H~\mathrm{d}\mathcal{L}^n,
\]
among all \(F\in\mathcal{M}^n\) with \(f\triangle E\Subset U\), then the following properties are satisfied.
\begin{itemize}
\item For each \(x\in U\cap\partial^*E\) there exists an open neighborhood \(V\subseteq U\) of \(x\) such that \(V\cap\partial^*E=V\cap\partial E\) can be represented as a rotated and translated graph of a \(C^{1,\alpha}\)-function; \(\alpha=\frac14(1-\frac{n}{p})\), and \(E\cap V\) is the rotated and translated subgraph of this function.
\item For all \(s\in(n-8,n]\), it holds \(\mathcal{H}^s\big((\partial E\setminus\partial^*E)\cap U\big)=0\) where \(\mathcal{H}^s\) stands for the counting measure for \(s<0\).
\end{itemize}
This is Massari's regularity theorem. The authors prove it here for all exponents \(\alpha<\frac{p-n}{p-1}\), and provide counterexamples confirming the optimality of this exponent.
Reviewer: Georgios Psaradakis (Kastoria)Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problemshttps://zbmath.org/1521.490362023-11-13T18:48:18.785376Z"Mazzoleni, Dario"https://zbmath.org/authors/?q=ai:mazzoleni.dario"Pellacci, Benedetta"https://zbmath.org/authors/?q=ai:pellacci.benedetta"Verzini, Gianmaria"https://zbmath.org/authors/?q=ai:verzini.gianmariaSummary: We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain \(\Omega \subset \mathbb{R}^N, N\geq 2\), within a suitable class of sign-changing weights. This problem arises in the study of the persistence of a species in population dynamics. Denoting with \(u\) the optimal eigenfunction and with \(D\) its superlevel set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of \(D\) tends to zero. We show that, when the measure of \(D\) is sufficiently small, \(u\) has a unique local maximum point lying on the boundary of \(\Omega\) and \(D\) is connected. Furthermore, the boundary of \(D\) intersects the boundary of the box \(\Omega\), and more precisely, \(\mathcal{H}^{N-1}(\partial D \cap \partial \Omega)\geq C|D|^{(N-1)/N}\) for some universal constant \(C> 0\). Though widely expected, these properties are still unknown if the measure of \(D\) is arbitrary.Triangles \& princesses \& bears, oh my! A journey from a puzzle to the Schrödinger equationhttps://zbmath.org/1521.510122023-11-13T18:48:18.785376Z"Duncan, David L."https://zbmath.org/authors/?q=ai:duncan.david-lThe journey starts at a puzzle involving three princesses and three hungry bears. Even though this puzzle is easily seen to be unsolvable, it leads to the following problem: Given two triangles \(\Delta\) and \(\Delta'\) with the same area in the Euclidean plane \(\mathbb R^2\), what is the ``shortest path'' from \(\Delta\) to \(\Delta'\) through equal-area triangles. By identifying \(\mathbb R^2\) with \(\mathbb C\), the field of complex numbers, and by considering ordered triangles, this problem is turned into a problem about geodesics on a particular hypersurface in \(\mathbb C^3\cong \mathbb R^6\). Solutions to the latter problem are exhibited in great detail. Next, these findings are generalised to ordered \(n\)-gons and, by a limit process \(n\to \infty\), to ``loops''. In this way, the reader finally is guided to the Schrödinger equation.
Reviewer: Hans Havlicek (Wien)Corrigendum to: ``On the \(1/H\)-flow by \(p\)-Laplace approximation: new estimates via fake distances under Ricci lower boundshttps://zbmath.org/1521.530682023-11-13T18:48:18.785376Z"Mari, Luciano"https://zbmath.org/authors/?q=ai:mari.luciano"Rigoli, Marco"https://zbmath.org/authors/?q=ai:rigoli.marco"Setti, Alberto G."https://zbmath.org/authors/?q=ai:setti.alberto-gSummary: We correct a mistake in our proof of Lemma 2.17 in our paper [ibid. 144, No. 3, 779--849 (2022; Zbl 1506.53100)]. Although we have to strengthen the assumptions therein and, accordingly, in Theorem 2.22, all of the results on the existence and properties of the IMCF are not affected. Minor changes, with no influence elsewhere in the paper, regard Lemma 3.3, Proposition 4.3 and Lemma 5.3A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropyhttps://zbmath.org/1521.530702023-11-13T18:48:18.785376Z"Chan, Pak-Yeung"https://zbmath.org/authors/?q=ai:chan.pak-yeung"Ma, Zilu"https://zbmath.org/authors/?q=ai:ma.zilu"Zhang, Yongjia"https://zbmath.org/authors/?q=ai:zhang.yongjiaThe authors show that an ancient Ricci flow with uniformly bounded Nash entropy has uniformly bounded \(\nu\)-functional. Consequently, on such an ancient solution, there are uniform logarithmic Sobolev and Sobolev inequalities. As an application of the Sobolev inequality, they also prove a certain volume growth lower bound for steady gradient Ricci solitons without any curvature positivity. Under the additional condition that the scalar curvature attains its maximum, the authors show that the steady gradient Ricci soliton must have quadratic volume growth without imposing any non-collapsed condition.
Reviewer: Abimbola Abolarinwa (Lagos)Ricci flow on manifolds with boundary with arbitrary initial metrichttps://zbmath.org/1521.530712023-11-13T18:48:18.785376Z"Chow, Tsz-Kiu Aaron"https://zbmath.org/authors/?q=ai:chow.tsz-kiu-aaronThis paper is concerned with investigating the short time existence and uniqueness of the Ricci flow on a compact manifold \((M,g)\) with boundary \(\partial M\). That is, finding a solution to the following PDE with arbitrary smooth initial data \(g_0\):
\[
\begin{cases} \frac{\partial}{\partial t} g(t) = -2\mathrm{Ric}(g(t))&\text{on }M \times (0,T],\\
A_{g(t)} = 0 &\text{on } \partial M \times (0,T],\\
g(0) = g_0, \end{cases}
\]
where \(A_{g(t)}\) denotes the second fundamental form of the boundary \(\partial M\) with respect to the metric \(g(t)\). Past work on this problem required, for instance, that the boundary of the initial data \(g_0\) be totally geodesic. However, the author is able to remove this assumption.
The strategy of the proof is to consider the double \(\widetilde{M}\) of the compact manifold \(M\) with boundary. Since \(\widetilde{M}\) is a closed manifold, one would hope that standard results and techniques for Ricci flow could be used. However, the initial metric on \(\widetilde{M}\) is only Hölder continuous. In order to prove short-time existence and uniqueness for a Ricci flow starting at such rough initial data, the author uses a by now standard and powerful strategy of considering the Ricci-De Turck flow and then transferring the existence and uniqueness of such a flow to the desired Ricci flow. That each of these two flows induces a solution to the other additionally requires knowledge of a particular solution to the harmonic map heat flow. Given the roughness of the initial data in the aforementioned situations, a lot of detailed analysis, which includes dealing with suitable weighted parabolic Hölder spaces, is conducted by the author. This all culminates in uses of the Banach fixed point theorem to prove the desired results.
As a consequence of the work mentioned above, the author is able to prove that various curvature conditions are preserved along the Ricci flow, provided the boundary \(\partial M\) of the initial data satisfies certain curvature conditions. The list of these preserved conditions is the following:
(1) If \((M,g_0)\) has a convex boundary, then having positive curvature operator, being PIC\(1\), and being PIC\(2\) are all preserved under the flow;
(2) If \((M,g_0)\) has a two-convex boundary, then having PIC is preserved under the flow;
(3) If \((M,g_0)\) has mean convex boundary, then positive scalar curvature is preserved along the flow.
Reviewer: Louis Yudowitz (Stockholm)Regularity and curvature estimate for List's flow in four dimensionhttps://zbmath.org/1521.530742023-11-13T18:48:18.785376Z"Wu, Guoqiang"https://zbmath.org/authors/?q=ai:wu.guoqiangThis paper is devoted to List's flow, which is a triple $(M,g(t),\varphi (t))_{t\in (0,T)}$ satisfying
\begin{align*}
\partial_{t}g(t) &=-2\text{Ric}(g(t))+2d\varphi (t)\otimes d\varphi (t), \\
\partial_{t}\varphi (t) &=\Delta_{g(t)}\varphi (t), \\
g(0) &=g_{0},\quad\varphi (0)=\varphi_{0},
\end{align*}
where $\varphi (t):M\rightarrow \mathbb{R}$ are smooth functions, $g_{0}$ is a fixed Riemannian metric, and $\varphi_{0}$ is a fixed smooth function on a compact $n$-dimensional Riemannian manifold $M$ without boundary. Before the work by \textit{B. List} [Commun. Anal. Geom. 16, No. 5, 1007--1048 (2008; Zbl 1166.53044)], the Ricci flow system for a Riemannian metric $\partial_{t}g=-2\mathrm{Ric}(g)$ has been used with great
success for the construction of canonical metrics on Riemannian manifolds of low dimension. B. List has developed a corresponding theory for canonical objects with a certain physical interpretation. He proved the existence of an entropy $E$ such that the stationary points of List's flow are solutions to the static Einstein vacuum equations, and studied the extended parabolic system
\begin{align*}
\partial_{t}g &=-2\text{Ric}(g)+2\alpha_{n}d\varphi \otimes d\varphi , \\
\partial_{t}\varphi &=\Delta_{g}\varphi ,
\end{align*}
which is equivalent to the gradient flow of $E$. For applications on noncompact asymptotically flat manifolds, he proved short time existence on complete manifolds in the case when $\varphi $ is a smooth function from $M$ to $\mathbb{R}$.
In this paper the author studies List's flow on a compact manifold such that the scalar curvature is bounded. He establishes a time derivative bound for solutions to the heat equation, and derives the existence of a cutoff function (with good properties) whose time derivative and Laplacian are bounded. This can be seen as a parabolic version of Cheeger-Colding's cutoff function in the setting of Ricci lower bound.
Based on the above results, the author proves a backward pseudolocality theorem for the List's flow in dimension four. In this, the author needs to prove the $L^{\infty }$ estimate for subsolutions to nonhomogeneous linear heat equations along List's flow using Moser iteration method.
As an application, the author obtains that the $L^{2}$-norm of the Riemannian curvature operator is bounded and also gets the limit behavior of the List's flow. More precisely, based on an $L^{2}$-bound on the Riemannian curvature operator and a backward pseudolocality result, the author shows that if $M$ is a compact $4$-dimensional Riemannian manifold, $(M,g(t),\varphi (t)) $ is a List's flow on $[0,T)$, and if the trace of the Ricci tensor $S$ satisfies $|S|\leq 1$ on $M\times \lbrack 0,T),$ and $|\varphi_{0}|\leq 1$, then $(M,g(t),\varphi (t))$ converges to an orbifold in the Cheeger-Gromov sense as $t\rightarrow T$.
Reviewer: Boubaker-Khaled Sadallah (Algier)Partial regularity for harmonic maps into spheres at a singular or degenerate free boundaryhttps://zbmath.org/1521.580062023-11-13T18:48:18.785376Z"Moser, Roger"https://zbmath.org/authors/?q=ai:moser.roger"Roberts, James"https://zbmath.org/authors/?q=ai:roberts.james-w|roberts.james-a|roberts.james-lMotivated by the study of fractional harmonic mappings, the authors provide regularity results for a class of free boundary harmonic mappings on certain domains exhibiting degenerate features.
Reviewer: Dumitru Motreanu (Perpignan)On the existence of solutions of a critical elliptic equation involving Hardy potential on compact Riemannian manifoldshttps://zbmath.org/1521.580082023-11-13T18:48:18.785376Z"Terki, Fatima Zohra"https://zbmath.org/authors/?q=ai:terki.fatima-zohra"Maliki, Youssef"https://zbmath.org/authors/?q=ai:maliki.youssefIn this paper, the authors study the existence of weak solutions to a certain non-linear elliptic equation on punctured compact Riemannian manifolds. This equation is a natural generalization of e.g. the equation for the scalar curvature in the Yamabe problem.
More precisely, let \((M,g)\) be an \(n\geq 3\) dimensional closed oriented Riemannian manifold with injectivity radius \(\delta_g>0\). Take a fixed point \(p\in M\) and define the truncated distance function \(\rho_p\) about this point to be the usual distance function in the geodesic ball \(B_{\delta_g}(p)\) and to be the constant \(\delta_g\) in the complementum \(M\setminus B_{\delta_g}(p)\). Let \(f,h\) be further smooth functions on \(M\) and consider the following nonlinear 2nd order scalar PDE on the punctured space \(M\setminus\{p\}\):
\[
\Delta_gu-\frac{h}{\rho_p^2}u=f\vert u\vert^{2^*-2}u
\]
where \(2^*=\frac{2n}{n-2}\) is the critical Sobolev exponent. Note that this equation gives back the Yamabe equation if the Hardy potential \(\frac{h}{\rho_p^2}\) is specified to \(\frac{n-2}{4(n-1)}\mathrm{Scal}_g\).
The authors study the existence of weak solutions to this equation over \(M\), more precisely solutions \(u\in L^2_1(M,g)\) or in different notation \(u\in H^2_1(M,g)\). For the precise technical statement cf. Theorems 1 and 2 in the paper.
Reviewer: Gábor Etesi (Budapest)On the boundary complex of the \(k\)-Cauchy-Fueter complexhttps://zbmath.org/1521.580102023-11-13T18:48:18.785376Z"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.18Summary: The \(k\)-Cauchy-Fueter complex, \(k=0,1,\ldots\), in quaternionic analysis are the counterpart of the Dolbeault complex in the theory of several complex variables. In this paper, we construct explicitly boundary complexes of these complexes on boundaries of domains, corresponding to the tangential Cauchy-Riemann complex in complex analysis. They are only known boundary complexes outside of complex analysis that have interesting applications to the function theory. As an application, we establish the Hartogs-Bochner extension for \(k\)-regular functions, the quaternionic counterpart of holomorphic functions. These boundary complexes have a very simple form on a kind of quadratic hypersurfaces, which have the structure of right-type nilpotent Lie groups of step two. They allow us to introduce the quaternionic Monge-Ampère operator and open the door to investigate pluripotential theory on such groups. We also apply abstract duality theorem to boundary complexes to obtain the generalization of Malgrange's vanishing theorem and the Hartogs-Bochner extension for \(k\)-CF functions, the quaternionic counterpart of CR functions, on this kind of groups.Integrability in the weak noise theoryhttps://zbmath.org/1521.600162023-11-13T18:48:18.785376Z"Tsai, Li-Cheng"https://zbmath.org/authors/?q=ai:tsai.li-chengSummary: We consider the variational problem associated with the Freidlin-Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation (SHE). For a general class of initial-terminal conditions, we show that a minimizer of this variational problem exists, and any minimizer solves a system of imaginary-time Nonlinear Schrödinger equations. This system is integrable. Utilizing the integrability, we prove that the formulas from the physics work (see [\textit{A. Krajenbrink} and \textit{P. Le Doussal}, Phys. Rev. Lett. 127, No. 6, Article ID 064101, 8 p. (2021; \url{doi:10.1103/PhysRevLett.127.064101})]) hold for every minimizer of the variational problem. As an application, we consider the Freidlin-Wentzell LDP for the SHE with the delta initial condition. Under a technical assumption on the poles of the reflection coefficients, we prove the explicit expression for the one-point rate function that was predicted in the physics works (see [\textit{P. Le Doussal} et al., Phys. Rev. Lett. 117, No. 7, Article ID 070403, 5 p. (2016; \url{doi:10.1103/PhysRevLett.117.070403}); Krajenbrink and Le Doussal, loc. cit.]). Under the same assumption, we give detailed pointwise estimates of the most probable shape in the upper-tail limit.The Allen-Cahn equation with generic initial datumhttps://zbmath.org/1521.600272023-11-13T18:48:18.785376Z"Hairer, Martin"https://zbmath.org/authors/?q=ai:hairer.martin"Lê, Khoa"https://zbmath.org/authors/?q=ai:le.khoa|le.khoa-n"Rosati, Tommaso"https://zbmath.org/authors/?q=ai:rosati.tommaso-cornelisThe authors consider the Allen-Cahn equation \(\partial_{t}u- \Delta u = u- u^{3}\) with a rapidly mixing Gaussian field as an initial condition. They study the scaling limit of such an equation and show that after a suitably long time the dynamics are well approximated by a certain class of Gaussian nodal sets which evolve under a mean curvature flow. While ideally one would like to consider the white noise initial data, this is not possible in this setting since the scaling exponent of the white noise is below a critical exponent for which one might expect any form of a local well-posedness for the equation. The authors show that if the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to a mean curvature flow.
Reviewer: Maria Gordina (Storrs)Moment intermittency in the PAM with asymptotically singular noisehttps://zbmath.org/1521.600282023-11-13T18:48:18.785376Z"Lamarre, Pierre Yves Gaudreau"https://zbmath.org/authors/?q=ai:gaudreau-lamarre.pierre-yves"Ghosal, Promit"https://zbmath.org/authors/?q=ai:ghosal.promit"Liao, Yuchen"https://zbmath.org/authors/?q=ai:liao.yuchenSummary: Let \(\xi\) be a singular Gaussian noise on \(\mathbb{R}^d\) that is either white, fractional, or with the Riesz covariance kernel; in particular, there exists a scaling parameter \(\omega >0\) such that \(c^{\omega /2}\xi (c\cdot )\) is equal in distribution to \(\xi\) for all \(c>0\). Let \((\xi_\varepsilon )_{\varepsilon >0}\) be a sequence of smooth mollifications such that \(\xi_\varepsilon \to \xi\) as \(\varepsilon \to 0\). We study the asymptotics of the moments of the parabolic Anderson model (PAM) with noise \(\xi_\varepsilon\) as \(\varepsilon \to 0\), both for large (i.e., \(t\to \infty )\) and fixed times \(t\). This approach makes it possible to study the moments of the PAM with regular and singular noises in a unified fashion, as well as interpolate between the two settings. As corollaries of our main results, we obtain the following:
\begin{itemize}
\item[1] When \(\zeta\) is subcritical (i.e., \(0 < \omega < 2\)), our results extend the known large-time moment and tail asymptotics for the Stratonovich PAM with noise \(\zeta\). Our method of proof clarifies the role of the maximizers of the
variational problems (known as Hartree ground states) that appear in these moment asymptotics in describing the geometry of intermittency. We take this opportunity to prove the existence and study the properties of the Hartree ground state with a fractional kernel, which we believe is of independent interest.
\item[2] When \(\zeta\) is critical or supercritical (i.e., \(\omega = 2\) or \(\omega > 2\)), our results provide a new interpretation of the moment blowup phenomenon observed
in the Stratonovich PAM with noise \(\zeta\). That is, we uncover that the latter is related to an intermittency effect that occurs in the PAM with noise \(\zeta_\varepsilon\) as \(\varepsilon \to 0\) for fixed finite times \(t > 0\).
\end{itemize}Stability analysis for a class of stochastic delay nonlinear systems driven by G-Lévy processhttps://zbmath.org/1521.600292023-11-13T18:48:18.785376Z"Ma, Li"https://zbmath.org/authors/?q=ai:ma.li.3"Li, Yujing"https://zbmath.org/authors/?q=ai:li.yujing"Zhu, Quanxin"https://zbmath.org/authors/?q=ai:zhu.quanxinThe authors study regularity and stability of solutions to a class of stochastic delay differential equations driven by G-Lévy processes. The notion of such processes was introduced about 15 years ago as an extension of G-Brownian motion with a sublinear expectation. One of the first results in the article is a Burkholder-Davis-Gundy (BDG) inequality for the jump measure. Then the BDG inequality is used to show the existence and uniqueness of solutions under non-Lipschitz condition. Under local Lipschitz and one-sided polynomial growth conditions, the authors in addition establish quasi-sure exponential stability and the \(p\)th moment exponential stability of the solution.
Reviewer: Maria Gordina (Storrs)Invariant measure for 2D stochastic Cahn-Hilliard-Navier-Stokes equationshttps://zbmath.org/1521.600302023-11-13T18:48:18.785376Z"Qiu, Zhaoyang"https://zbmath.org/authors/?q=ai:qiu.zhaoyang"Wang, Huaqiao"https://zbmath.org/authors/?q=ai:wang.huaqiao"Huang, Daiwen"https://zbmath.org/authors/?q=ai:huang.daiwenThis paper focuses on the invariant measure for stochastic Cahn-Hilliard-Navier-Stokes equations in two-dimensional spaces. Applying the Maslowski-Seidler method, the authors establish the existence of invariant measure in state space \(L^2_x\times H^1\) with the weak topology. Furthermore, they prove the existence of global pathwise solutions using the stochastic compactness argument.
This paper is innovative and interesting. It provides important theoretical tools and methods for studying the dynamic behavior of the stochastic Cahn-Hilliard-Navier-Stokes equations.
Reviewer: Guanggan Chen (Chengdu)On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noisehttps://zbmath.org/1521.600322023-11-13T18:48:18.785376Z"Tang, Hao"https://zbmath.org/authors/?q=ai:tang.haoThe paper is devoted to study of the stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise. The main results are associated with local solution, blow-up criterion, and global existence. For the multidimensional case, the interplay between stability on exiting times and continuous dependence of solution on initial data is studied. The paper also establishes two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type noise only involving the gradient operator.
Reviewer: Anatoliy Swishchuk (Calgary)SPDE bridges with observation noise and their spatial approximationhttps://zbmath.org/1521.600342023-11-13T18:48:18.785376Z"di Nunno, Giulia"https://zbmath.org/authors/?q=ai:di-nunno.giulia"Ortiz-Latorre, Salvador"https://zbmath.org/authors/?q=ai:ortiz-latorre.salvador"Petersson, Andreas"https://zbmath.org/authors/?q=ai:petersson.andreasThis paper is devoted to introduce SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. In this article, the SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.
Reviewer: Hossam A. Ghany (al-Qāhira)Quantifying and managing uncertainty in piecewise-deterministic Markov processeshttps://zbmath.org/1521.600412023-11-13T18:48:18.785376Z"Cartee, Elliot"https://zbmath.org/authors/?q=ai:cartee.elliot"Farah, Antonio"https://zbmath.org/authors/?q=ai:farah.antonio"Nellis, April"https://zbmath.org/authors/?q=ai:nellis.april"Van Hook, Jacob"https://zbmath.org/authors/?q=ai:van-hook.jacob"Vladimirsky, Alexander"https://zbmath.org/authors/?q=ai:vladimirsky.alexanderSummary: In piecewise-deterministic Markov processes (PDMPs) the state of a finite-dimensional system evolves continuously, but the evolutive equation may change randomly as a result of discrete switches. A running cost is integrated along the corresponding piecewise-deterministic trajectory up to the termination to produce the \textit{cumulative cost} of the process. We address three natural questions related to uncertainty in cumulative cost of PDMP models: (1) how to compute the cumulative distribution function (CDF) of the cumulative cost when the switching rates are fully known; (2) how to accurately bound the CDF when the switching rates are uncertain; and (3) assuming the PDMP is controlled, how to select a control to optimize that CDF. In all three cases, our approach requires posing a system of suitable hyperbolic partial differential equations, which are then solved numerically on an augmented state space. We illustrate our method using simple examples of trajectory planning under uncertainty for several one-dimensional and two-dimensional first-exit time problems. In the appendix, we also apply this method to a model of fish harvesting in an environment with random switches in carrying capacity.Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacianhttps://zbmath.org/1521.600462023-11-13T18:48:18.785376Z"Bañuelos, Rodrigo"https://zbmath.org/authors/?q=ai:banuelos.rodrigo"Mariano, Phanuel"https://zbmath.org/authors/?q=ai:mariano.phanuel-a"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.5Summary: For domains in \(\mathbb{R}^d\), \(d\geq 2\), we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power \(p>0\) and the supremum over all starting points of the \(p\)-moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of \(p\) and that for \(p \geq 1\), the upper bound is asymptotically sharp as \(d\to \infty \). For all \(p>0\), we prove the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For this class of domains we conjecture that the cube is extremal.Multilevel quasi-Monte Carlo for optimization under uncertaintyhttps://zbmath.org/1521.650072023-11-13T18:48:18.785376Z"Guth, Philipp A."https://zbmath.org/authors/?q=ai:guth.philipp-a"Van Barel, Andreas"https://zbmath.org/authors/?q=ai:van-barel.andreasSummary: This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo (MLQMC) method to the estimation of the gradient is investigated, with a circulant embedding method used to sample the stochastic field. A novel regularity analysis of the adjoint variable is essential for the MLQMC estimation of the gradient in combination with the samples generated using the circulant embedding method. A rigorous cost and error analysis shows that a randomly shifted quasi-Monte Carlo method leads to a faster rate of decay in the root mean square error of the gradient than the ordinary Monte Carlo method, while considering multiple levels substantially reduces the computational effort. Numerical experiments confirm the improved rate of convergence and show that the MLQMC method outperforms the multilevel Monte Carlo method and single level quasi-Monte Carlo method.Accelerated simulation of Boltzmann-BGK equations near the diffusive limit with asymptotic-preserving multilevel Monte Carlohttps://zbmath.org/1521.650082023-11-13T18:48:18.785376Z"Løvbak, Emil"https://zbmath.org/authors/?q=ai:lovbak.emil"Samaey, Giovanni"https://zbmath.org/authors/?q=ai:samaey.giovanniSummary: Kinetic equations model the position-velocity distribution of particles subject to transport and collision effects. Under a diffusive scaling, these combined effects converge to a diffusion equation for the position density in the limit of an infinite collision rate. Despite this well-defined limit, numerical simulation is expensive when the collision rate is high but finite, as small time steps are then required. In this work, we present an asymptotic-preserving multilevel Monte Carlo particle scheme that makes use of this diffusive limit to accelerate computations. In this scheme, we first sample the diffusive limiting model to compute a biased initial estimate of a quantity of interest, using large time steps. We then perform a limited number of finer simulations with transport and collision dynamics to correct the bias. The efficiency of the multilevel method depends on being able to perform correlated simulations of particles on a hierarchy of discretization levels. We present a method for correlating particle trajectories and present both an analysis and numerical experiments. We demonstrate that our approach significantly reduces the cost of particle simulations in high-collisional regimes, compared with prior work, indicating significant potential for adopting these schemes in various areas of active research.Approximation of SPDE covariance operators by finite elements: a semigroup approachhttps://zbmath.org/1521.650102023-11-13T18:48:18.785376Z"Kovács, Mihály"https://zbmath.org/authors/?q=ai:kovacs.mihaly"Lang, Annika"https://zbmath.org/authors/?q=ai:lang.annika"Petersson, Andreas"https://zbmath.org/authors/?q=ai:petersson.andreasSummary: The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition is proven. This formula is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates are derived in the trace class and Hilbert-Schmidt norms with numerical simulations illustrating the results.On the rational Levin quadrature for evaluation of highly oscillatory integrals and its applicationhttps://zbmath.org/1521.650242023-11-13T18:48:18.785376Z"Ma, Junjie"https://zbmath.org/authors/?q=ai:ma.junjie(no abstract)Maximum principle preserving and unconditionally stable scheme for a conservative Allen-Cahn equationhttps://zbmath.org/1521.650722023-11-13T18:48:18.785376Z"Choi, Yongho"https://zbmath.org/authors/?q=ai:choi.yongho"Kim, Junseok"https://zbmath.org/authors/?q=ai:kim.junseokSummary: In this study, we present a novel conservative Allen-Cahn (CAC) equation and its maximum principle preserving and unconditionally stable numerical method. There have been many research works of the numerical methods for the CAC equation. To conserve the total mass, many mathematical models for the CAC equation introduced Lagrange multipliers which are added to the original Allen-Cahn equation. Therefore, some of the methods do not preserve the maximum principle, i.e., it is possible to have values greater than the maximum and smaller than the minimum values of the admissible solutions. In this study, we propose a novel CAC equation with a new Lagrange multiplier which is a power exponent to the concentration so that the maximum principle strictly holds. Furthermore, we describe the proposed numerical algorithm in detail and present several computational experiments to validate the superior performance of the proposed scheme.Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problemhttps://zbmath.org/1521.650732023-11-13T18:48:18.785376Z"Feng, Libo"https://zbmath.org/authors/?q=ai:feng.libo"Liu, Fawang"https://zbmath.org/authors/?q=ai:liu.fawang"Anh, Vo V."https://zbmath.org/authors/?q=ai:anh.vo-v"Qin, Shanlin"https://zbmath.org/authors/?q=ai:qin.shanlin(no abstract)An efficient operator-splitting radial basis function-generated finite difference (RBF-FD) scheme for image noise removal based on nonlinear total variation modelshttps://zbmath.org/1521.650742023-11-13T18:48:18.785376Z"Mazloum, J."https://zbmath.org/authors/?q=ai:mazloum.jalil"Hadian Siahkal-Mahalle, B."https://zbmath.org/authors/?q=ai:hadian-siahkal-mahalle.behrang(no abstract)A local radial basis function-finite difference (RBF-FD) method for solving 1D and 2D coupled Schrödinger-Boussinesq (SBq) equationshttps://zbmath.org/1521.650752023-11-13T18:48:18.785376Z"Oruç, Ömer"https://zbmath.org/authors/?q=ai:oruc.omerSummary: In this study, one-dimensional (1D) and two-dimensional (2D) coupled Schrödinger-Boussinesq (SBq) equations are examined numerically. A local meshless method based on radial basis function-finite difference (RBF-FD) method for spatial approximation is devised. We use polyharmonic splines as radial basis function along with augmented polynomials. By using polyharmonic splines we avoid to choose optimal shape parameter which requires special algorithms in meshless methods. For temporal discretization, low-storage ten-stage fourth-order explicit strong stability preserving Runge Kutta method is used which gives more flexibility on temporal step width. \(L_\infty\) and \(L_2\) error norms are calculated to show accuracy of the proposed method. Further, conserved quantities are monitoried during numerical simulations to see how good the proposed method preserves them. Stability of the proposed method is dicussed numerically. Some codes are developed in Julia programming language to achieve more speed up in numerical simulations. Obtained results and their comparison with some studies such as wavelet, difference schemes and Fourier spectral methods available in literature verify the efficiency and reliability of the proposed method.Numerical solutions of the Boussinesq equation with nonlinear restoring forcehttps://zbmath.org/1521.650762023-11-13T18:48:18.785376Z"Vucheva, Veselina"https://zbmath.org/authors/?q=ai:vucheva.veselina"Vassilev, Vassil M."https://zbmath.org/authors/?q=ai:vassilev.vassil-m"Kolkovska, Natalia"https://zbmath.org/authors/?q=ai:kolkovska.natalia-tSummary: In this work we consider the Boussinesq equation with nonlinear restoring force. Equations of this type model the transverse or longitudinal vibration of an elastic rod subject to a constant tangential follower force and laying on a nonlinear elastic foundation due to which a cubic term appears in addition to the linear terms corresponding to a purely Winkler-Pasternak foundation. The dynamical behavior of such mechanical systems is not well-studied in the current literature. Here, we give exact solitary wave solutions (solitons) of the regarded nonlinear equation in explicit analytic form. We propose and study finite difference schemes to solve the considered problem. The nonlinear terms are approximated in two different ways. Both schemes have second order of approximation in space and time steps. The extensive numerical experiments show second order of convergence for single solitary wave and the interaction between two solitary waves.
For the entire collection see [Zbl 1511.65004].Generalized finite difference method with irregular mesh for a class of three-dimensional variable-order time-fractional advection-diffusion equationshttps://zbmath.org/1521.650772023-11-13T18:48:18.785376Z"Wang, Zhaoyang"https://zbmath.org/authors/?q=ai:wang.zhaoyang"Sun, HongGuang"https://zbmath.org/authors/?q=ai:sun.hongguangSummary: Fractional advection-diffusion equation, as a generalization of classical advection-diffusion equation, has been always mentioned to simulate anomalous diffusion in porous media. This work introduces a meshless generalized finite difference method (GFDM) to solve a class of three-dimensional variable-order time fractional advection-diffusion equation (TFADE) in finite domains. Three examples with known analytic solutions in different domains are given to demonstrate that the method is accurate and stable. To reduce computational and storage cost, we discretize the time derivative terms of TFADE by a fast finite difference method (FFDM) based on sum-of-exponentials (SOE) approximation. Meanwhile, discretizing space derivative terms, GFDM generates a linear equation set including function values of neighboring nodes with various weight coefficients. Then the partial derivatives of TFADE are indicated as the linear system above. Also, this paper investigates the irregular mesh in the finite spatial domain, which is more closely meets the description of practice problems. Numerical results indicate that models with irregular mesh can also be simulated by GFDM which maintains high accuracy. Furthermore, the method is stable and accurate in solving three-dimensional irregular domain problems, where the relative errors can be less than 0.01\%. This paper shows that FFDM based on SOE approximation can improve computational efficiency, and GFDM can flexibly and efficiently solve three-dimensional variable-order and variable-coefficient TFADE.Numerical simulation of the soliton solutions for a complex modified Korteweg-de Vries equation by a finite difference methodhttps://zbmath.org/1521.650782023-11-13T18:48:18.785376Z"Xu, Tao"https://zbmath.org/authors/?q=ai:xu.tao"Zhang, Guowei"https://zbmath.org/authors/?q=ai:zhang.guowei.1|zhang.guowei"Wang, Liqun"https://zbmath.org/authors/?q=ai:wang.liqun"Xu, Xiangmin"https://zbmath.org/authors/?q=ai:xu.xiangmin"Li, Min"https://zbmath.org/authors/?q=ai:li.min.9Summary: In this paper, a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modified Korteweg-de Vries (MKdV) equation (which is equivalent to the Sasa-Satsuma equation) with the vanishing boundary condition. It is proved that such a numerical scheme has the second-order accuracy both in space and time, and conserves the mass in the discrete level. Meanwhile, the resulting scheme is shown to be unconditionally stable via the von Nuemann analysis. In addition, an iterative method and the Thomas algorithm are used together to enhance the computational efficiency. In numerical experiments, this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation. The numerical accuracy, mass conservation and linear stability are tested to assess the scheme's performance.SUSHI for a Bingham flow type problemhttps://zbmath.org/1521.650792023-11-13T18:48:18.785376Z"Aboussi, Wassim"https://zbmath.org/authors/?q=ai:aboussi.wassim"Benkhaldoun, Fayssal"https://zbmath.org/authors/?q=ai:benkhaldoun.fayssal"Bradji, Abdallah"https://zbmath.org/authors/?q=ai:bradji.abdallahSummary: We establish a nonlinear finite volume scheme for a Bingham Flow Type Problem. The equation is a nonlinear parabolic one and it is a simplified version of the Bingham visco-plastic flow model [\textit{E. J. Dean} et al., J. Non-Newton. Fluid Mech. 142, No. 1--3, 36--62 (2007; Zbl 1107.76061), pp. 38--40]. The space discretization is performed using SUSHI (Scheme using Stabilization and Hybrid Interfaces) developed in [\textit{R. Eymard} et al., IMA J. Numer. Anal. 30, No. 4, 1009--1043 (2010; Zbl 1202.65144)] whereas the time discretization is uniform. We first prove a discrete \textit{a priori} estimate. We then, prove the existence of at least one discrete solution using the Brouwer fixed point theorem. The uniqueness of the approximate solution is shown as well. We subsequently prove error estimates of order one in time and space in \(L^\infty (L^2)\) and \(L^2(H^1)\)-discrete norms.
For the entire collection see [Zbl 1511.65004].SUSHI for a non-linear time fractional diffusion equation with a time independent delayhttps://zbmath.org/1521.650812023-11-13T18:48:18.785376Z"Benkhaldoun, Fayssal"https://zbmath.org/authors/?q=ai:benkhaldoun.fayssal"Bradji, Abdallah"https://zbmath.org/authors/?q=ai:bradji.abdallahSummary: We establish a linear implicit finite volume scheme for a non-linear time fractional diffusion equation with a time independent delay in any space dimension. The fractional order derivative is given in the Caputo sense. The discretization in space is performed using the SUSHI ((Scheme Using stabilized Hybrid Interfaces) developed in [\textit{R. Eymard} et al., IMA J. Numer. Anal. 30, No. 4, 1009--1043 (2010; Zbl 1202.65144)], whereas the discretization in time is given by a constrained time step-size. The approximation of the fractional order derivative is given by \(L1\)-formula.
We prove rigorously new convergence results in \(L^\infty (L^2)\) and \(L^2(H^1_0)\)-discrete norms. The order is proved to be optimal in space and it is \(k^{2-\alpha }\) in time, with \(k\) is the constant time step and \(\alpha\) is the fractional order of the Caputo derivative.
This paper is a continuation of some of our previous works which dealt either with only the linear fractional PDEs (Partial Differential Equations) without delays, e.g. [\textit{A. Bradji}, Springer Proc. Math. Stat. 323, 305--314 (2020; Zbl 1454.65074); Comput. Math. Appl. 79, No. 2, 500--520 (2020; Zbl 1443.65268); C. R., Math., Acad. Sci. Paris 356, No. 4, 439--448 (2018; Zbl 1447.65071); \textit{A. Bradji} and \textit{J. Fuhrmann}, Lect. Notes Comput. Sci. 10187, 33--45 (2017; Zbl 1371.65090)], or with only time dependent PDEs (the time derivative is given in the usual sense) with delays, e.g. [\textit{F. Benkhaldoun} et al., Lect. Notes Comput. Sci. 13127, 498--506 (2022; Zbl 1487.65127); \textit{F. Benkhaldoun} and \textit{A. Bradji}, Springer Proc. Math. Stat. 323, 315--324 (2020; Zbl 1454.65071); \textit{A. Bradji} and \textit{T. Ghoudi}, Lect. Notes Comput. Sci. 11189, 351--359 (2019; Zbl 1416.65290)].
For the entire collection see [Zbl 1511.65004].Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systemshttps://zbmath.org/1521.650822023-11-13T18:48:18.785376Z"Jung, Jonathan"https://zbmath.org/authors/?q=ai:jung.jonathan"Perrier, Vincent"https://zbmath.org/authors/?q=ai:perrier.vincentThis article deals with the study of the long-time behavior of cell-centered finite volume approximations for general symmetrizable first-order linear hyperbolic systems on a bounded domain. Two types of boundary conditions are considered: wall boundary conditions and Steger-Warming boundary conditions. Since the continuous system is not dissipative, its long-time behavior study is a difficult problem, and the set of all possible long-time limits may be large and difficult to describe. The aim of the authors is then to investigate this problem in the discrete setting, with a stable finite volume discretization. Indeed, in the discrete case, it is possible to prove the existence of a long-time limit for the numerical scheme provided that the considered numerical flux is sufficiently dissipative.
The authors first study the semidiscrete numerical scheme (discrete in space, continuous in time), which is a linear finite-dimensional system. Its long-time behavior is known, and depends only on the spectral structure of the corresponding matrix. The study is then conducted for the fully discrete scheme, with a forward or backward Euler time discretization. Some tight sufficient conditions are given for ensuring the existence of a long-time limit for the finite volume discretization of this general first-order hyperbolic system. The convergence rate to the long-time limit is driven by the spectral gap of the space discretization, and only the numerical dissipation is responsible for the existence of the long-time limit.
Then, these general results are applied to different stable numerical schemes for the first-order wave system: Rusanov, Godunov, Godunov with a centered discretization of the pressure gradient, and LMAAP (low Mach acoustic accuracy preserving) stabilization proposed in [\textit{P. Bruel} et al., J. Comput. Phys. 378, 723--759 (2019; Zbl 1416.76139)]. Hypotheses are checked for each of these stabilizations and both considered boundary conditions.
Finally, several numerical experiments are performed on the long-time limit of the scheme for the wave system, with the different introduced stabilizations, on triangular and quadrangular meshes, confirming the obtained theoretical results.
Reviewer: Marianne Bessemoulin-Chatard (Nantes)Half boundary method for two-dimensional steady-state nonlinear convection-diffusion equationshttps://zbmath.org/1521.650832023-11-13T18:48:18.785376Z"Meng, Xiangyuan"https://zbmath.org/authors/?q=ai:meng.xiangyuan"Huang, Mei"https://zbmath.org/authors/?q=ai:huang.mei"Wang, Boxue"https://zbmath.org/authors/?q=ai:wang.boxue"Li, Yaodi"https://zbmath.org/authors/?q=ai:li.yaodi"Cheng, Yanting"https://zbmath.org/authors/?q=ai:cheng.yanting"Morita, Chihiro"https://zbmath.org/authors/?q=ai:morita.chihiro(no abstract)Balance-characteristic scheme as applied to the shallow water equations over a rough bottomhttps://zbmath.org/1521.650842023-11-13T18:48:18.785376Z"Goloviznin, V. M."https://zbmath.org/authors/?q=ai:goloviznin.v-m"Isakov, V. A."https://zbmath.org/authors/?q=ai:isakov.v-aThe CABARET scheme is used for the numerical solution of the one-dimensional shallow water equations over a rough bottom. The scheme involves conservative and flux variables, whose values at a new time level are calculated by applying the characteristic properties of the shallow water equations. The scheme is verified using a series of test and model problems.
Reviewer: Qifeng Zhang (Hangzhou)A localized hybrid kernel meshless technique for solving the fractional Rayleigh-Stokes problem for an edge in a viscoelastic fluidhttps://zbmath.org/1521.650852023-11-13T18:48:18.785376Z"Avazzadeh, Zakieh"https://zbmath.org/authors/?q=ai:avazzadeh.zakieh"Nikan, Omid"https://zbmath.org/authors/?q=ai:nikan.omid"Anh Tuan Nguyen"https://zbmath.org/authors/?q=ai:anh-tuan-nguyen."Van Tien Nguyen"https://zbmath.org/authors/?q=ai:van-tien-nguyen.(no abstract)An iteration-free semi-Lagrangian meshless method for Burgers' equationshttps://zbmath.org/1521.650862023-11-13T18:48:18.785376Z"Ma, Liping"https://zbmath.org/authors/?q=ai:ma.liping"Zhao, Lijing"https://zbmath.org/authors/?q=ai:zhao.lijing"Wang, Xiaodong"https://zbmath.org/authors/?q=ai:wang.xiaodong.6|wang.xiaodong.4|wang.xiaodong.1|wang.xiaodong.5|wang.xiaodong.8|wang.xiaodong(no abstract)Numerical solution of an inverse source problem for a time-fractional PDE via direct meshless local Petrov-Galerkin methodhttps://zbmath.org/1521.650872023-11-13T18:48:18.785376Z"Molaee, Tahereh"https://zbmath.org/authors/?q=ai:molaee.tahereh"Shahrezaee, Alimardan"https://zbmath.org/authors/?q=ai:shahrezaee.alimardan(no abstract)A meshless method to solve the variable-order fractional diffusion problems with fourth-order derivative termhttps://zbmath.org/1521.650882023-11-13T18:48:18.785376Z"Safari, Farzaneh"https://zbmath.org/authors/?q=ai:safari.farzaneh"Jing, Li"https://zbmath.org/authors/?q=ai:jing.li"Lu, Jun"https://zbmath.org/authors/?q=ai:lu.jun"Chen, Wen"https://zbmath.org/authors/?q=ai:chen.wen(no abstract)Fast multigrid reduction-in-time for advection via modified semi-Lagrangian coarse-grid operatorshttps://zbmath.org/1521.650902023-11-13T18:48:18.785376Z"De Sterck, Hans"https://zbmath.org/authors/?q=ai:de-sterck.hans"Falgout, Robert D."https://zbmath.org/authors/?q=ai:falgout.robert-d"Krzysik, Oliver A."https://zbmath.org/authors/?q=ai:krzysik.oliver-aSummary: Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs) but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. The key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-Lagrangian-based coarse-grid operator that leads to fast and scalable multilevel time integration of linear advection PDEs. The coarse-grid operator is composed of a semi-Lagrangian discretization followed by a correction term, with the correction designed so that the leading-order truncation error of the composite operator is approximately equal to that of the ideal coarse-grid operator. Parallel results show substantial speed-ups over sequential time integration for variable-wave-speed advection problems in one and two spatial dimensions, and using high-order discretizations up to order five. The proposed approach establishes the first practical method that provides small and scalable MGRIT iteration counts for advection problems.The direct meshless local Petrov-Galerkin technique with its error estimate for distributed-order time fractional cable equationhttps://zbmath.org/1521.650922023-11-13T18:48:18.785376Z"Habibirad, Ali"https://zbmath.org/authors/?q=ai:habibirad.ali"Hesameddini, Esmail"https://zbmath.org/authors/?q=ai:hesameddini.esmail"Azin, Hadis"https://zbmath.org/authors/?q=ai:azin.hadis"Heydari, Mohammad Hossein"https://zbmath.org/authors/?q=ai:heydari.mohammadhosseinSummary: Distributed-order fractional calculus is a quickly growing concept of the more general area of fractional calculus that has significant and extensive usage for designing complex systems. This work is used the direct meshless local Petrov-Galerkin (DMLPG) technique for the numerical solution of the distributed-order time fractional Cable equation. DMLPG implements a generalized moving least square (GMLS) process to discretize the equation in space variables. By using this scheme, the test function is approximated via the values at nodes, directly. Thus, this algorithm passes integration with the MLS shape functions substituting with a more inexpensive integration than polynomials. Here, the distributed integral is discretized by the \(M\)-point Gauss-Legendre quadrature rule. Then, the finite difference scheme is applied to approximate the fractional derivative discretization. Also, the unconditionally stability and rate of convergence \(O ( \tau^{2 - \max \{ \alpha , \beta \}} )\) of the time-discrete technique are demonstrated. Moreover, the current method converts the problem into a system of linear algebraic equations. To demonstrate the capability and flexibility of our scheme, some examples with different geometric domains are supposed in two-dimensional cases.On the application of a hierarchically semi-separable compression for space-fractional parabolic problems with varying time stepshttps://zbmath.org/1521.650942023-11-13T18:48:18.785376Z"Slavchev, Dimitar"https://zbmath.org/authors/?q=ai:slavchev.dimitar"Margenov, Svetozar"https://zbmath.org/authors/?q=ai:margenov.svetozar-dSummary: Anomalous (fractional) diffusion is observed when the Brownian motion hypotheses are violated. It is modeled with the fractional Laplace operator, which can be defined in several ways. In this work we use the integral definition with the Riesz potential. For the discretization in space we apply the finite element method and for the discretization in time -- a backward Euler scheme with varying time steps. The fractional Laplacian is a non-local operator and the arising stiffness matrix is dense. The time dependent problem is reduced to solving a sequence of linear systems whose matrices are constructed from the stiffness matrix, lumped mass matrix and the time step. When the time step changes we must refactorize the matrix before solving the current system. If the time step doesn't change we can solve with the matrix factorized on a previous time step change. When utilizing the generic method using a block LU factorization, the computational complexity of the forward elimination is \(O(n^3)\) and \(O(n^2)\) of the backward substitution. In this work we develop an alternative method based on the hierarchically semi-separable (HSS) compression. With this method we compress the matrix at the beginning only. The HSS compression has a computational complexity \(O(n^2r)\). Then, when the time step changes we need to apply ULV-like factorization with computational complexity of \(O(nr^2)\). The solution step with the factorized matrix at each time step has computational complexity of \(O(nr)\). Here, \(r\) is the maximum off-diagonal rank of the approximate matrix, which is computed during the compression process. For suitable problems \(r\) is much smaller than the number of unknowns \(n\). The numerical experiments presented show the advantages of the developed HSS compression based solution method.
For the entire collection see [Zbl 1511.65004].Mass-, energy-, and momentum-preserving spectral scheme for Klein-Gordon-Schrödinger system on infinite domainshttps://zbmath.org/1521.650972023-11-13T18:48:18.785376Z"Guo, Shimin"https://zbmath.org/authors/?q=ai:guo.shimin"Mei, Liquan"https://zbmath.org/authors/?q=ai:mei.liquan"Yan, Wenjing"https://zbmath.org/authors/?q=ai:yan.wenjing"Li, Ying"https://zbmath.org/authors/?q=ai:li.ying.6This paper proposes a finite difference / spectral method for the approximation of the solutions of the Klein-Gordon-Schrödinger system on infinite domains. More exactly, this paper considers the following system:
\begin{align*}
i \frac{\partial u}{\partial t} + \frac{\kappa_1}{2} \Delta u + \gamma u v = 0 \\
\frac{\partial^2v}{\partial t^2} - \kappa_2 \Delta v + \mu^2 v - \gamma |u|^2 = 0 \\
u(x, 0) = u_0, \quad v(x,0) = v_0, \quad \frac{\partial v}{\partial t}(x,0) = v_0^\diamond \\
\lim_{|x| \to \infty} u = \lim_{|x| \to \infty} v = 0, \quad t \ge 0.
\end{align*}
Combining the exponential scalar variable and Lagrange multiplier technique, the proposed numerical scheme preserves the mass, the momentum and the energy of the system. Numerical simulations illustrate the accuracy of the approximation method and the conservation of the system's invariants.
Reviewer: Nicolae Cîndea (Aubière)An efficient approach for solving nonlinear multidimensional Schrödinger equationshttps://zbmath.org/1521.650992023-11-13T18:48:18.785376Z"Karabaş, Neslişah İmamoğlu"https://zbmath.org/authors/?q=ai:karabas.neslisah-imamoglu"Korkut, Sıla Övgü"https://zbmath.org/authors/?q=ai:korkut.sila-ovgu"Tanoğlu, Gamze"https://zbmath.org/authors/?q=ai:tanoglu.gamze-b"Aziz, Imran"https://zbmath.org/authors/?q=ai:aziz.imran"Siraj-ul-Islam"https://zbmath.org/authors/?q=ai:siraj-ul-islam.Summary: An efficient numerical method is proposed for the solution of the nonlinear cubic Schrödinger equation. The proposed method is based on the Fréchet derivative and the meshless method with radial basis functions. An important characteristic of the method is that it can be extended from one-dimensional problems to multi-dimensional ones easily. By using the Fréchet derivative and Newton-Raphson technique, the nonlinear equation is converted into a set of linear algebraic equations which are solved iteratively. Numerical examples reveal that the proposed method is efficient and reliable with respect to the accuracy and stability.An RBF-FD method for the time-fractional advection-dispersion equation with nonlinear source termhttps://zbmath.org/1521.651012023-11-13T18:48:18.785376Z"Londoño, Mauricio A."https://zbmath.org/authors/?q=ai:londono.mauricio-a"Giraldo, Ramón"https://zbmath.org/authors/?q=ai:giraldo.ramon"Rodríguez-Cortés, Francisco J."https://zbmath.org/authors/?q=ai:rodriguez-cortes.francisco-jSummary: Fractional advection-dispersion equations have proved to be useful for modeling a wide range of problems in environmental and engineering sciences. In this work, we adapt a Radial Basis Function-generated Finite Difference (RBF-FD) method to obtain approximated numerical solutions of the initial-boundary value problem of the time-fractional advection-dispersion equation with variable coefficients and nonlinear source. We use a strategy of minimization of the local truncation error in approximating the initial condition to find appropriate local shape parameters for the Gaussian RBF. For discretizing the fractional time derivative, in the Caputo's sense, we use a scheme of \(( 4 - \alpha )\) th-order, where \(\alpha \in (0,1)\) is the order of the fractional derivative. We evaluate the performance of the RBF-FD method for different 2-dimensional problems on domains with complex geometries in which a high precision is exhibited of the solutions found. Particularly, we test our method for approximating solutions of Fisher's equation with fractional time derivative, where the source that depends on the solution, is approximated by a linearization with respect to time, and we obtain a rate of convergence of second order in time.A new high-order nine-point stencil, based on integrated-RBF approximations, for the first biharmonic equationhttps://zbmath.org/1521.651022023-11-13T18:48:18.785376Z"Mai-Duy, N."https://zbmath.org/authors/?q=ai:mai-duy.nam"Strunin, D."https://zbmath.org/authors/?q=ai:strunin.dmitry-v"Karunasena, W."https://zbmath.org/authors/?q=ai:karunasena.w(no abstract)An efficient localized meshless technique for approximating nonlinear sinh-Gordon equation arising in surface theoryhttps://zbmath.org/1521.651042023-11-13T18:48:18.785376Z"Nikan, O."https://zbmath.org/authors/?q=ai:nikan.omid"Avazzadeh, Z."https://zbmath.org/authors/?q=ai:avazzadeh.zakiehSummary: This paper adopts an efficient localized meshless technique for computing the solution of the nonlinear sinh-Gordon equation (NShGE). The NShGE is one useful description for many natural processes such as solid state physics, surface theory, fluid dynamics, nonlinear optics and dislocation in materials. In the proposed method, at the first step, a second-order accurate formulation is implemented to obtain the temporal discretization. At the second step, a localized collocation meshless technique based on the radial basis function partition of unity is proposed to derive the spatial discretization. A major drawback associated with global collocation techniques is the computational cost due to resulting dense algebraic system. The localized technique tackles the ill-conditioning inherent in global collocation techniques and reduces the associated computational cost. It is shown that the proposed method is stable and second-order convergent with respect to the time variable. Numerical results and comparisons illustrate the high accuracy of the proposed method.Soliton wave solutions of nonlinear mathematical models in elastic rods and bistable surfaceshttps://zbmath.org/1521.651052023-11-13T18:48:18.785376Z"Nikan, O."https://zbmath.org/authors/?q=ai:nikan.omid"Avazzadeh, Z."https://zbmath.org/authors/?q=ai:avazzadeh.zakieh"Rasoulizadeh, M. N."https://zbmath.org/authors/?q=ai:rasoulizadeh.mohammad-navaz(no abstract)Coupled Kansa and hybrid optimization methodological approach for Kolmogorov-Feller equationshttps://zbmath.org/1521.651062023-11-13T18:48:18.785376Z"Salleh, Ihsane"https://zbmath.org/authors/?q=ai:salleh.ihsane"Belkourchia, Yassin"https://zbmath.org/authors/?q=ai:belkourchia.yassin"Azrar, Lahcen"https://zbmath.org/authors/?q=ai:azrar.lahcen(no abstract)Numerical approximation of time-dependent fractional convection-diffusion-wave equation by RBF-FD methodhttps://zbmath.org/1521.651082023-11-13T18:48:18.785376Z"Zhang, Xindong"https://zbmath.org/authors/?q=ai:zhang.xindong"Yao, Lin"https://zbmath.org/authors/?q=ai:yao.linSummary: In this paper, a method based on radial basis function finite difference (RBF-FD) is developed for solving the time fractional convection-diffusion-wave equation (TFCDWE). We first approximate the equation by a scheme of order \(O(\tau+h^2)\), where \(\tau,h\) are the time step size and spatial step size, respectively. We prove the stability and convergence of the discrete scheme, then the multiquadric RBF-FD approach is used to approximate the spatial derivatives. The aim of this paper is to show that the RBF-FD method is useful for solving our mentioned equation when the shape parameter selection is appropriate. The proposed method can be applied to complex domain, and has the advantages of mesh-free and simple procedure. Finally, numerical examples are proposed to verify the correctness of our previous theoretical analysis and to demonstrate the superiority of the RBF-FD method.Two space-time methods for solving Burgers' equationhttps://zbmath.org/1521.651102023-11-13T18:48:18.785376Z"Cao, Yanhua"https://zbmath.org/authors/?q=ai:cao.yanhua"Li, Nan"https://zbmath.org/authors/?q=ai:li.nan"Zhang, Zitong"https://zbmath.org/authors/?q=ai:zhang.zitong"Fan, Zizhu"https://zbmath.org/authors/?q=ai:fan.zizhuSummary: Burgers' equation is a kind of quasi-linear important partial differential equation appeared in many fields. The nonlinear term contained in this equation results in difficulty in finding its high precision numerical solution. Most existed research focused on one-dimensional and two-dimensional Burgers' systems. Seldom research studied the three-dimensional Burgers' problem. In this paper, two space-time methods based on the polynomial particular solutions (ST-MPPS) and the polynomial used as basis functions directly, have been proposed to solve three-dimensional Burgers' equation at different final time with different Reynolds numbers. The numerical examples show that the greatest attraction of the ST-MPPS is its high precision and robustness. When using the space-time polynomial functions method, some interesting phenomenon appears with the Reynolds number increasing.Continuous time limit of the stochastic ensemble Kalman inversion: strong convergence analysishttps://zbmath.org/1521.651172023-11-13T18:48:18.785376Z"Blömker, Dirk"https://zbmath.org/authors/?q=ai:blomker.dirk"Schillings, Claudia"https://zbmath.org/authors/?q=ai:schillings.claudia"Wacker, Philipp"https://zbmath.org/authors/?q=ai:wacker.philipp"Weissmann, Simon"https://zbmath.org/authors/?q=ai:weissmann.simonThe ensemble Kalman filter (EnKF) for inverse problems, also known as ensemble Kalman inversion (EKI) is considered. The main focus of this work is to theoretically verify the convergence of the discrete EKI method to its continuous time formulation. The article is organized as follows. Section 1 is an Introduction. A brief overview of the existing literature, Mathematical setup, EKI: The EnKF applied to inverse problems, and an Outline of the paper are given in Subsections 1.1--1.4. In Section 2, the authors' general numerical approximation results for stochastic differential equations are presented, which are then applied to the solution of general nonlinear inverse problems with the EKI method in Section 3. The application to linear inverse problems is presented in Section 4. Some conclusions are presented in the last Section 5. Discussing possible further directions to take are given. Most of the authors' proofs are shifted to the appendix in order to keep the focus on the key contribution presented in this article.
Reviewer: Temur A. Jangveladze (Tbilisi)Boundary integral formulation of the standard eigenvalue problem for the 2-D Helmholtz equationhttps://zbmath.org/1521.651202023-11-13T18:48:18.785376Z"Karimaghaei, M."https://zbmath.org/authors/?q=ai:karimaghaei.m"Phan, A.-V."https://zbmath.org/authors/?q=ai:phan.anh-vuSummary: In this paper, a boundary integral formulation is presented for obtaining the standard eigenvalue problem for the two-dimensional (2-D) Helmholtz equation. The formulation is derived by using the series expansions of zero-order Bessel functions for the fundamental solution to the Helmholtz equation. The proposed approach leads to a series of new fundamental functions which are independent of the wave number \(k\) of the Helmholtz equation. The coefficient matrix of the resulting homogeneous system of boundary element equations is of the form of a polynomial matrix in \(k\) which allows a much faster search for the eigenvalues by scanning \(k\) over an interval of interest or the standard eigenvalue problem to be formulated for directly solving for the eigenvalues without resort to iterative methods. The proposed technique was used to solve some known problems with available analytical solutions: 2-D domains with circular and rectangular geometries under Dirichlet and/or Neumann boundary conditions. The outcomes demonstrate that the proposed approach is computationally efficient and highly accurate.Robust BPX preconditioner for fractional Laplacians on bounded Lipschitz domainshttps://zbmath.org/1521.651232023-11-13T18:48:18.785376Z"Borthagaray, Juan Pablo"https://zbmath.org/authors/?q=ai:borthagaray.juan-pablo"Nochetto, Ricardo H."https://zbmath.org/authors/?q=ai:nochetto.ricardo-h"Wu, Shuonan"https://zbmath.org/authors/?q=ai:wu.shuonan"Xu, Jinchao"https://zbmath.org/authors/?q=ai:xu.jinchaoSummary: We propose and analyze a robust Bramble-Pasciak-Xu (BPX) preconditioner for the integral fractional Laplacian of order \(s \in (0, 1)\) on bounded Lipschitz domains. Compared with the standard BPX preconditioner, an additional scaling factor \(1-\widetilde{\gamma}^s\), for some fixed \(\widetilde{\gamma} \in (0, 1)\), is incorporated to the coarse levels. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain uniformly bounded with respect to both the number of levels and the fractional power.On the finite element approximation of a semicoercive Stokes variational inequality arising in glaciologyhttps://zbmath.org/1521.651242023-11-13T18:48:18.785376Z"de Diego, Gonzalo G."https://zbmath.org/authors/?q=ai:g-de-diego.gonzalo"Farrell, Patrick E."https://zbmath.org/authors/?q=ai:farrell.patrick-e|farrell.patrick-emmet"Hewitt, Ian J."https://zbmath.org/authors/?q=ai:hewitt.ian-jIn this work, the problem of a marine ice sheet resting on a bedrock and sliding into the ocean, where it goes afloat is considered. The suitable contact conditions transform the instantaneous Stokes problem into a variational inequality. The article is outlined as follows. Section 1 is an Introduction. In Section 2, the Stokes variational inequality and its mixed formulation are presented. A Korn-type inequality is proved and is demonstrated that the mixed formulation is well-posed. In Section 3, a family of finite element approximations of the mixed problem is analyzed and error estimates in terms of best approximation results for the velocity, pressure, and Lagrange multiplier are presented. In Section 4, a concrete finite element scheme involving quadratic elements for the velocity and piecewise-constant elements for the pressure and the Lagrange multiplier is introduced. Then error estimates for this scheme are presented and a problem with a manufactured solution to calculate convergence rates and compare these with the authors' estimates is solved. Numerical results are reported to validate the error estimates. Some conclusions are given in Section 5. Finally, Appendices A, B, B.1, B.2, and B.3 with fixing equivalence of formulations and technical results on finite element spaces are given.
Reviewer: Temur A. Jangveladze (Tbilisi)Least-squares finite elements for distributed optimal control problemshttps://zbmath.org/1521.651252023-11-13T18:48:18.785376Z"Führer, Thomas"https://zbmath.org/authors/?q=ai:fuhrer.thomas"Karkulik, Michael"https://zbmath.org/authors/?q=ai:karkulik.michaelSummary: We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is \(\inf\)-\(\sup\) stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented.Local maximum-entropy approximation based stabilization methods for the convection diffusion problemshttps://zbmath.org/1521.651312023-11-13T18:48:18.785376Z"Peddavarapu, Sreehari"https://zbmath.org/authors/?q=ai:peddavarapu.sreehari"Srinivasan, Raghuraman"https://zbmath.org/authors/?q=ai:srinivasan.raghuraman(no abstract)Elastic transmission eigenvalues and their computation via the method of fundamental solutionshttps://zbmath.org/1521.651362023-11-13T18:48:18.785376Z"Kleefeld, A."https://zbmath.org/authors/?q=ai:kleefeld.andreas"Pieronek, L."https://zbmath.org/authors/?q=ai:pieronek.lukasSummary: A stabilized version of the fundamental solution method to catch ill-conditioning effects is investigated with focus on the computation of complex-valued elastic interior transmission eigenvalues in two dimensions for homogeneous and isotropic media. Its algorithm can be implemented very shortly and adopts to many similar partial differential equation-based eigenproblems as long as the underlying fundamental solution function can be easily generated. We develop a corroborative approximation analysis which also implicates new basic results for transmission eigenfunctions and present some numerical examples which together prove successful feasibility of our eigenvalue recovery approach.Boundary metrics on soliton moduli spaceshttps://zbmath.org/1521.700362023-11-13T18:48:18.785376Z"Sutcliffe, Paul"https://zbmath.org/authors/?q=ai:sutcliffe.paul-j|sutcliffe.paul-mSummary: The geodesic approximation is a powerful method for studying the dynamics of BPS solitons. However, there are systems, such as BPS monopoles in three-dimensional hyperbolic space, where this approach is not applicable because the moduli space metric defined by the kinetic energy is not finite. In the case of hyperbolic monopoles, an alternative metric has been defined using the abelian connection on the sphere at infinity, but its relation to the dynamics of hyperbolic monopoles is unclear. Here this metric is placed in a more general context of boundary metrics on soliton moduli spaces. Examples are studied in systems in one and two space dimensions, where it is much easier to compare the results with simulations of the full nonlinear field theory dynamics. It is found that geodesics of the boundary metric provide a reasonable description of soliton dynamics.Mixed boundary conditions as limits of dissipative boundary conditions in dynamic perfect plasticityhttps://zbmath.org/1521.740282023-11-13T18:48:18.785376Z"Babadjian, Jean-Francois"https://zbmath.org/authors/?q=ai:babadjian.jean-francois"Llerena, Randy"https://zbmath.org/authors/?q=ai:llerena.randySummary: This paper addresses the well posedness of a dynamical model of perfect plasticity with mixed boundary conditions for general closed and convex elasticity sets. The proof relies on an asymptotic analysis of the solution of a perfect plasticity model with relaxed dissipative boundary conditions obtained by \textit{J.-F. Babadjian} and \textit{V. Crismale} [J. Math. Pures Appl. (9) 148, 75--127 (2021; Zbl 1461.74010)]. One of the main issues consists in extending the measure theoretic duality pairing between stresses and plastic strains, as well as a convexity inequality to a more general context where deviatoric stresses are not necessarily bounded. Complete answers are given in the pure Dirichlet and pure Neumann cases. For general mixed boundary conditions, partial answers are given in dimension 2 and 3 under additional geometric hypothesis on the elasticity set and the reference configuration.Fluid-plate interaction with Kelvin-Voigt damping and bending moment at the interface: well-posedness, spectral analysis, uniform stabilityhttps://zbmath.org/1521.740532023-11-13T18:48:18.785376Z"Mahawattege, Rasika"https://zbmath.org/authors/?q=ai:mahawattege.rasika"Triggiani, Roberto"https://zbmath.org/authors/?q=ai:triggiani.robertoSummary: We consider a fluid-plate interaction model where the two dimensional plate is subject to viscoelastic (strong) damping, as it occurs in some biological systems [\textit{N. Özkaya} et al., Fundamentals of biomechanics-equilibrium, motion, and deformation. New York, NY: Springer (2021)]. The strength of the Kelvin-Voigt damping is measured by a constant \(0 < \rho \leq 1\). Coupling occurs at the interface between the two media, where each component evolves. In this paper, we apply ``low'' physically hinged boundary interface conditions, which involve the bending moment operator for the plate. We prove four main results: (1) analyticity, on the natural energy space, of the corresponding contraction semigroup (and of its adjoint); (2) sharp location of the spectrum of its generator (and similarly of the adjoint generator), neither of which has compact resolvent, and in fact both of which have the point \(\lambda = -\frac{1}{\rho}\) in their respective continuous spectrum; (3) both original generator and its adjoint have the origin \(\lambda = 0\) as a common eigenvalue with a common, explicit, 1-dimensional eigenspace; (4) The subspace of codimension 1 obtained by the original energy space by factoring out the common 1-dimensional eigenspace is invariant under the action of the (here restricted) semigroup (or of its adjoint), and on such subspace both original and adjoint semigroups are uniformly stable.
For the entire collection see [Zbl 1517.47001].Matrix Green's function solution of closed-form singularity for functionally graded and transversely isotropic materials under circular ring force vectorhttps://zbmath.org/1521.740742023-11-13T18:48:18.785376Z"Xiao, Sha"https://zbmath.org/authors/?q=ai:xiao.sha"Yue, Zhongqi Quentin"https://zbmath.org/authors/?q=ai:yue.zhongqi-quentin(no abstract)Remarks on homogenization and \(3D\)-\(2D\) dimension reduction of unbounded energies on thin filmshttps://zbmath.org/1521.741862023-11-13T18:48:18.785376Z"Anza Hafsa, Omar"https://zbmath.org/authors/?q=ai:anza-hafsa.omar"Mandallena, Jean-Philippe"https://zbmath.org/authors/?q=ai:mandallena.jean-philippeThe authors consider a heterogeneous thin film occupying in a reference configuration the bounded and open subset \(\Sigma _{\varepsilon }=\Sigma \times \left[ -\frac{\varepsilon ^{\gamma }}{2},\frac{\varepsilon ^{\gamma } }{2}\right] \) of \(\mathbb{R}^{3}\), where \(\gamma \in ]0,\infty \lbrack \) is fixed, \(\varepsilon >0\), and \(\Sigma \) is a Lipschitz, open and bounded subset of \(\mathbb{R}^{2}\). The stored-energy density of the material filling \(\Sigma _{\varepsilon }\) is a Borel measurable function \(W:\mathbb{R} ^{2}\times \mathbb{M}^{3\times 3}\rightarrow \lbrack 0,\infty ]\) satisfying: \(W\) is \(p\)-coercive with \(p>1\), i.e. there exists \(C>0\) such that \( W(x,F)\geq C\left\vert F\right\vert ^{p}\) for all \((x,F)\in \mathbb{R} ^{2}\times \mathbb{M}^{3\times 3}\), \(W\) is \(1\)-periodic with respect to \(x\), i.e., \(W(x+z,F)=W(x,F)\) for all \((x,F)\in \mathbb{R}^{2}\times \mathbb{M} ^{3\times 3}\) and all \(z\in \mathbb{Z}^{2}\), \(W(x,F)\rightarrow \infty \) as \( \det F\rightarrow 0\), \(W\) is \(p\)-ample, i.e., there exists \(c>0\) such that \( \mathcal{Z}W(x,F)\leq c(1+\left\vert F\right\vert ^{p})\) for all \((x,F)\in \mathbb{R}^{2}\times \mathbb{M}^{3\times 3}\), where \(\mathcal{Z}W:\mathbb{R} ^{2}\times \mathbb{M}^{3\times 3}\rightarrow \lbrack 0,\infty ]\) is defined by \(\mathcal{Z}W(x,F)=\inf\left\{ \int_{Y}W(x,F+\nabla \varphi (x,x_{3}))dxdx_{3}:\varphi \in W_{0}^{1,\infty }(]-\frac{1}{2},\frac{1}{2} [^{3};\mathbb{R}^{3})\right\} \), there exists \(\lambda \in \mathcal{L}_{2}\) such that for every \(x,x^{\prime }\in \mathbb{R}^{2}\) and every \(F\in \mathbb{M}^{3\times 3}\), \(W(x,F)\leq \left\vert \lambda (x)-\lambda (x^{\prime })\right\vert (1+W(x^{\prime },F))+W(x^{\prime },F)\), and there exist a finite family \(\{V_{j}\}_{j\in J}\) of open disjoint subsets of \( \mathbb{R}^{2}\), with \(\mathcal{L}^{2}(\partial V_{j})=0\) for all \(j\in J\) and \(\mathcal{L}^{2}(\mathbb{R}^{2}\setminus \cup _{j\in J}V_{j})=0\), and a finite family \(\{H_{j}:\mathbb{M}^{3\times 3}\rightarrow \lbrack 0,\infty ]\}_{j\in J}\) of Borel measurable functions such that \(W(x,F)=\sum_{j\in J}1_{V_{j}}(x)H_{j}(F)\). Here \(\mathcal{L}_{2}\) is the class of \(\lambda \in L^{\infty }(\mathbb{R}^{2};[0,\infty \lbrack )\) such that \(\lambda \) is continuous almost everywhere with respect to \(\mathcal{L}^{2}\). The authors define the energy functional of the material \(E_{\varepsilon }:W^{1,p}(\Sigma _{\varepsilon };\mathbb{R}^{3})\rightarrow \lbrack 0,\infty ]\) (with \(p>1\)) as \(E_{\varepsilon }(u)=\frac{1}{\varepsilon ^{\gamma }} \int_{\Sigma _{\varepsilon }}W(\frac{x}{\varepsilon },\nabla u(x,x_{3})dxdx_{3}\).
The main result of the paper proves that if \(W\) satisfies the above hypotheses, the energy \(E_{\varepsilon }\) \(\Gamma (\pi )\)-converges to \(\overline{E}:W^{1,p}(\Sigma _{\varepsilon };\mathbb{R} ^{3})\rightarrow \lbrack 0,\infty ]\) defined as \(\overline{E} (v):=\int_{\Sigma }\overline{W}(\nabla v(x))dx\), with \(\overline{W}:\mathbb{M }^{3\times 2}\rightarrow \lbrack 0,\infty ]\), which is given by three expressions involving minimizations, according to the position of \( \gamma \) with respect to 1. The authors first recall the definition of the \( \Gamma (\pi )\)-convergence introduced by \textit{G. Anzellotti} et al. [Asymptotic Anal. 9, No. 1, 61--100 (1994; Zbl 0811.49020)], and which is used for dimension reduction problems in mechanics. The proof of the main result first relies on a similar result obtained in a bounded case and which is deduced applying tools obtained by \textit{A. Braides} et al. [Indiana Univ. Math. J. 49, No. 4, 1367--1404 (2000; Zbl 0987.35020)] and by \textit{Y.C. Shu} in [Arch. Ration. Mech. Anal. 153, No. 1, 39--90 (2000; Zbl 0959.74043)], then proving that \(\mathcal{Z}W(x,.)\) is quasiconvex, whence \(p\)-locally Lipschitz. The note ends with the description of examples.
Reviewer: Alain Brillard (Riedisheim)A finite point method for the fractional cable equation using meshless smoothed gradientshttps://zbmath.org/1521.742312023-11-13T18:48:18.785376Z"Li, Xiaolin"https://zbmath.org/authors/?q=ai:li.xiaolin.1"Li, Shuling"https://zbmath.org/authors/?q=ai:li.shulingSummary: This paper presents a meshless finite point method (FPM) for the numerical analysis of the fractional cable equation. A second-order time discrete scheme is proposed to approximate both integer-order and fractional-order time derivatives. Then, based on the stabilized moving least squares approximation and the meshless smoothed gradients, a new implementation of the FPM is provided to enhance the accuracy and convergence rate in space. Theoretical error of the FPM is analyzed. Numerical results verify the efficiency of the method and show that the method can gain second-order accuracy in time and fourth-order accuracy in space.An LT-BEM formulation for problems of anisotropic functionally graded materials governed by transient diffusion-convection-reaction equationhttps://zbmath.org/1521.742892023-11-13T18:48:18.785376Z"Azis, M. I."https://zbmath.org/authors/?q=ai:azis.mohammad-ivan(no abstract)The direct interpolation boundary element method and the domain superposition technique applied to piecewise Helmholtz's problems with internal heterogeneityhttps://zbmath.org/1521.742902023-11-13T18:48:18.785376Z"Barcelos, Hercules de Melo"https://zbmath.org/authors/?q=ai:de-melo-barcelos.hercules"Loeffler, Carlos Friedrich"https://zbmath.org/authors/?q=ai:loeffler.carlos-friedrich"Lara, Luciano de Oliveira Castro"https://zbmath.org/authors/?q=ai:lara.luciano-de-oliveira-castroSummary: This work presents the combination of the direct interpolation boundary element method (DIBEM) and the domain superposition technique (DST) to address piecewise inhomogeneous two dimensional Helmholtz problems, in which the internal constitutive property in each sector varies smoothly according to a known function. The domain integrals generated by the medium's heterogeneity are transformed into boundary integrals according to the DIBEM strategy where radial basis functions are used. The DST is applied to generate the influence coefficients related to the several sectors and compute them in the final matrix system. Thus, the proposed methodology preserves the main features and advantages of the Boundary Element Method. Concerning the evaluation of the numerical results, three different tests are performed, considering regular and irregular domains. For each example, benchmarks are generated by correlate simulations using the Finite Element Method.DEM-BEM coupling in time domain for one-dimensional wave propagationhttps://zbmath.org/1521.742912023-11-13T18:48:18.785376Z"Barros, Guilherme"https://zbmath.org/authors/?q=ai:barros.guilherme"Pereira, Andre"https://zbmath.org/authors/?q=ai:pereira.andre-maues-brabo"Rojek, Jerzy"https://zbmath.org/authors/?q=ai:rojek.jerzy"Thoeni, Klaus"https://zbmath.org/authors/?q=ai:thoeni.klaus(no abstract)A multi-domain BEM based on dual interpolation boundary face method for 3D elasticity problemhttps://zbmath.org/1521.742992023-11-13T18:48:18.785376Z"Chai, Pengfei"https://zbmath.org/authors/?q=ai:chai.pengfei"Zhang, Jianming"https://zbmath.org/authors/?q=ai:zhang.jianming"Xiao, Rongxiong"https://zbmath.org/authors/?q=ai:xiao.rongxiong"He, Rui"https://zbmath.org/authors/?q=ai:he.rui"Lin, WeiCheng"https://zbmath.org/authors/?q=ai:lin.weicheng(no abstract)Boundary element procedure for computation of internal directional derivatives in homogeneous Laplace's problems solved by the finite element methodhttps://zbmath.org/1521.743102023-11-13T18:48:18.785376Z"de Melo Barcelos, Hercules"https://zbmath.org/authors/?q=ai:de-melo-barcelos.hercules"Loeffler, Carlos Friedrich"https://zbmath.org/authors/?q=ai:loeffler.carlos-friedrich"Lara, Luciano de Oliveira Castro"https://zbmath.org/authors/?q=ai:lara.luciano-de-oliveira-castro"Mansur, Webe João"https://zbmath.org/authors/?q=ai:mansur.webe-joao(no abstract)Boundary element modeling of fractional nonlinear generalized photothermal stress wave propagation problems in FG anisotropic smart semiconductorshttps://zbmath.org/1521.743172023-11-13T18:48:18.785376Z"Fahmy, Mohamed Abdelsabour"https://zbmath.org/authors/?q=ai:fahmy.mohamed-abdelsabourSummary: The main aim of this article is to develop an efficient boundary element method (BEM) modeling of the fractional nonlinear generalized photo thermal stress wave propagation problems in the context of functionally graded (FG) anisotropic smart semiconductors. Due to nonlinearity, fractional order heat conduction and strongly anisotropy of mechanical properties, the governing equations system of such problems is often very difficult to solve using classical analytical methods. Therefore, a reliable and efficient coupling scheme based on BEM was proposed to address this challenge, where, the Cartesian transformation method (CTM) has been implemented to calculate the domain integrals, and the generalized modified shift-splitting (GMSS) has been implemented for solving the linear systems arising from BEM. The calculation findings are depicted in graphical forms to display the impacts of temperature-dependent, anisotropy, piezoelectric, graded parameter and fractional parameter on the nonlinear photo thermal stress wave propagation in the considered structure. The numerical findings confirm the consistency and efficacy of the developed modeling methodology.Localized singular boundary method for solving Laplace and Helmholtz equations in arbitrary 2D domainshttps://zbmath.org/1521.743602023-11-13T18:48:18.785376Z"Wang, Fajie"https://zbmath.org/authors/?q=ai:wang.fajie"Chen, Zengtao"https://zbmath.org/authors/?q=ai:chen.zengtao"Li, Po-Wei"https://zbmath.org/authors/?q=ai:li.po-wei"Fan, Chia-Ming"https://zbmath.org/authors/?q=ai:fan.chia-mingSummary: In this research, the localized singular boundary method (LSBM) is proposed to solve the Laplace and Helmholtz equations in 2D arbitrary domains. In the traditional SBM, the resultant matrix system is a dense matrix, and it is unsuited for solving the large-scale problems. As a localized domain-type meshless method, a local subdomain for every node can be composed by its own and several nearest nodes. To each of the subdomains, the SBM formulation is applied to derive an implicit expression of the variable at each node in conjunction with the moving least-square approximation. To satisfy the boundary conditions at every boundary node and the governing equation at every node, a sparse linear algebraic system can be obtained. Thus, the numerical solutions at all nodes can be achieved by solving it. Compared with the traditional SBM, the LSBM involves only the origin intensity factor on a circular boundary associated with Dirichlet boundary conditions. It can also effectively avoid the boundary layer effect in the conventional SBM. Furthermore, the proposed LSBM requires less memory storage and computational cost due to the sparse and banded matrix system. Several numerical examples are tested to verify the accuracy and stability of the proposed LSBM.On the static, vibration, and transient responses of micro-plates made of materials with different microstructureshttps://zbmath.org/1521.743912023-11-13T18:48:18.785376Z"Roque, C. M. C."https://zbmath.org/authors/?q=ai:roque.c-m-c"Żur, Krzysztof Kamil"https://zbmath.org/authors/?q=ai:zur.krzysztof-kamil(no abstract)A meshless collocation method for solving the inverse Cauchy problem associated with the variable-order fractional heat conduction model under functionally graded materialshttps://zbmath.org/1521.744162023-11-13T18:48:18.785376Z"Hu, Wen"https://zbmath.org/authors/?q=ai:hu.wen"Fu, Zhuojia"https://zbmath.org/authors/?q=ai:fu.zhuojia"Tang, Zhuochao"https://zbmath.org/authors/?q=ai:tang.zhuochao"Gu, Yan"https://zbmath.org/authors/?q=ai:gu.yan(no abstract)RANS-based numerical simulation of shock wave/turbulent boundary layer interaction induced by a blunted fin normal to a flat platehttps://zbmath.org/1521.760172023-11-13T18:48:18.785376Z"Kolesnik, Elizaveta"https://zbmath.org/authors/?q=ai:kolesnik.elizaveta"Smirnov, Evgueni"https://zbmath.org/authors/?q=ai:smirnov.evgueni-n|smirnov.evgueni-m"Smirnovsky, Alexander"https://zbmath.org/authors/?q=ai:smirnovsky.alexander-aSummary: Results of refined RANS-based numerical simulation of shock wave/turbulent boundary layer interaction induced by a blunted fin mounted on a flat plate are presented. The RANS equations for compressible viscous gas flow closed by a (varied) eddy-viscosity turbulence model were solved using the finite volume method implemented in the unstructured-grid parallelized in-house code SINF/Flag-S developed at the SPbPU. Detailed numerical data concerning time-averaged characteristics of the separation region and the shock wave/turbulent boundary layer interaction are acquired, including identification of the horse-shoe vortices and their influence on the pressure and shear stress distributions on the streamlined surfaces. A thorough comparison with measurements presented in literature were conducted. Methodical data on the influence of grid resolution and turbulence model on the predicted structure of the viscous-inviscid interaction are obtained and recommendations for achieving a high-quality resolution of the viscous effects and the complex shock-wave pattern of the flow are worked out.Solution of two dual problems of gluing vorter and potential flows by M. A. Goldshtick variational methodhttps://zbmath.org/1521.760462023-11-13T18:48:18.785376Z"Vaĭnshteĭn, Isaak I."https://zbmath.org/authors/?q=ai:vainshtein.isaak-iSummary: A general problem of motion of incompressible liquid with vortex zones with different constant vorticity is formulated. It is considered the M. A. Goldshtic variational method of the research of dual problems for flows with vortex and potential areas that describe the model of separated flows and the model of ideal liquid motion in a field of Coriolis forces. It is proved the existence of the second nontrivial solution to the M. A. Goldshtick problem.Highest cusped waves for the Burgers-Hilbert equationhttps://zbmath.org/1521.760552023-11-13T18:48:18.785376Z"Dahne, Joel"https://zbmath.org/authors/?q=ai:dahne.joel"Gómez-Serrano, Javier"https://zbmath.org/authors/?q=ai:gomez-serrano.javierSummary: In this paper we prove the existence of a periodic highest, cusped, traveling wave solution for the Burgers-Hilbert equation \(f_t + f f_x = {\mathbf{H}}[f]\), and give its asymptotic behaviour at 0. The proof combines careful asymptotic analysis and a computer-assisted approach.The interaction of a mode-1 internal solitary wave with a step and the generation of mode-2 waveshttps://zbmath.org/1521.760662023-11-13T18:48:18.785376Z"Liu, Zihua"https://zbmath.org/authors/?q=ai:liu.zihua"Grimshaw, Roger"https://zbmath.org/authors/?q=ai:grimshaw.roger-h-j"Johnson, Edward"https://zbmath.org/authors/?q=ai:johnson.edward-rSummary: In this study, we examine the transformation of a mode-1 internal solitary wave incident on a bottom step, and the consequent generation of mode-2 internal solitary waves. A linear long-wave theory of mode coupling in the vicinity of the step is used to estimate the mode-1 and mode-2 wave reflection and transmission coefficients, and hence the energy fluxes. Away from the step, the wave evolution of the transmitted and reflected waves is simulated by the Korteweg-de Vries equation. Specific calculations are made using a three-layer fluid model. Three different regimes based on the layer thicknesses are examined and discussed in detail for either depression or elevation mode-1 incident waves. The common features found are that the transmitted waves (mainly mode-1) are the dominant part; most of the incident energy is transmitted and only a small part is reflected. The amplitudes of the generated mode-2 waves and the reflected mode-1 waves increase when either the upper- or middle-layer thickness increases. When the lower layer is thin enough, the amplitude of the transmitted mode-2 wave can be larger than the mode-1 waves, and the reflected energy can increase considerably which we infer may be due to a blocking effect of the step on the lower layer. The evolution away from the step is either fission into several solitary waves, or the development of a rarefaction wave followed by an undular bore, depending on the relative signs of the wave amplitudes and the nonlinear coefficient in the Korteweg-de Vries equation.Complete integrability of a new class of Hamiltonian hydrodynamic type systemshttps://zbmath.org/1521.760762023-11-13T18:48:18.785376Z"Makridin, Z. V."https://zbmath.org/authors/?q=ai:makridin.zakhar-v"Pavlov, M. V."https://zbmath.org/authors/?q=ai:pavlov.maxim-vSummary: In this paper, we consider a new class of Hamiltonian hydrodynamic type systems whose conservation laws are polynomial with respect to one of the field variables.On the local pressure expansion for the Navier-Stokes equationshttps://zbmath.org/1521.760792023-11-13T18:48:18.785376Z"Bradshaw, Zachary"https://zbmath.org/authors/?q=ai:bradshaw.zachary"Tsai, Tai-Peng"https://zbmath.org/authors/?q=ai:tsai.tai-pengThe authors discuss relations between distributional, mild and Leray solutions to the three-dimensional Navier-Stokes equations via an analysis of the so-called ``local pressure expansion''. This concept, based on delicate properties of BMO solutions of the Poisson equation, leads to a comparison of various local regularity properties of Navier-Stokes equations solution, independent of Lemarié-Rieusset's Littlewood-Paley decomposition of the pressure. Various applications to uniqueness and regularity questions are also discussed.
Reviewer: Piotr Biler (Wrocław)Asymptotic behavior of weak solutions to the inhomogeneous Navier-Stokes equationshttps://zbmath.org/1521.760802023-11-13T18:48:18.785376Z"Han, Pigong"https://zbmath.org/authors/?q=ai:han.pigong"Liu, Chenggang"https://zbmath.org/authors/?q=ai:liu.chenggang"Lei, Keke"https://zbmath.org/authors/?q=ai:lei.keke"Wang, Xuewen"https://zbmath.org/authors/?q=ai:wang.xuewenThe main result is the time decay in the \(L^2\) norm of weak solutions of the nonhomogeneous incompressible Navier-Stokes equations in the whole space \(\mathbb R^n\), \(n=2,3\). The obtained decay rates are optimal since they coincide with the decay rates for the homogeneous equations known for nearly 40 years.
Reviewer: Piotr Biler (Wrocław)On the Euler\(+\)Prandtl expansion for the Navier-Stokes equationshttps://zbmath.org/1521.760812023-11-13T18:48:18.785376Z"Kukavica, Igor"https://zbmath.org/authors/?q=ai:kukavica.igor"Nguyen, Trinh T."https://zbmath.org/authors/?q=ai:nguyen.trinh-t"Vicol, Vlad"https://zbmath.org/authors/?q=ai:vicol.vlad-c"Wang, Fei"https://zbmath.org/authors/?q=ai:wang.fei.2The vanishing viscosity limit for the Navier-Stokes equations in the half-plane supplemented with the Dirichlet boundary condition is studied. The authors justify the validity of the Euler-Prandtl approximation for a class of initial data via vorticity analysis in an \(L^1\)-type norm.
Reviewer: Piotr Biler (Wrocław)Determination of the 3D Navier-Stokes equations with dampinghttps://zbmath.org/1521.760822023-11-13T18:48:18.785376Z"Shi, Wei"https://zbmath.org/authors/?q=ai:shi.wei"Yang, Xinguang"https://zbmath.org/authors/?q=ai:yang.xinguang"Yan, Xingjie"https://zbmath.org/authors/?q=ai:yan.xingjieIn this paper, the authors considered the determination of trajectories for the three-dimensional Navier-Stokes equations with nonlinear damping subject to periodic boundary condition. By making use of the energy estimate of Galerkin approximated equation, the finite number of determining modes and asymptotically determined functionals have been obtained via the Grashof numbers for the non-autonomous and autonomous damped Navier-Stokes fluid flow respectively.
Reviewer: Changxing Miao (Beijing)Direct and inverse problems on the joint movement of the three viscous liquids in the flat layershttps://zbmath.org/1521.760902023-11-13T18:48:18.785376Z"Lemeshkova, Elena N."https://zbmath.org/authors/?q=ai:lemeshkova.elena-nikolaevnaSummary: The exact stationary decision of the problem about the joint movement of the three viscous liquids in the flat layers has been found. The decision of the direct and inverse non-stationary problem has been given in the form of the final analytical formulas using the method of Laplas transformation. The following statement has been proved: if a gradient of the pressure in one liquid has a final limit, then the decision is located on a stationary mode. Also for a problem about the ``the flooded layer'' movement it has been shown that velocities converge to the different constants with the time growth.Study of the spatial transition in a plane channel flowhttps://zbmath.org/1521.760922023-11-13T18:48:18.785376Z"Machaca Abregu, William I."https://zbmath.org/authors/?q=ai:machaca-abregu.william-i"Dari, Enzo A."https://zbmath.org/authors/?q=ai:dari.enzo-a"Teruel, Federico E."https://zbmath.org/authors/?q=ai:teruel.federico-eSummary: This study presents DNS results of the laminar-turbulent spatial transition in a plane channel flow. The transition is achieved imposing at the inlet the most unstable modes of the associated Orr-Sommerfeld and Squire eigenvalue problems. First, a study of the dependence of the transition on the intensity of the perturbations is presented. For \(Re = 5000\), eleven simulations employing different amplitudes of the Tollmien-Schlichting and oblique waves were analyzed to find that the variation of the friction Reynolds number and shape factor downstream the departure of the transition is roughly independent on the amplitude of the perturbations and that the location of the peak in the friction Reynolds number is strongly dependent on the amplitude of each wave. This implies that, for the type of perturbations simulated here, the transitional phenomenon is essentially delayed or accelerated by the amplitude of the perturbations. Second, two cases with well different amplitude of perturbations are compared in detail. Results show that in both cases the following stages can be identified: quasi-linear stage, late stage, \textit{spike} stage, peak transitional zone, post-transitional zone and fully turbulent zone. Moreover, downstream the first state of the \textit{spike} stage, both cases are essentially equal despite the fact that both transitions are separated by 50 channel half-height diameters in the streamwise coordinate. Finally, the physical phenomenon of the peak zone in the friction Reynolds number is explained considering the coherent vortices packet found across the height of the channel in the super-late stage of the transition.On the initial-boundary problem for thermocapillary motion of an emulsion in spacehttps://zbmath.org/1521.760942023-11-13T18:48:18.785376Z"Petrova, Anna G."https://zbmath.org/authors/?q=ai:petrova.anna-georgevnaSummary: The paper is devoted to the study of the initial-boundary problem for thermocapillary motion of an emulsion in closed bounded domain with sufficiently smooth boundary in the absence of gravity. With the use of Tikhonov-Shauder fixed point theorem the local in time solvability to the problem with zero mean volume velocity of the mixture and zero heat flux on the boundary is proved.A fast, decomposed pressure correction method for an intrusive stochastic multiphase flow solverhttps://zbmath.org/1521.760962023-11-13T18:48:18.785376Z"Turnquist, Brian"https://zbmath.org/authors/?q=ai:turnquist.brian-p"Owkes, Mark"https://zbmath.org/authors/?q=ai:owkes.markSummary: Solution of the pressure Poisson equation is often the most expensive aspect of solving the incompressible form of Navier-Stokes. For a single phase deterministic model the pressure calculation is costly. Expanded to an intrusive stochastic multiphase framework, the simulation expense grows dramatically due to coupling between the stochastic pressure field and stochastic density. To address this issue in a deterministic framework, \textit{M. S. Dodd} and \textit{A. Ferrante} [J. Comput. Phys. 273, 416--434 (2014; Zbl 1351.76161)] discuss a decomposed pressure correction method which utilizes an estimated pressure field and constant density to modify the standard pressure correction method. The resulting method is useful for improving computational cost for one-fluid formulations of multiphase flow calculations. In this paper, we extend the decomposed pressure correction method to intrusive uncertainty quantification of multiphase flows. The work improves upon the original formulation by modifying the estimated pressure field. The new method is assessed in terms of accuracy and reduction in computational cost with oscillating droplet, damped surface wave, and atomizing jet test cases where we find convergence of results with the proposed method to those of a traditional pressure correction method and analytic solutions, where appropriate.Linear inviscid damping in Sobolev and Gevrey spaceshttps://zbmath.org/1521.761312023-11-13T18:48:18.785376Z"Zillinger, Christian"https://zbmath.org/authors/?q=ai:zillinger.christianIn recent years, the asymptotic stability of the two-dimensional incompressible Euler equation
\[\partial_tv+v\cdot\nabla v+\nabla p=0\]
near shear flow solutions \(v = (U(y),0)\) has been an area of very active research.
In [Arch. Ration. Mech. Anal. 235, No. 2, 1327--1355 (2020; Zbl 1434.35079)], \textit{H. Jia} established linear inviscid damping in Gevrey spaces for compactly supported Gevrey regular shear flows in a finite channel. In the present article, the author provides an alternative short proof of stability in Gevrey spaces for those flows which admit an approach by a Fourier-based Lyapunov functional. Furthermore, in the setting of a finite channel, one does not need to assume compact support but only pertubations vanishing to infinite order. The author also establishes Sobolev stability results for perturbations vanishing to finite order.
Reviewer: Raphaël Danchin (Paris)GPU-accelerated DNS of compressible turbulent flowshttps://zbmath.org/1521.762242023-11-13T18:48:18.785376Z"Kim, Youngdae"https://zbmath.org/authors/?q=ai:kim.youngdae"Ghosh, Debojyoti"https://zbmath.org/authors/?q=ai:ghosh.debojyoti"Constantinescu, Emil M."https://zbmath.org/authors/?q=ai:constantinescu.emil-m"Balakrishnan, Ramesh"https://zbmath.org/authors/?q=ai:balakrishnan.rameshSummary: This paper explores strategies to transform an existing CPU-based high-performance computational fluid dynamics solver, \textsc{HyPar}, for compressible flow simulations on emerging exascale heterogeneous (CPU+GPU) computing platforms. The scientific motivation for developing a GPU-enhanced version of \textsc{HyPar} is to simulate canonical turbulent flows at the highest resolution possible on such platforms. We show that optimizing memory operations and thread blocks results in 200x speedup of computationally intensive kernels compared with a CPU core. Using multiple GPUs and CUDA-aware MPI communication, we demonstrate both strong and weak scaling of our GPU-based \textsc{HyPar} implementation on the NVIDIA Volta V100 GPUs. We simulate the decay of homogeneous isotropic turbulence in a triply periodic box on grids with up to \(102 4^3\) points (5.3 billion degrees of freedom) and on up to 1,024 GPUs. We compare the wall times for CPU-only and CPU+GPU simulations. The results presented in the paper are obtained on the Summit and Lassen supercomputers at Oak Ridge and Lawrence Livermore National Laboratories, respectively.The meshless local Petrov-Galerkin method based on moving Taylor polynomial approximation to investigate unsteady diffusion-convection problems of anisotropic functionally graded materials related to incompressible flowhttps://zbmath.org/1521.763082023-11-13T18:48:18.785376Z"Abbaszadeh, Mostafa"https://zbmath.org/authors/?q=ai:abbaszadeh.mostafa"Dehghan, Mehdi"https://zbmath.org/authors/?q=ai:dehghan.mehdi"Azis, Mohammad Ivan"https://zbmath.org/authors/?q=ai:azis.mohammad-ivanSummary: This paper concerns to a meshless local Petrov-Galerkin (MLPG) method for studying the unsteady diffusion-convection problems of anisotropic functionally graded materials. A new version of MLPG method based on the moving Taylor polynomial approximation is developed to discrete the spatial variable. Then, we obtain a system of ODEs which depends to the time variable. A strong stability preserving (SSP) Runge-Kutta idea is provided to solve the final ODEs with enough accuracy and stability. Also, the grading function which defines the variable elastic coefficient can be any types of continuous functions. The developed numerical formulation is applied for different examples of non-rectangular domains to check its accuracy.On explicit discontinuous Galerkin methods for conservation lawshttps://zbmath.org/1521.763402023-11-13T18:48:18.785376Z"Huynh, H. T."https://zbmath.org/authors/?q=ai:huynh.hieu-trung|huynh.huu-thanh|huynh.hung-t|huynh.huu-tueSummary: The concept of projection applied to explicit discontinuous Galerkin (DG) schemes is investigated. The two explicit DG methods in this study are based on a `predictor-corrector' formulation, the first introduced by \textit{F. Lörcher} et al. [J. Sci. Comput. 32, No. 2, 175--199 (2007; Zbl 1143.76047)] and \textit{G. Gassner} et al. [J. Sci. Comput. 34, No. 3, 260--286 (2008; Zbl 1218.76027)] called space-time expansion discontinuous Galerkin or STE-DG scheme,
and the second, introduced independently by
the author (Huynh 2006)
[``High-order space-time methods for conservation laws'', in: Proceedings of the 21st AIAA computational fluid dynamics conference, 2013. Reston, VI: American Institute of Aeronautics and Astronautics (AIAA). Article ID 2013-2432, 36 p. (2013; \url{doi:10.2514/6.2013-2432})]
called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction takes place; the two methods differ, however, in the evaluation of the projections in the space-time volume integrals. The STE-DG scheme evaluates these in a straightforward manner, whereas the moment scheme employs a successive procedure with each moment update uses the results by the lower-order updates. The trade-off is that the moment scheme has the disadvantage of a more elaborate corrector step and the significant advantage of a CFL (Courant-Friedrichs-Lewy) condition of 1 for all (degree) \(p\) and accuracy order of \(2 p + 1\) (super accuracy property) for one-dimensional (1D) advection. In contrast, the STE-DG method is accurate to only the expected order of \(p + 1\) and has a more restrictive CFL condition. Due to the predictor-corrector formulation that does not involve methods of characteristics, these schemes extend easily to systems of equations in multiple dimensions. Concerning the case of two spatial dimensions (2D), for an advection using a Cartesian grid, when the flow does not align with the axes, especially when it is along a diagonal direction, the CFL conditions for the moment schemes also become restrictive and need improvement as will be shown by Fourier (von Neumann) analyses.Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation lawshttps://zbmath.org/1521.763442023-11-13T18:48:18.785376Z"Kuzmin, Dmitri"https://zbmath.org/authors/?q=ai:kuzmin.dmitri"Quezada de Luna, Manuel"https://zbmath.org/authors/?q=ai:de-luna.manuel-quezadaSummary: This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. We prove that the standard CG method is entropy conservative for the square entropy. In general, the rate of entropy production/dissipation depends on the residual of the governing equation and on the accuracy of the finite element approximation to the entropy variable. The inclusion of linear high-order stabilization generates an additional source/sink in the entropy budget equation. To balance the amount of entropy production in each cell, we construct entropy-dissipative element contributions using a coercive bilinear form and a parameter-free entropy viscosity coefficient. The entropy stabilization term is high-order consistent, and optimal convergence behavior is achieved in practice. To enforce preservation of local bounds in addition to entropy stability, we use the Bernstein basis representation of the finite element solution and a new subcell flux limiting procedure. The underlying inequality constraints ensure the validity of localized entropy conditions and local maximum principles. The benefits of the proposed modifications are illustrated by numerical results for linear and nonlinear test problems.Subcell limiting strategies for discontinuous Galerkin spectral element methodshttps://zbmath.org/1521.763632023-11-13T18:48:18.785376Z"Rueda-Ramírez, Andrés M."https://zbmath.org/authors/?q=ai:rueda-ramirez.andres-mauricio"Pazner, Will"https://zbmath.org/authors/?q=ai:pazner.will-e"Gassner, Gregor J."https://zbmath.org/authors/?q=ai:gassner.gregor-jSummary: We present a general family of subcell limiting strategies to construct robust high-order accurate nodal discontinuous Galerkin (DG) schemes. The main strategy is to construct compatible low order finite volume (FV) type discretizations that allow for convex blending with the high-order variant with the goal of guaranteeing additional properties, such as bounds on physical quantities and/or guaranteed entropy dissipation. For an implementation of this main strategy, four main ingredients are identified that may be combined in a flexible manner: (i) a nodal high-order DG method on Legendre-Gauss-Lobatto nodes, (ii) a compatible robust subcell FV scheme, (iii) a convex combination strategy for the two schemes, which can be element-wise or subcell-wise, and (iv) a strategy to compute the convex blending factors, which can be either based on heuristic troubled-cell indicators, or using ideas from flux-corrected transport methods. By carefully designing the metric terms of the subcell FV method, the resulting methods can be used on unstructured curvilinear meshes, are locally conservative, can handle strong shocks efficiently while directly guaranteeing physical bounds on quantities such as density, pressure or entropy. We further show that it is possible to choose the four ingredients to recover existing methods such as a provably entropy dissipative subcell shock-capturing approach or a sparse invariant domain preserving approach. We test the versatility of the presented strategies and mix and match the four ingredients to solve challenging simulation setups, such as the KPP problem (a hyperbolic conservation law with non-convex flux function), turbulent and hypersonic Euler simulations, and MHD problems featuring shocks and turbulence.Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes III. Code optimization via tensor contractionhttps://zbmath.org/1521.763782023-11-13T18:48:18.785376Z"You, Hojun"https://zbmath.org/authors/?q=ai:you.hojun"Kim, Chongam"https://zbmath.org/authors/?q=ai:kim.chongamSummary: The present study deals with the code optimization and its implementation of the direct reconstruction method (DRM) using the complete-search tensor contraction (CsTC) framework to extract the best performance of high-order methods on modern computing architectures. DRM was originally proposed to overcome severe computational costs of the physical domain-based discontinuous Galerkin (DG) method on mixed-curved meshes. In this work, the performance of DRM is further enhanced through the code optimization via the CsTC technique. Required kernels for tensor operations in the DRM solution algorithm are analyzed and optimized by completely searching all candidates of GEMM (General Matrix Multiplication) subroutines. The computational performance is thoroughly examined by simulating a turbulent flow over a circular cylinder at \(R e_D = 3900\) by DG-\textit{P3} and -\textit{P5} approximations. Compared to a quadrature-based approach with the full integration, the optimized DRM significantly reduces the memory requirements and the number of floating-point operations to compute the DG residual on a linear mesh as well as high-order curved meshes. On a \textit{P3}-mesh, the optimized DRM provides \(13.74 \times\) and \(23.03 \times\) speed-ups in DG-\textit{P3} and -\textit{P5}, respectively, while the amount of memory required is reduced to \(1 / 16.6\) and \(1 / 19.9\). On a linear mesh, it even yields \(1.25 \times\) and \(1.12 \times\) speed-ups in DG-\textit{P3} and -\textit{P5}, respectively. The memory requirement is reduced to \(1 / 1.27\) and \(1 / 1.15 ,\) respectively. In particular, it is observed that the optimized DRM on a \textit{P3}-mesh performs better than the optimized quadrature-based method on a \textit{P1}-mesh.A data-driven shock capturing approach for discontinuous Galerkin methodshttps://zbmath.org/1521.763802023-11-13T18:48:18.785376Z"Yu, Jian"https://zbmath.org/authors/?q=ai:yu.jian.1"Hesthaven, Jan S."https://zbmath.org/authors/?q=ai:hesthaven.jan-sSummary: We propose a data-driven artificial viscosity model for shock capturing in discontinuous Galerkin methods. The proposed model trains a multi-layer feedforward network to map from the element-wise solution to a smoothness indicator, based on which the amount of artificial viscosity is determined. The data set for the training of the network is obtained using canonical functions. The compactness of the data set, which is critical to the success of training the network, is ensured by normalization and the adjustment of the range of the smoothness indicator. The network is able to recover the expected smoothness much more reliably than its traditional counterpart, i.e. the averaged modal decay model. Several smooth and non-smooth test cases are considered to investigate the performance of this data-driven model. Convergence tests show that the proposed model recovers the accuracy of the corresponding inviscid schemes for smooth regions. For a wide range of non-smooth flows, the model is shown to suppress spurious oscillations well.Parallel defect-correction methods for incompressible flows with friction boundary conditionshttps://zbmath.org/1521.763882023-11-13T18:48:18.785376Z"Zheng, Bo"https://zbmath.org/authors/?q=ai:zheng.bo.1|zheng.bo"Shang, Yueqiang"https://zbmath.org/authors/?q=ai:shang.yueqiangSummary: We study three parallel defect-correction methods based on finite element approximations for the incompressible Navier-Stokes problem with friction boundary conditions and high Reynolds numbers in this work, where a fully overlapping domain decomposition is considered for parallelization. In the proposed methods, with a global multiscale grid that builds a fine grid around its own subdomain and coarse elsewhere, we iteratively solve an artificial viscosity nonlinear variational inequality problem in a defect step, and then compute the residual by the linearized variational inequality problems in the \(r\)-step corrections. The studied methods are easy to implement on the basis of the existing Navier-Stokes solver and possess less communication complexity. We provide a rigorously theoretical derivation for the error estimates of the one-step correction solutions from the proposed methods under some stable conditions, and derive scalings of the algorithmic parameters. We demonstrate by a series of numerical experiments that the velocity and pressure errors computed by our parallel defect-correction methods are comparable to those of the standard defect-correction method, while our present methods reduce the computational cost.A unified asymptotic preserving and well-balanced scheme for the Euler system with multiscale relaxationhttps://zbmath.org/1521.763922023-11-13T18:48:18.785376Z"Arun, K. R."https://zbmath.org/authors/?q=ai:arun.k-r"Krishnan, M."https://zbmath.org/authors/?q=ai:krishnan.m-hari|krishnan.murugappa|krishnan.mangala-sunder|krishnan.musaravakkam-s|krishnan.m-m|krishnan.m-nikhil|krishnan.mekala|krishnan.mayuram-s|krishnan.manojkumar"Samantaray, S."https://zbmath.org/authors/?q=ai:samantaray.sauravSummary: The design and analysis of a unified asymptotic preserving (AP) and well-balanced scheme for the Euler Equations with gravitational and frictional source terms is presented in this paper. The asymptotic behaviour of the Euler system in the limit of zero Mach and Froude numbers, and large friction is characterised by an additional scaling parameter. Depending on the values of this parameter, the Euler system relaxes towards a hyperbolic or a parabolic limit equation. Standard Implicit-Explicit Runge-Kutta schemes are incapable of switching between these asymptotic regimes. We propose a time semi-discretisation to obtain a unified scheme which is AP for the two different limits. A further reformulation of the semi-implicit scheme can be recast as a fully-explicit method in which the mass update contains both hyperbolic and parabolic fluxes. A space-time fully-discrete scheme is derived using a finite volume framework. A hydrostatic reconstruction strategy, an upwinding of the sources at the interfaces, and a careful choice of the central discretisation of the parabolic fluxes are used to achieve the well-balancing property for hydrostatic steady states. Results of several numerical case studies are presented to substantiate the theoretical claims and to verify the robustness of the scheme.High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance lawshttps://zbmath.org/1521.763932023-11-13T18:48:18.785376Z"Berberich, Jonas P."https://zbmath.org/authors/?q=ai:berberich.jonas-p"Chandrashekar, Praveen"https://zbmath.org/authors/?q=ai:chandrashekar.praveen"Klingenberg, Christian"https://zbmath.org/authors/?q=ai:klingenberg.christianSummary: We introduce a general framework for the construction of well-balanced finite volume methods for hyperbolic balance laws. We use the phrase \textit{well-balancing} in a broader sense, since our proposed method can be applied to exactly follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions and not only time independent solutions. The solution has to be known a priori, either as an analytical expression or as discrete data. The proposed framework modifies the standard finite volume approach such that the well-balancing property is obtained and in case the method is high order accurate, this is maintained under our modification. We present numerical tests for the compressible Euler equations with and without gravity source term and with different equations of state, and for the equations of compressible ideal magnetohydrodynamics.Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic fieldhttps://zbmath.org/1521.764002023-11-13T18:48:18.785376Z"Brull, Stéphane"https://zbmath.org/authors/?q=ai:brull.stephane"Dubroca, Bruno"https://zbmath.org/authors/?q=ai:dubroca.bruno"Lhébrard, Xavier"https://zbmath.org/authors/?q=ai:lhebrard.xavierSummary: The present paper concerns the study of the nonconservative bitemperature Euler system with transverse magnetic field. We firstly introduce an underlying conservative kinetic model coupled to Maxwell equations. The nonconservative bitemperature Euler system with transverse magnetic field is then established from this kinetic model by hydrodynamic limit. Next we present the derivation of a finite volume method to approximate weak solutions. It is obtained by solving a relaxation system of Suliciu type, and is similar to HLLC type solvers. The solver is shown in particular to preserve positivity of density and internal energies. Moreover we use a local minimum entropy principle to prove discrete entropy inequalities, ensuring the robustness of the scheme.Numerical simulation of three-fluid Rayleigh-Taylor instability using an enhanced volume-of-fluid (VOF) model: new benchmark solutionshttps://zbmath.org/1521.764242023-11-13T18:48:18.785376Z"Garoosi, Faroogh"https://zbmath.org/authors/?q=ai:garoosi.faroogh"Mahdi, Tew-Fik"https://zbmath.org/authors/?q=ai:mahdi.tew-fikSummary: The main objective of the present study is to introduce two novel benchmark solutions namely: two-dimensional three-fluid Rayleigh-Taylor Instability problems, aiming to provide an up-to-date data set and a unique fundamental insight into morphology and hydrodynamic behavior of coupled Rayleigh-Taylor-Kelvin-Helmholtz instability phenomenon. To this end, the Volume-Of-Fluid (VOF) model is adopted to probe the complex configurations and kinetic processes of highly nonlinear multi-fluid flow problems with large topological changes and moving interfaces. However, to improve the performance and accuracy of the classical VOF model and preserve monotonicity for the density and viscosity, a novel high-order bounded advection scheme is first proposed in the context of the Total Variation Diminishing and Normalized Variable Diagram (TVD-NVD) constraints and then is utilized for the discretization of the convection terms in the Navier-Stokes and transport equations. To further increase the accuracy of the numerical simulations, the second-order PLIC-ELVIRA is implemented for the reconstruction of the physical discontinuity between phases and the determination of its curvature. Furthermore, to enhance the consistency and stability of the classical VOF model in handling incompressible multi-fluid flows, a novel semi-iterative pressure-velocity coupling algorithm is constructed by the combination of the standard PISO and SIMPLEC algorithms and is then applied to ensure mass conservation in each grid cell. To demonstrate the versatility and robustness of the proposed model in dealing with the multiphase flows involving large interface deformation and breaking phenomena, a series of canonical test cases such as dam-break over a dry bed with and without stationary obstacle, 2D three-fluid rising bubble, two-fluid and three-fluid Rayleigh-Taylor Instability are adopted. In the last stage, an improved VOF model is applied to solve two new three-fluid Rayleigh-Taylor Instability benchmark problems on a staggered grid system. The results of this study can provide a wide panorama on the improvements of standard VOF model and may be utilized as benchmark solutions for validation of various CFD tools or simply to understand more complex related multi-fluid flows.A robust hybrid unstaggered central and Godunov-type scheme for Saint-Venant-Exner equations with wet/dry frontshttps://zbmath.org/1521.764412023-11-13T18:48:18.785376Z"Li, Dingfang"https://zbmath.org/authors/?q=ai:li.dingfang"Dong, Jian"https://zbmath.org/authors/?q=ai:dong.jianSummary: We aim to propose a robust hybrid unstaggered central and Godunov-type scheme based on hydrostatic reconstruction (HR) for Saint-Venant-Exner equations with wetting and drying transitions. The discretization of the bed slope source term is based on the HR method to preserve the still water steady-state solution. The hydrodynamic model described by the Saint-Venant system is numerically solved using the well-balanced unstaggered central scheme proposed in [\textit{J. Dong} and \textit{D. F. Li}, Appl. Math. Comput. 372, Article ID 124992, 18 p. (2020; Zbl 1433.76105)]. The morphodynamic model described by the Exner equation is numerically solved by using a Lax-Friedrichs method. In solving the morphodynamic model, we proposed a novel ``cut-off'' function to guarantee the nonlinear stability and preserve the stationary solution. The present scheme can exactly obtain the stationary solution and is capable of guaranteeing the positivity of the water depth. The key contributions of this work are the well-balanced property at the wet-dry fronts and the stability of the current scheme in solving two physical models that have relatively either strong or weak interactions. Finally, we use several classical problems of the system to demonstrate these properties.Third-order scale-independent WENO-Z scheme to achieve optimal order at critical pointshttps://zbmath.org/1521.764432023-11-13T18:48:18.785376Z"Li, Qin"https://zbmath.org/authors/?q=ai:li.qin.1"Huang, Xiao"https://zbmath.org/authors/?q=ai:huang.xiao"Yan, Pan"https://zbmath.org/authors/?q=ai:yan.pan"Duan, Yi"https://zbmath.org/authors/?q=ai:duan.yi"You, Yancheng"https://zbmath.org/authors/?q=ai:you.yanchengSummary: Our past work has shown that when the critical points occur within grid intervals, the relations of accuracy of the smoothness indicators of weighted essentially non-oscillatory schemes (WENO) reported by Jiang and Shu differ from those that are obtained by assuming that the critical points occur on the grid nodes. The global smoothness indicator in the WENO-Z scheme might accordingly differ from the original one. We use this understanding to first discuss several issues regarding current improvements to third-order WENO-Z (e.g., WENO-NP3, -F3, -NN3, and -PZ3), i.e., numerical results with scale dependence, the validity of the analysis by assuming that the critical points occur on the nodes, and sensitivity in terms of the computational time step and initial conditions through the examination of the order of convergence. Numerical simulations and analyses were used to highlight defects in these improvements that occur either due to the scale dependence of the results, or the failure to recover the optimal order when the critical points occur on the half-nodes. Following this, a generic analysis that assumes that the first-order critical points occur within the grid intervals is provided. The theoretical results thus derived are used to propose two scale-independent third-order WENO-Z schemes that can be used to attain the optimal order at the critical points. The first scheme is obtained by extending a downstream smoothness indicator to derive a new global smoothness indicator and incorporating it into the mapping function. The second scheme is achieved by extending another smoothness indicator and using a different global indicator. The following validations are chosen and tested: the typical 1D problem of scalar advections, and 1D and 2D problems based on Euler's equations. The results verify the capability of the proposed schemes to recover the optimal order at the critical points. Moreover, the first of the above two proposed schemes outperforms the improved third-order WENO-Z scheme in terms of numerical resolution and robustness, which is usually favored by applications.A kinetic flux difference splitting method for compressible flowshttps://zbmath.org/1521.764662023-11-13T18:48:18.785376Z"Shrinath, K. S."https://zbmath.org/authors/?q=ai:shrinath.k-s"Maruthi, N. H."https://zbmath.org/authors/?q=ai:maruthi.n-h"Raghurama Rao, S. V."https://zbmath.org/authors/?q=ai:raghurama-rao.s-v"Vasudeva Rao, Veeredhi"https://zbmath.org/authors/?q=ai:vasudeva-rao.veeredhiSummary: A low diffusive flux difference splitting based kinetic scheme is developed based on a discrete velocity Boltzmann equation, with a novel three velocity model. While two discrete velocities are used for upwinding, the third discrete velocity is utilized to introduce appropriate additional numerical diffusion only in the expansion regions, identified using relative entropy (Kullback-Liebler divergence) at the cell-interface, along with the estimation of physical entropy. This strategy provides an interesting alternative to entropy fix, which is typically needed for low diffusive schemes. Grid-aligned steady discontinuities are captured exactly by fixing the primary numerical diffusion such that flux equivalence leads to zero numerical diffusion across discontinuities. Results for bench-mark test problems are presented for inviscid and viscous compressible flows.A high-resolution scheme for axisymmetric hydrodynamics based on the 2D GRP solvershttps://zbmath.org/1521.764982023-11-13T18:48:18.785376Z"Zhu, Zijin"https://zbmath.org/authors/?q=ai:zhu.zijin"Cui, Qingjie"https://zbmath.org/authors/?q=ai:cui.qingjie"Ni, Guoxi"https://zbmath.org/authors/?q=ai:ni.guoxiSummary: In the simulation of axisymmetric fluids, imposing an appropriate numerical boundary condition at the symmetrical axis \(r=0\) is an inevitable difficulty. To maintaining the conservation property, we choose to keep its form coincided with the numerical scheme on interior control volumes. Then we obtain necessary interface values in both scenarios from solving the two-dimensional generalized Riemann problems (2D GRPs) and their one-sided versions. Therein the effects of transversal variation and geometrical source are specifically emphasized to exhibit the genuine multi-dimensionality of the entire algorithm. In the construction of the one-sided 2D GRP solver, we have to face the difficulty brought by the singularity of the source at \(r=0\). The ingenious combination of acoustic approximation, symmetry argument, and L'Hospital's rule is proposed in obtaining a pivotal limiting value there. Several challenging tests are provided to demonstrate the effectiveness and robustness of our approach.High resolution central scheme using a new upwind slope limiter for hyperbolic conservation lawshttps://zbmath.org/1521.765252023-11-13T18:48:18.785376Z"Cai, Zhenyu"https://zbmath.org/authors/?q=ai:cai.zhenyu"Li, Decai"https://zbmath.org/authors/?q=ai:li.decai"Hu, Yang"https://zbmath.org/authors/?q=ai:hu.yang"Li, Mingjun"https://zbmath.org/authors/?q=ai:li.mingjun"Meng, Xiangshen"https://zbmath.org/authors/?q=ai:meng.xiangshenSummary: In this paper, a new high resolution central-upwind scheme for hyperbolic conservation laws is proposed, based on a new upwind biased slope limiter. The new scheme is defined as a correction of KT central scheme by applying the new slope limiter. The new limiter uses more upwind information for the piecewise linear interpolations, so that it has an upwind nature. The main features of the new scheme are high resolution and nonoscillatory. At the same time, it retains the efficiency and simplicity of central scheme. The total-variation diminishing (TVD) property and maximum principle of the new scheme are proved. The numerical experiments, including one-dimensional, two-dimensional Riemann problems and double Mach reflection problem for Euler equations, show the desired resolution and robustness of the new scheme.One dimensional hybrid WENO-AO method using improved troubled cell indicator based on extreme pointhttps://zbmath.org/1521.765322023-11-13T18:48:18.785376Z"Chen, Li Li"https://zbmath.org/authors/?q=ai:chen.lili"Huang, Cong"https://zbmath.org/authors/?q=ai:huang.congSummary: One dimensional troubled cell indicator, which is based on the extreme point of approximated polynomial, was proposed by
\textit{J. Zhu} and \textit{J. Qiu} [SIAM J. Sci. Comput. 39, No. 3, A1089--A1113 (2017; Zbl 1366.65081)].
This troubled cell indicator performs well, but still has two shortcomings. First, it calculates the extreme point of high-degree polynomial, however the calculation is not simple. The reason is that, the explicit formula of extreme point of quartic polynomial has been complex, the one of higher degree polynomial becomes more complex or even is hard to be obtained, so has to be calculated by using numerical method. Second, a complex or without explicit formula of extreme point will bring tremendous difficulty to analyze why the troubled cell indicator can capture the major character of solution. In order to overcome these shortcomings, we first improve the troubled cell indicator, then use it to design the hybrid WENO-AO method for solving one dimensional hyperbolic conservation laws.High-order mapped WENO methods with improved efficiencyhttps://zbmath.org/1521.765432023-11-13T18:48:18.785376Z"Hu, Fuxing"https://zbmath.org/authors/?q=ai:hu.fuxingSummary: This paper is devised to improve the efficiency of mapped WENO (WENO-M) methods. The WENO-M methods apply a mapping function on the nonlinear weights to recover the optimal accuracy at extremal points and meanwhile improve the resolution of numerical solutions at non-smooth regions. But this mapping process increases the computational costs as well. To improve the efficiency of WENO-M methods, we develop a simple mapping function which is applied on the indicators of smoothness (not on the nonlinear weights). The mapped indicators of smoothness are sufficient to obtain the optimal accuracy at extremal points. The computational costs are also deceased since the nonlinear weights are only required to be computed once in each WENO reconstruction. Finally, several numerical tests confirm that the improved WENO-M (WENO-IM) methods have the similar numerical accuracy with WENO-M methods, but less computational costs.A fifth-order nonlinear spectral difference scheme for hyperbolic conservation lawshttps://zbmath.org/1521.765542023-11-13T18:48:18.785376Z"Lin, Yu"https://zbmath.org/authors/?q=ai:lin.yu"Chen, Yaming"https://zbmath.org/authors/?q=ai:chen.yaming"Deng, Xiaogang"https://zbmath.org/authors/?q=ai:deng.xiaogangSummary: We develop in this paper a fifth-order nonlinear spectral difference method for solving hyperbolic conservation laws, whose solutions often admit discontinuities. To avoid instability caused by the Gibbs phenomenon arising from interpolation across discontinuities, a fifth-order nonlinear interpolation scheme is proposed within a single cell, keeping the compactness of the original linear spectral difference method. Some numerical results are also presented to demonstrate the accuracy and effectiveness of the proposed method.Improvement of the WENO-Z+ schemehttps://zbmath.org/1521.765652023-11-13T18:48:18.785376Z"Luo, Xin"https://zbmath.org/authors/?q=ai:luo.xin"Wu, Song-ping"https://zbmath.org/authors/?q=ai:wu.songpingSummary: The WENO-Z+ scheme [\textit{F. Acker} et al., J. Comput. Phys. 313, 726--753 (2016; Zbl 1349.65260)] was obtained by adding a new term into the WENO-Z weights. The added term has the role of raising the weights of less-smooth substencils. The WENO-Z+ scheme achieves superior results in the test problems that involve only a single wave component. For problems containing multiscale structures, however, the results of WENO-Z+ show little improvement compared with those of WENO-Z. In this short note, we investigate the reason why WENO-Z+ has little improvement in solving the multiscale problems and propose a set of modifications to it. A series of numerical tests confirm that the new scheme, which is named WENO-Z+I, has significantly improved multiscale resolution. The multiscale resolution of the fifth-order WENO-Z+I scheme is even significantly better than that of the seventh-order WENO-Z scheme. We also find that the WENO-Z+ type schemes can achieve better spectral properties than the corresponding linear upwind scheme.On the supremum of the steepness parameter in self-adjusting discontinuity-preserving schemeshttps://zbmath.org/1521.765802023-11-13T18:48:18.785376Z"Ruan, Yucang"https://zbmath.org/authors/?q=ai:ruan.yucang"Tian, Baolin"https://zbmath.org/authors/?q=ai:tian.baolin"Zhang, Xinting"https://zbmath.org/authors/?q=ai:zhang.xinting"He, Zhiwei"https://zbmath.org/authors/?q=ai:he.zhiweiSummary: Self-adjusting steepness (SAS)-based schemes preserve various structures in the compressible flows. These schemes provide a range of desired behaviors depending on the steepness-adjustable limiters with the steepness measured by a steepness parameter. These properties include either second-order accuracy with exact steepness infima that are theoretically given or having anti-diffusive/compression properties with a larger steepness parameter. Nevertheless, the supremum of the steepness parameter has not been determined theoretically yet. In this study, we demonstrate that any anti-diffusive limiter should be limited by Ultra-bee limiter according to Sweby's total variation diminishing (TVD) condition. Two typical steepness-adjustable limiters are analyzed in detail including the tangent of hyperbola for interface capturing (THINC) limiter and the steepness-adjustable harmonic (SAH) limiter. Applying this constraint, we derive for the first time two inequalities which the steepness parameters much satisfy. Furthermore, we obtain the analytical expression of the Courant-Friedrichs-Lewy (CFL) number-dependent supremum of the steepness parameter. Using this solution, we then propose supremum-determined SAS schemes. These schemes are further extended to solve the compressible Euler equations. The results of typical numerical tests confirm our theoretical conclusions and show that the final schemes are capable of sharply capturing contact discontinuities and minimizing numerical oscillations.Construction and application of several new symmetrical flux limiters for hyperbolic conservation lawhttps://zbmath.org/1521.765922023-11-13T18:48:18.785376Z"Tang, Shujiang"https://zbmath.org/authors/?q=ai:tang.shujiang"Li, Mingjun"https://zbmath.org/authors/?q=ai:li.mingjunSummary: The construction of limiter functions is a crucial factor in total-variation-diminishing (TVD) schemes to achieve high resolution and numerical stability. In this paper, three symmetrical limiter functions are constructed by introducing the MAX function into the classical van Albada, van Leer, and PR-\( \kappa\) limiters. The analysis and numerical results demonstrate that the MUSCL scheme equipped with the proposed limiters satisfies the sufficient conditions for second-order convergence in smooth regions and exhibits lower dissipation and better resolution than the MUSCL scheme utilizing classic limiters corresponding to both smooth and discontinuous solutions.Numerical simulation of real gas one-component two-phase flow using a Roe-based schemehttps://zbmath.org/1521.765942023-11-13T18:48:18.785376Z"Tegethoff, Katharina"https://zbmath.org/authors/?q=ai:tegethoff.katharina"Schuster, Sebastian"https://zbmath.org/authors/?q=ai:schuster.sebastian"Brillert, Dieter"https://zbmath.org/authors/?q=ai:brillert.dieterSummary: The objective of this paper is to derive and test a set of algebraic equations describing the fluid flow based on the Euler equations. The derivation is based on the finite volume approach and the benefits of Roe's approach are preserved. A further objective is that the set of algebraic equations can be solved by incorporating an equation of state as complex as for example the EOS for \(\mathrm{CO}_2\) given by Span and Wager albeit this equation of state is not used during the derivation. The particular innovation stems from the assumption made during the derivation. Only small changes of fluid properties are assumed at first. This allows to use the equation of state of a thermally perfect gas (specific heat capacity constant) to derive Roe's matrix. Effectively this leads to the set of equations already derived by Roe. But the determination of any state variables and in particular the Roe-averaged speed of sound is done by means of the previously chosen EOS. Thereafter, the obtained set of equations is utilised to calculate flows with small to large gradients. It turns out that the derived scheme gives results in good agreement even for Sod's problem and flows in Laval nozzles with the compressibility factor changing from 1.5 to 0.5 across the shock. Subsequently, the solver is used for predicting condensing steam flows. Besides the calculation of thermodynamic properties also the radii of droplets are derived. The results obtained could be used for optimising the numerical stability of flow prediction in steam turbines. The integration of the EOS for \(\mathrm{CO}_2\) given by Span and Wager in the Roe-based solver could also be applied to the investigation of Joule cycles operated with Carbon Dioxide in a supercritical state, thus increasing the accuracy. Due to the novel way of integrating an arbitrary equation of state into a numerical scheme, it also represents a contribution to current developments in the field of computational fluid dynamics.Fifth order AWENO finite difference scheme with adaptive numerical diffusion for Euler equationshttps://zbmath.org/1521.765962023-11-13T18:48:18.785376Z"Wang, Yinghua"https://zbmath.org/authors/?q=ai:wang.yinghua"Don, Wai Sun"https://zbmath.org/authors/?q=ai:don.wai-sun"Wang, Bao-Shan"https://zbmath.org/authors/?q=ai:wang.baoshanSummary: In solving hyperbolic conservation laws using the fifth-order characteristic-wise alternative WENO finite-difference scheme (AWENO) with Z-type affine-invariant nonlinear Ai-weights, the classical local Lax-Friedrichs flux (LLF) is modified with an adaptive numerical diffusion (ND) coefficient to form an adaptive LLF flux (LLF-M). The adaptive ND coefficient depends nonlinearly on the local scale-independent smoothness measures on the pressure, density, dilation, and vorticity. The feature sensor combines the well-known Durcos' sensor on the dilation and vorticity of the velocity field and Jameson's sensor on the density and pressure. Based on the measure of the feature sensor, the ND coefficient of the LLF-M flux is designed to transit smoothly and quickly from a set minimum to the maximum, the local spectral radius of the eigenvalues of the Jacobian of the flux. Hence, the modified AWENO scheme improves the resolution of small-scale structures due to a substantial reduction of excessive dissipation while capturing discontinuities essentially non-oscillatory (ENO-property). The performance of the improved AWENO scheme is validated in one- and two-dimensional benchmark problems with discontinuities and smooth small-scale structures. The results show that the new adaptive LLF-M flux improves resolution, captures fine-scale structures, and is robust in a long-term simulation.An efficient fifth-order finite difference multi-resolution WENO scheme for inviscid and viscous flow problemshttps://zbmath.org/1521.765972023-11-13T18:48:18.785376Z"Wang, Zhenming"https://zbmath.org/authors/?q=ai:wang.zhenming"Zhu, Jun"https://zbmath.org/authors/?q=ai:zhu.jun"Tian, Linlin"https://zbmath.org/authors/?q=ai:tian.linlin"Yang, Yuchen"https://zbmath.org/authors/?q=ai:yang.yuchen"Zhao, Ning"https://zbmath.org/authors/?q=ai:zhao.ningSummary: In this paper, a more efficient fifth-order finite difference multi-resolution WENO scheme is designed to solve the compressible inviscid and viscous flow problems based on the original multi-resolution WENO scheme presented in
[\textit{J. Zhu} and \textit{C.-W. Shu}, J. Comput. Phys. 375, 659--683 (2018; Zbl 1416.65286)].
This new improved method inherits all the following features of the original multi-resolution WENO scheme
[Zhu and Shu, loc. cit.]:
(1) a series of unequal-sized central stencils are used to obtain the spatial reconstruction polynomials; (2) any positive numbers whose sum is one can be set as the linear weights; (3) smaller \(L^1\) and \(L^\infty\) errors could be obtained for smooth problems; (4) it is easier to extend to the unstructured finite volume framework. However, different from the original multi-resolution WENO scheme, this improved method simplifies the reconstruction process, significantly improves the computational efficiency, and has greater engineering application potential. Numerical results show that the two types of fifth-order finite difference multi-resolution WENO schemes have similar results, but the CPU time of this new multi-resolution WENO scheme is about 0.7 times that of the original one. Moreover, the new fifth-order multi-resolution WENO scheme with a small increase in the computational cost shows less dissipation error than the classical WENO scheme
[\textit{G.-S. Jiang} and \textit{C.-W. Shu}, J. Comput. Phys. 126, No. 1, 202--228 (1996; Zbl 0877.65065)],
and can capture more subtle flow structures for solving inviscid and viscous flow problems on the same mesh level. Several benchmark inviscid and viscous problems are illustrated to verify the above conclusions and the improved performance of this fifth-order multi-resolution WENO scheme.An efficient smoothness indicator mapped WENO scheme for hyperbolic conservation lawshttps://zbmath.org/1521.766072023-11-13T18:48:18.785376Z"Zhang, Xin"https://zbmath.org/authors/?q=ai:zhang.xin.29"Yan, Chao"https://zbmath.org/authors/?q=ai:yan.chao"Qu, Feng"https://zbmath.org/authors/?q=ai:qu.feng.1Summary: The mapping function method is a common approach to improve the accuracy of WENO type schemes. However, with the demand of the accuracy improvement higher, the mapping function becomes more and more complex, and the calculation cost also increases significantly. In this study, a novel smoothness indicator mapping method called WENO-ISM is proposed based on the WENO scheme. It employs a simple function which amplifies the original smoothness to improve the scheme's accuracy. In addition to the difference of the mapping function, the superior innovation of the designed scheme is that the mapping object is changed to the smoothness indicator instead of the nonlinear weight calculated from the WENO-JS scheme. The underlying idea of this method is to use this mapping function to directly regard the optimal weight of the stencil in the relatively smooth region as the nonlinear weight. For these, the calculation cost can be greatly reduced without loss of accuracy. In the meanwhile, the simple mapping function itself is with higher accuracy and lower cost. Numerical experiments with one-dimensional linear advection and ADR analysis show that this scheme is superior to other mapped WENO schemes in accuracy and especially in computation cost. Furthermore, improved results of this scheme are obtained by several typical one-dimensional and two-dimensional numerical cases.New mapped unequal-sized trigonometric WENO scheme for hyperbolic conservation lawshttps://zbmath.org/1521.766082023-11-13T18:48:18.785376Z"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.131"Zhu, Jun"https://zbmath.org/authors/?q=ai:zhu.junSummary: This paper designs a new finite difference mapped unequal-sized trigonometric weighted essentially non-oscillatory (MUS-TWENO) scheme for solving hyperbolic conservation laws, highly oscillation problems, and some extreme problems containing low density, low pressure, or low energy. A new mapping function and associated new mapped nonlinear weights are proposed to reduce the difference between the linear weights and nonlinear weights in trigonometric polynomial space. It could get smaller numerical errors and obtain optimal fifth-order convergence with a tiny \(\varepsilon\) even near critical points in smooth regions when simulating some highly oscillatory problems. This new MUS-TWENO scheme uses three unequal-sized stencils to design three unequal degree trigonometric polynomials and the sophisticated optimal linear weights can be set as any positive numbers on condition that their summation is one. It is the first time that we can reconstruct a high degree trigonometric polynomial over the whole big stencil, while many classical high-order WENO spatial reconstructions only reconstruct the values at the boundary points or discrete quadrature points. Extensive benchmark examples including highly oscillatory problems and some extreme problems are used to testify the good representations of this new finite difference MUS-TWENO scheme.Intrusive generalized polynomial chaos with asynchronous time integration for the solution of the unsteady Navier-Stokes equationshttps://zbmath.org/1521.766322023-11-13T18:48:18.785376Z"Bonnaire, P."https://zbmath.org/authors/?q=ai:bonnaire.p"Pettersson, P."https://zbmath.org/authors/?q=ai:pettersson.peter|pettersson.per|pettersson.paul"Silva, C. F."https://zbmath.org/authors/?q=ai:silva.camilo-fSummary: Generalized polynomial chaos provides a reliable framework for many problems of uncertainty quantification in computational fluid dynamics. However, it fails for long-time integration of unsteady problems with stochastic frequency. In this work, the asynchronous time integration technique, introduced in previous works to remedy this issue for systems of ODEs, is applied to the Kármán vortex street problem. For this purpose, we make use of a stochastic clock speed that provides the phase shift between the realizations and enables the simulation of an in-phase behavior. Results of the proposed method are validated against Monte Carlo simulations and show good results for statistic fields and point-wise values such as phase portraits, as well as PDFs of the limit cycle. We demonstrate that low-order expansions are sufficient to meet the demands for some statistic measures. Therefore, computational costs are still competitive with those of the standard form of intrusive generalized polynomial chaos (igPC) and its non-intrusive counterpart (NigPC).A Fourier transformation based UGKS for Vlasov-Poisson equations in cylindrical coordinates \((r, \theta)\)https://zbmath.org/1521.766872023-11-13T18:48:18.785376Z"Ni, Anchun"https://zbmath.org/authors/?q=ai:ni.anchun"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.35"Ni, Guoxi"https://zbmath.org/authors/?q=ai:ni.guoxi"Chen, Yibing"https://zbmath.org/authors/?q=ai:chen.yibingSummary: A Fourier transformation based unified gas-kinetic scheme (UGKS) is proposed to simulate the kinetic behaviors of plasma in cylindrical coordinates \((r, \theta, v_r, v_\theta)\), the model is depicted by Vlasov-Poisson equations coupled with Bhatnagar-Gross-Krook (BGK) collision term. Fourier transformation is applied to the equations for \(\theta\) and a series of equations are gotten. Based on Strang-splitting strategy, Vlasov equations are divided into two parts, the transport-collision part solved by a multiscale gas-kinetic scheme, and acceleration part solved by Runge-Kutta method. The algorithm is applied on charge separation problem at plasma edge and Z-pinch configuration. Numerical results show our scheme can give a clear kinetic picture of plasma in 2D phase space, and also can capture the process from non-equilibrium to equilibrium state by Coulomb collisions.Discrete effects on the source term for the lattice Boltzmann modelling of one-dimensional reaction-diffusion equationshttps://zbmath.org/1521.766992023-11-13T18:48:18.785376Z"Silva, Goncalo"https://zbmath.org/authors/?q=ai:silva.goncalo|silva.goncalo-bSummary: This work presents a detailed numerical analysis of one-dimensional, time-dependent (linear) reaction-diffusion type equations modelled with the lattice Boltzmann method (LBM), using the two-relaxation-time (TRT) scheme, for the D1Q3 lattice. The interest behind this study is twofold. First, because it applies to the description of many engineering problems, such as the mass transport in membranes, the heat conduction in fins, or the population growth in biological systems. Second, because this study also permits understanding the general effect of solution-dependent sources in LBM, where this problem offers a simple, yet non-trivial, canonical groundwork. Without recurring to perturbative techniques, such as the Chapman-Enskog expansion, we exactly derive the macroscopic numerical scheme that is solved by the LBM-TRT model with a solution-dependent source and show that it obeys a four-level explicit finite difference structure. In the steady-state limit, this scheme reduces to a second-order finite difference approximation of the stationary reaction-diffusion equation that, due to artefacts from the source term discretization, may operate with an effective diffusion coefficient of negative value, although still remaining stable. Such a surprising result is demonstrated through an exact stability analysis that proves the unconditional stability of the LBM-TRT model with a solution-dependent source, in line with the already proven source-less pure diffusion case [\textit{Y. Lin} et al., ``Multiple-relaxation-time lattice Boltzmann model-based four-level finite-difference scheme for one-dimensional diffusion equations'', Phys. Rev. E (3) 104, No. 1, Article ID 015312, 14 p. (2021; \url{doi:10.1103/PhysRevE.104.015312})]. This proof enlarges the confidence over the LBM-TRT model robustness also for the (linear) reaction-diffusion problem class. Finally, a truncation error analysis is performed to disclose the structure of the leading order errors. From this knowledge, two strategies are proposed to improve the scheme accuracy from second- to fourth-order. One exclusively based on the tuning of the LBM-TRT scheme free-parameters, namely the two relaxation rates and the lattice weight coefficient, and the other based on the redefinition of the structure of the relaxation rates, where the leading order truncation error is absorbed into one of the relaxation rates, liberating the other to improve additional features of the scheme. Numerical tests presented in the last part of the work support the ensemble of theoretical findings.Perturbation, symmetry analysis, Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanicshttps://zbmath.org/1521.767362023-11-13T18:48:18.785376Z"Wang, Gangwei"https://zbmath.org/authors/?q=ai:wang.gangwei"Wazwaz, Abdul-Majid"https://zbmath.org/authors/?q=ai:wazwaz.abdul-majidSummary: In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg-de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain Bäcklund transformation associated with truncated Painlevé expansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal Bäcklund transformations of the derived conservation laws for the first time.Navier-Stokes-Fourier system with phase transitionshttps://zbmath.org/1521.767742023-11-13T18:48:18.785376Z"Watson, Stephen J."https://zbmath.org/authors/?q=ai:watson.stephen-jSummary: We consider the Navier-Stokes-Fourier (\(\mathcal{N}SF\)) system for a class of compressible fluids that exhibit a \textit{gas-liquid} phase transition at low temperatures. For the initial-boundary value problem corresponding to thermally insulated end-points that are held at a constant pressure, we establish the existence and uniqueness of temporally global classical solutions. A novel feature of the analysis presented here is the derivation of \textit{uniform} point-wise apriori estimates on the specific volume, which refines the non-uniform estimates framework developed in [the author, Arch. Ration. Mech. Anal. 153, No. 1, 1--37 (2000; Zbl 0996.74032)].Electron acoustic solitary waves in unmagnetized nonthermal plasmashttps://zbmath.org/1521.768072023-11-13T18:48:18.785376Z"Khalid, Muhammad"https://zbmath.org/authors/?q=ai:khalid.muhammad-zeeshan|khalid.muhammad-saif-ullah|khalid.muhammad-usman"Khan, Aqil"https://zbmath.org/authors/?q=ai:khan.aqil"Khan, Mohsin"https://zbmath.org/authors/?q=ai:khan.mohsin-r|khan.mohsin-s"Khan, Daud"https://zbmath.org/authors/?q=ai:khan.daud"Ahmad, Sheraz"https://zbmath.org/authors/?q=ai:ahmad.sheraz"Ata-ur-Rahman"https://zbmath.org/authors/?q=ai:ata-ur-rahman.Summary: Electron acoustic (EA) solitary waves (SWs) are studied in an unmagnetized plasma consisting of hot electrons (following Cairns-Tsalli distribution), inertial cold electrons, and stationary ions. By employing a reductive perturbation technique (RPT), the nonlinear Korteweg-de Vries (KdV) equation is derived and its SW solution is analyzed. Here, the effects of plasma parameters such as the nonextensivity parameter (\(q\)), the nonthermality of electrons (\(\alpha\)), and the cold-to-hot electron density ratio (\(\beta\)) are investigated.Bounds on heat transfer by incompressible flows between balanced sources and sinkshttps://zbmath.org/1521.768212023-11-13T18:48:18.785376Z"Song, Binglin"https://zbmath.org/authors/?q=ai:song.binglin"Fantuzzi, Giovanni"https://zbmath.org/authors/?q=ai:fantuzzi.giovanni"Tobasco, Ian"https://zbmath.org/authors/?q=ai:tobasco.ianSummary: Internally heated convection involves the transfer of heat by fluid motion between a distribution of sources and sinks. Focusing on the balanced case where the total heat added by the sources matches the heat taken away by the sinks, we obtain \textit{a priori} bounds on the minimum mean thermal dissipation \(\langle|\nabla T|^2\rangle\) as a measure of the inefficiency of transport. In the advective limit, our bounds scale with the inverse mean kinetic energy of the flow. The constant in this scaling law depends on the source-sink distribution, as we explain both in a pair of examples involving oscillatory or concentrated heating and cooling, and via a general asymptotic variational principle for optimizing transport. Key to our analysis is the solution of a pure advection equation, which we do to find examples of extreme heat transfer by cellular and `pinching' flows. When the flow obeys a momentum equation, our bound is re-expressed in terms of a flux-based Rayleigh number \(R\) yielding \(\langle|\nabla T|^2\rangle \geq CR^{-\alpha}\). The power \(\alpha\) is 0, 2/3 or 1 depending on the arrangement of the sources and sinks relative to gravity.Multifield variational formulations of diffusion initial boundary value problemshttps://zbmath.org/1521.768232023-11-13T18:48:18.785376Z"de Anda Salazar, Jorge"https://zbmath.org/authors/?q=ai:de-anda-salazar.jorge"Heuzé, Thomas"https://zbmath.org/authors/?q=ai:heuze.thomas"Stainier, Laurent"https://zbmath.org/authors/?q=ai:stainier.laurentSummary: We present two multifield and one single-field variational principles for the initial boundary value problem of diffusion. Chemical potential and concentration appear as conjugate variables in the multifield formulations. The main importance of the proposed formulations is the approach used to generate the variational principles, where the framework of \textit{Generalized Standard Materials} is used for constitutive laws while natural boundary conditions and the balance of mass are used as constraints of the optimization problem. This approach allows to derive such principles for multiphysic problems in a generic manner. A detailed derivation and analysis of the formulations are presented, where it can be seen their equivalence with the most common strong and weak forms of the problem using Fick's laws along with the logarithmic mass action law. From the stationarity condition with respect to the mass flux of the initially proposed functional, two main relations are identified. First, the chemical potential appears as the opposite of the Lagrange multiplier that allows to enforce the balance of mass and natural boundary conditions. Second, a conjugate relation is found for a given substance between its mass flux and the opposite of the gradient of its chemical potential. Furthermore, to reduce the number of variables of the initial variational principle, a field reduction is applied, reaching the model presented by \textit{C. Miehe} et al. [Int. J. Numer. Methods Eng. 99, No. 10, 737--762 (2014; Zbl 1352.74110)] for Fickean diffusion. Nevertheless, the aforementioned relations cannot be derived from the reduced model. Finally, a numerical implementation is presented for completeness where we compare the performance of the proposed formulations against the usual weak form.On the process of filtration of fractional viscoelastic liquid foodhttps://zbmath.org/1521.768362023-11-13T18:48:18.785376Z"Meng, Yahui"https://zbmath.org/authors/?q=ai:meng.yahui"Li, Botong"https://zbmath.org/authors/?q=ai:li.botong"Si, Xinhui"https://zbmath.org/authors/?q=ai:si.xinhui"Chen, Xuehui"https://zbmath.org/authors/?q=ai:chen.xuehui"Liu, Fawang"https://zbmath.org/authors/?q=ai:liu.fawangSummary: In the process of filtration, fluid impurities precipitate/accumulate; this results in an uneven inner wall of the filter, consequently leading to non-uniform suction/injection. The Riemannian-Liouville fractional derivative model is used to investigate viscoelastic incompressible liquid food flowing through a permeable plate and to generalize Fick's law. Moreover, we consider steady-state mass balance during ultrafiltration on a plate surface, and a fractional-order concentration boundary condition is established, thereby rendering the problem real and complex. The governing equation is numerically solved using the finite difference algorithm. The effects of the fractional constitutive models, generalized Reynolds number, generalized Schmidt number, and permeability parameter on the velocity and concentration fields are compared. The results show that an increase in fractional-order \(\alpha\) in the momentum equation leads to a decrease in the horizontal velocity. Anomalous diffusion described by the fractional derivative model weakens the mass transfer; therefore, the concentration decreases with increasing fractional derivative \(\gamma\) in the concentration equation.Existence and uniqueness for a convective phase change model with temperature-dependent viscosityhttps://zbmath.org/1521.768852023-11-13T18:48:18.785376Z"Belhamadia, Y."https://zbmath.org/authors/?q=ai:belhamadia.youssef"Deteix, J."https://zbmath.org/authors/?q=ai:deteix.jean"Jaffal-Mourtada, B."https://zbmath.org/authors/?q=ai:jaffal-mourtada.basma"Yakoubi, D."https://zbmath.org/authors/?q=ai:yakoubi.drissSummary: In this article, we consider a class of phase change model with temperature-dependent viscosity, convection and mixed boundary conditions on a bounded domain that reflect melting and solidification in a variety of real-world applications, such as metal casting and crystal growth. The mathematical model, which is based on the enthalpy formulation, takes into consideration the thermophysical differences between the liquid and solid states. The moving liquid-solid interface is explicitly fulfilled as the energy and momentum equations are solved over the full physical domain. Under particular assumptions, we derive various a priori estimates and prove well-posedness results. Numerical simulation of the model employed in the paper is presented as an illustration of an example of a melting problem.A Beale-Kato-Majda criterion for free boundary incompressible ideal magnetohydrodynamicshttps://zbmath.org/1521.769022023-11-13T18:48:18.785376Z"Fu, Jie"https://zbmath.org/authors/?q=ai:fu.jie"Hao, Chengchun"https://zbmath.org/authors/?q=ai:hao.chengchun"Yang, Siqi"https://zbmath.org/authors/?q=ai:yang.siqi"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.243Summary: We prove a continuation criterion for the free boundary problem of three-dimensional incompressible ideal magnetohydrodynamic (MHD) equations in a bounded domain, which is analogous to the theorem given in [\textit{J. T. Beale} et al., Commun. Math. Phys. 94, 61--66 (1984; Zbl 0573.76029)]. We combine the energy estimates of our previous works [\textit{C. Hao} and \textit{T. Luo}, Arch. Ration. Mech. Anal. 212, No. 3, 805--847 (2014; Zbl 1293.35244)] on incompressible ideal MHD and some analogous estimates in [\textit{D. Ginsberg}, SIAM J. Math. Anal. 53, No. 3, 3366--3384 (2021; Zbl 1473.35667)] to show that the solution can be continued as long as the curls of the magnetic field and velocity, the second fundamental form and injectivity radius of the free boundary and some norms of the pressure remain bounded, provided that the Taylor-type sign condition holds.
{\copyright 2023 American Institute of Physics}Optical solitons for the decoupled nonlinear Schrödinger equation using Jacobi elliptic approachhttps://zbmath.org/1521.780142023-11-13T18:48:18.785376Z"Sabi'u, Jamilu"https://zbmath.org/authors/?q=ai:sabiu.jamilu"Tala-Tebue, Eric"https://zbmath.org/authors/?q=ai:tala-tebue.eric"Rezazadeh, Hadi"https://zbmath.org/authors/?q=ai:rezazadeh.hadi"Arshed, Saima"https://zbmath.org/authors/?q=ai:arshed.saima"Bekir, Ahmet"https://zbmath.org/authors/?q=ai:bekir.ahmetSummary: Most of the important aspects of soliton propagation through optical fibers for transcontinental and transoceanic long distances can best be described using the nonlinear Schrödinger equation. Optical solitons are electromagnetic waves that span in nonlinear dispersive media and permit the stress and intensity to stay unaltered as a result of the delicate balance between dispersion and nonlinearity effects. However, this study exploited the Jacobi elliptic method and obtained different soliton solutions of the decoupled nonlinear Schrödinger equation with ease. Discussions about the obtained solutions were made with the aid of some 3D graphs.Simulation of Maxwell equation based on an ADI approach and integrated radial basis function-generalized moving least squares (IRBF-GMLS) method with reduced order algorithm based on proper orthogonal decompositionhttps://zbmath.org/1521.780162023-11-13T18:48:18.785376Z"Ebrahimijahan, Ali"https://zbmath.org/authors/?q=ai:ebrahimijahan.ali"Dehghan, Mehdi"https://zbmath.org/authors/?q=ai:dehghan.mehdi"Abbaszadeh, Mostafa"https://zbmath.org/authors/?q=ai:abbaszadeh.mostafa(no abstract)Comments on: ``Thermal solitons along wires with flux-limited lateral exchange''https://zbmath.org/1521.800072023-11-13T18:48:18.785376Z"Jordan, P. M."https://zbmath.org/authors/?q=ai:jordan.pedro-mSummary: A derivation error in the article [\textit{M. Sciacca} et al., J. Math. Phys. 62, No. 10, Article ID 101503, 14 p. (2021; Zbl 1480.80007)] cited in the title of this Comment is pointed out and corrected. In addition, the Maxwell-Cattaneo based model assumed therein is extended to include expected Joule heating effects; an alternative theory of second-sound that allows the same modeling to be performed, but with fewer assumptions, is noted and applied; and the difference between ordinary solitary waves and solitons is recalled.
{\copyright 2023 American Institute of Physics}Response to: ``Comments on: ``Thermal solitons along wires with flux-limited lateral exchange''''https://zbmath.org/1521.800122023-11-13T18:48:18.785376Z"Sciacca, M."https://zbmath.org/authors/?q=ai:sciacca.michele"Alvarez, F. X."https://zbmath.org/authors/?q=ai:alvarez.f-xavier"Jou, D."https://zbmath.org/authors/?q=ai:jou.david"Bafaluy, J."https://zbmath.org/authors/?q=ai:bafaluy.javierSummary: {\copyright 2023 American Institute of Physics}
Response to the comment [\textit{P. M. Jordan}, J. Math. Phys. 64, No. 9, Article ID 094101, 3 p. (2023; Zbl 1521.80007)].Simulations of dendritic solidification via the diffuse approximate methodhttps://zbmath.org/1521.800182023-11-13T18:48:18.785376Z"Najafi, Mahboubeh"https://zbmath.org/authors/?q=ai:najafi.mahboubeh"Dehghan, Mehdi"https://zbmath.org/authors/?q=ai:dehghan.mehdi(no abstract)An unconditionally stable fast high order method for thermal phase change modelshttps://zbmath.org/1521.800342023-11-13T18:48:18.785376Z"Wang, Weiwen"https://zbmath.org/authors/?q=ai:wang.weiwen"Azaiez, Mejdi"https://zbmath.org/authors/?q=ai:azaiez.mejdi"Xu, Chuanju"https://zbmath.org/authors/?q=ai:xu.chuanjuSummary: Thermal phase change problems arise in a large number of applications. In this paper, we consider a phase field model instead of the classical Stefan model to describe phenomena, which may appear in some complex phase change problems such as dendritic crystal growth, phase transformations in metallic alloys, etc. Our aim is to propose efficient and accurate schemes for the model, which is the coupling of a heat transfer equation and a phase field equation. The schemes are constructed based on an auxiliary variable approach for the phase field equation and semi-implicit treatment for the heat transfer equation. The main novelty of the paper consists in: (i) construction of the efficient schemes, which only requires solving several second-order elliptic problems with constant coefficients; (ii) proof of the unconditional stability of the schemes; (iii) fast high order solver for the resulting equations at each time step. A series of numerical examples are presented to verify the theoretical claims and to illustrate the efficiency of our method. As far as we know, it seems this is the first attempt made for the thermal phase change model of this type.Entropy of quantum systems with linear dissipationhttps://zbmath.org/1521.810172023-11-13T18:48:18.785376Z"Kirchanov, V. S."https://zbmath.org/authors/?q=ai:kirchanov.v-s(no abstract)Parametric family of discrimination information in extended para-statistics of non-extensive systemshttps://zbmath.org/1521.810412023-11-13T18:48:18.785376Z"Zaripov, R. G."https://zbmath.org/authors/?q=ai:zaripov.rinat-g(no abstract)Eigenvalue curves for generalized MIT bag modelshttps://zbmath.org/1521.810602023-11-13T18:48:18.785376Z"Arrizabalaga, Naiara"https://zbmath.org/authors/?q=ai:arrizabalaga.naiara"Mas, Albert"https://zbmath.org/authors/?q=ai:mas.albert"Sanz-Perela, Tomás"https://zbmath.org/authors/?q=ai:sanz-perela.tomas"Vega, Luis"https://zbmath.org/authors/?q=ai:vega.luisSummary: We study spectral properties of Dirac operators on bounded domains \(\Omega \subset{\mathbb{R}}^3\) with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter \(\tau \in \mathbb{R}\); the case \(\tau = 0\) corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of \(\tau\), and we exploit this monotonicity to study the limits as \(\tau \rightarrow \pm \infty\). We prove that if \(\Omega\) is not a ball then the first positive eigenvalue is greater than the one of a ball with the same volume for all \(\tau\) large enough. Moreover, we show that the first positive eigenvalue converges to the mass of the particle as \(\tau \downarrow -\infty\), and we also analyze its first order asymptotics.Stationary hypergeometric solitons and their stability in a Bose-Einstein condensate with \(\mathcal{PT}\)-symmetric potentialhttps://zbmath.org/1521.810622023-11-13T18:48:18.785376Z"Bhatia, Sanjana"https://zbmath.org/authors/?q=ai:bhatia.sanjana"Goyal, Amit"https://zbmath.org/authors/?q=ai:goyal.amit"Jana, Soumendu"https://zbmath.org/authors/?q=ai:jana.soumendu"Kumar, C. N."https://zbmath.org/authors/?q=ai:kumar.c-nagarajaSummary: We report the existence of stationary nonlinear matter-waves in a trapped Bose-Einstein condensate subject to a \(\mathcal{PT}\)-symmetric Pöschl-Teller potential with a gain/loss profile. Exact nonlinear modes are obtained and their stability criteria are determined. The analysis shows that beyond a critical depth of confining potential well, the condensate wavefunction is stable against small fluctuations in the field. Analytical results obtained are in good agreement with the numerical simulation of the localized modes in the \(\mathcal{PT}\) symmetry regime. Employing the isospectral hamiltonian technique of supersymmetric quantum mechanics, we demonstrate a mechanism to control the shape of the Pöschl-Teller well and hence the intensity of the localized modes. Most importantly, our results reveal that even with a small fluctuation present in the trapping potential bearing dissipation, the system is robust enough to support stable propagation of nonlinear modes.Exact time-evolution of a generalized two-dimensional quantum parametric oscillator in the presence of time-variable magnetic and electric fieldshttps://zbmath.org/1521.810632023-11-13T18:48:18.785376Z"Büyükaşık, Şirin A."https://zbmath.org/authors/?q=ai:buyukasik.sirin-a"Çayiç, Zehra"https://zbmath.org/authors/?q=ai:cayic.zehraSummary: The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei-Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin-Man'ko-Trifonov and the Lewis-Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy-Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.
{\copyright 2022 American Institute of Physics}Exact solutions of the generalized Dunkl oscillator in the Cartesian systemhttps://zbmath.org/1521.810642023-11-13T18:48:18.785376Z"Dong, Shi-Hai"https://zbmath.org/authors/?q=ai:dong.shihai"Quezada, L. F."https://zbmath.org/authors/?q=ai:quezada.luis-fernando"Chung, W. S."https://zbmath.org/authors/?q=ai:chung.won-sang"Sedaghatnia, P."https://zbmath.org/authors/?q=ai:sedaghatnia.parisa"Hassanabadi, H."https://zbmath.org/authors/?q=ai:hassanabadi.hassanSummary: In this paper, we use the generalized Dunkl derivatives instead of the standard partial derivatives in the Schrödinger equation to obtain an explicit expression of the generalized Dunkl-Schrödinger equation in 3D. It was found that this generalized Dunkl-Schrödinger equation for the 3D harmonic oscillator is exactly solvable in the Cartesian coordinates. From the relevant commutation relations, it is evident that the symmetry possessed by the original Dunkl Harmonic oscillator is \textit{broken} by the generalized Dunkl derivative. Finally, we show that energy levels can be affected by considering a deformation parameter \(\varepsilon\).Variational approach to the Schrödinger equation with a delta-function potentialhttps://zbmath.org/1521.810662023-11-13T18:48:18.785376Z"Fernández, Francisco M."https://zbmath.org/authors/?q=ai:fernandez.francisco-mSummary: We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form \(H_g = H + g \delta (x)\), where \(\delta (x)\) is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh-Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.Schrödinger's equation from Snell's lawhttps://zbmath.org/1521.810692023-11-13T18:48:18.785376Z"Lima, Nathan"https://zbmath.org/authors/?q=ai:lima.nathan|lima.nathan-willig"Karam, Ricardo"https://zbmath.org/authors/?q=ai:karam.ricardo(no abstract)Dispersion chain of quantum mechanics equationshttps://zbmath.org/1521.810722023-11-13T18:48:18.785376Z"Perepelkin, E. E."https://zbmath.org/authors/?q=ai:perepelkin.evgeny-e"Sadovnikov, B. I."https://zbmath.org/authors/?q=ai:sadovnikov.b-i"Inozemtseva, N. G."https://zbmath.org/authors/?q=ai:inozemtseva.natalia-g"Korepanova, A. A."https://zbmath.org/authors/?q=ai:korepanova.a-aSummary: Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton-Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space (the Wigner function) and the de Broglie-Bohm {\guillemotleft}pilot wave{\guillemotright} theory. An example of solving the Schrödinger equation of the second rank (for the phase space) is analyzed, which, in contrast to the Wigner function, gives a positive distribution density function.Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the spherehttps://zbmath.org/1521.810732023-11-13T18:48:18.785376Z"Quesne, C."https://zbmath.org/authors/?q=ai:quesne.christianeSummary: We consider a family of extensions of the Kepler-Coulomb potential on a \(d\)-dimensional sphere and analyze it in a deformed supersymmetric framework, wherein the starting potential is known to exhibit a deformed shape invariance property. We show that the members of the extended family are also endowed with such a property, provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build quasi-exactly solvable potentials with known ground and first-excited states. Some explicit results are obtained for the first three members of the family. We then use a generating function method, wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function \(W_+(r)\) [and its accompanying function \(W_-(r)]\). From the results obtained for the latter for the first three family members, we propose some formulas for \(W_\pm(r)\) valid for the \(m\)th family member, depending on \(m+1\) constants \(a_0,a_1,\dots,a_m\). Such constants satisfy a system of \(m+1\) linear equations. Solving the latter allows us to extend the results up to the seventh family member and then to formulate a conjecture giving the general structure of the \(a_i\) constants in terms of the parameters of the problem.The number of eigenvalues of the three-particle Schrödinger operator on three dimensional latticehttps://zbmath.org/1521.810752023-11-13T18:48:18.785376Z"Khalkhuzhaev, A. M."https://zbmath.org/authors/?q=ai:khalkhuzhaev.ahmad-m"Abdullaev, J. I."https://zbmath.org/authors/?q=ai:abdullaev.janikul-i"Boymurodov, J. Kh."https://zbmath.org/authors/?q=ai:boymurodov.j-khSummary: We consider the three-particle discrete Schrödinger operator \(H_{\mu,\gamma}(\mathbf{K})\), \(\mathbf{K}\in\mathbb{T}^3\) associated to a system of three particles (two fermions and one another particle) interacting through zero range pairwise potential \(\mu>0\) on the three-dimensional lattice \(\mathbb{Z}^3.\) It is proved that there exist positive numbers \(\gamma_2>\gamma_1\) that the operator \(H_{\mu,\gamma}(\boldsymbol{\pi})\), \(\boldsymbol{\pi}=(\pi,\pi,\pi)\) for \(\gamma\in(0,\gamma_1)\) has no eigenvalue, for \(\gamma\in(\gamma_1,\gamma_2)\) has a simple eigenvalue and for \(\gamma>\gamma_2\) it has three eigenvalues lying below the essential spectrum for sufficiently large \(\mu.\)On the spectrum of the one-particle Schrödinger operator with point interactionhttps://zbmath.org/1521.810762023-11-13T18:48:18.785376Z"Kulzhanov, Utkir"https://zbmath.org/authors/?q=ai:kulzhanov.utkir"Muminov, Z. I."https://zbmath.org/authors/?q=ai:muminov.zahriddin-i"Ismoilov, Golibjon"https://zbmath.org/authors/?q=ai:ismoilov.golibjonSummary: We consider a one-dimensional one-particle quantum system interacted by two identical point interactions situated symmetrically with respect to the origin at the points \(\pm x_0\). The corresponding Schrödinger operator (energy operator) is constructed as a self-adjoint extension of the symmetric Laplace operator. An essential spectrum is described and the condition for the existence of the eigenvalue of the Schrödinger operator is studied. The main results of the work are based on the study of the operator extension spectrum of the operator \(h\).On the number and location of eigenvalues of the two particle Schrödinger operator on a latticehttps://zbmath.org/1521.810772023-11-13T18:48:18.785376Z"Lakaev, S. N."https://zbmath.org/authors/?q=ai:lakaev.saidakhmat-n"Khamidov, Sh. I."https://zbmath.org/authors/?q=ai:khamidov.shakhobiddin-iSummary: We study the discrete spectrum of the two-particle Schrödinger operator \(\widehat{H}_{\lambda\mu}(K)\), \(K\in\mathbb{T}^2,\) associated to the Bose-Hubbard Hamiltonian \(\widehat{\mathbb{H}}_{\lambda\mu}\) of a system of two identical bosons interacting on site and nearest-neighbor sites in the two dimensional lattice \(\mathbb{Z}^2\) with interaction magnitudes \(\lambda\in\mathbb{R}\) and \(\mu\in\mathbb{R},\) respectively. Under certain conditions on \(\lambda\), \(\mu\in\mathbb{R}\) we prove that the discrete Schrödinger operator \(\widehat{H}_{\lambda\mu}(0)\) can have zero, one, two or three eigenvalues below the bottom or above the top of the essential spectrum. Moreover, we show the conditions for existence of three eigenvalues, where two of them are situated below the bottom of the essential spectrum, and other one above its top.Using zero-energy states to explore how external boundaries affect the number of bound states in a quantum wellhttps://zbmath.org/1521.810792023-11-13T18:48:18.785376Z"Timberlake, Todd K."https://zbmath.org/authors/?q=ai:timberlake.todd-keene"Babione, Sarah E."https://zbmath.org/authors/?q=ai:babione.sarah-e(no abstract)A non-trivial PT-symmetric continuum Hamiltonian and its eigenstates and eigenvalueshttps://zbmath.org/1521.810802023-11-13T18:48:18.785376Z"Mead, Lawrence R."https://zbmath.org/authors/?q=ai:mead.lawrence-r"Lee, Sungwook"https://zbmath.org/authors/?q=ai:lee.sungwook"Garfinkle, David"https://zbmath.org/authors/?q=ai:garfinkle.davidSummary: In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, \(\mathcal{C}\), for this system. All calculations are performed analytically and without approximation.
{\copyright 2022 American Institute of Physics}Equations of motion governing the dynamics of the exceptional points of parameterically dependent nonhermitian Hamiltonianshttps://zbmath.org/1521.810812023-11-13T18:48:18.785376Z"Šindelka, Milan"https://zbmath.org/authors/?q=ai:sindelka.milan"Stránský, Pavel"https://zbmath.org/authors/?q=ai:stransky.pavel"Cejnar, Pavel"https://zbmath.org/authors/?q=ai:cejnar.pavelSummary: We study exceptional points (EPs) of a nonhermitian Hamiltonian \(\hat{H}(\lambda, \delta)\) whose parameters \(\lambda\in\mathbb{C}\) and \(\delta\in\mathbb{R}\). As the real control parameter \(\delta\) is varied, the \(k\)th EP (or \(k\)th cluster of simultaneously existing EPs) of \(\hat{H}(\lambda, \delta)\) moves in the complex plane of \(\lambda\) along a continuous trajectory, \(\lambda_k(\delta)\). Using an appropriate non-hermitian formalism (based upon the \(c\)-product and not upon the conventional Dirac product), we derive a self-contained set of equations of motion (EOM) for the trajectory \(\lambda_k(\delta)\), while interpreting \(\delta\) as the propagation time. Such EOM become of interest whenever one wishes to study the response of EPs to external perturbations or continuous parametric changes of the pertinent Hamiltonian. This is e.g. the case of EPs emanating from hermitian curve crossings/degeneracies (which turn into avoided crossings/near-degeneracies when the Hamiltonian parameters are continuously varied). The presented EOM for EPs have not only their theoretical merits, they possess also a substantial practical relevance. Namely, the just presented approach can be regarded even as an efficient numerical method, useful for generating EPs for a broad class of complex quantum systems encountered in atomic, nuclear and condensed matter physics. Performance of such a method is tested here numerically on a simple yet nontrivial toy model.The existence and asymptotics of eigenvalues of Schrödinger operator on two dimensional latticeshttps://zbmath.org/1521.810902023-11-13T18:48:18.785376Z"Boltaev, A. T."https://zbmath.org/authors/?q=ai:boltaev.a-t"Almuratov, F. M."https://zbmath.org/authors/?q=ai:almuratov.firdavsjon-mSummary: We study the spectral properties of the Schrödinger-type operator \(\widehat{H}_{\mu}:=\widehat{H}_0+\mu\widehat{V}\), \(\mu>0\), associated to a one-particle system in two dimensional lattice \({\mathbb{Z}}^2,\) where the non-perturbed operator \(\widehat{H}_0\) is a convolution-type operator generated by a Hopping matrix \(\widehat{\varepsilon}:{\mathbb{Z}}^d\to{\mathbb{C}}\) and the potential \(\widehat{V}\) is the multiplication operator by a function \(\widehat{v}\) such that \(\widehat{v}(0)=a\), \(\widehat{v}(x)=b\) for \(|x|=1\) and \(\widehat{v}(x)=0\) for \(|x|\geq 2,\) where \(a,b\in{\mathbb{R}}\setminus\{0\}.\) Under certain regularity assumption on \(\widehat{\varepsilon},\) we establish the existence or non-existence of eigenvalues of \(\widehat{H}_{\mu}\) lying below the essential spectrum. Moreover, in the case of existence we study the convergent expansion of eigenvalues for sufficiently small and positive \(\mu.\)Upper bound for the diameter of a tree in the quantum graph theoryhttps://zbmath.org/1521.810912023-11-13T18:48:18.785376Z"Boyko, O. P."https://zbmath.org/authors/?q=ai:boyko.o-p"Martynyuk, O. M."https://zbmath.org/authors/?q=ai:martynyuk.olga-m"Pivovarchik, V. M."https://zbmath.org/authors/?q=ai:pivovarchik.vyacheslav-nSummary: We study two Sturm-Liouville spectral problems on an equilateral tree with continuity and Kirchhoff conditions at the internal vertices and Neumann conditions at the pendant vertices (first problem) and with Dirichlet conditions at the pendant vertices (second problem). The spectrum of each of these problems consists of infinitely many normal (isolated Fredholm) eigenvalues. It is shown that if we know the asymptotics of eigenvalues, then it is possible to estimate the diameter of a tree from above for each of these problems.Nonlinear optical properties in \(GaAs/GA_{0.7}Al_{0.3}As\) spherical quantum dots with Like-Deng-Fan-Eckart potentialhttps://zbmath.org/1521.810962023-11-13T18:48:18.785376Z"Chang, Ceng"https://zbmath.org/authors/?q=ai:chang.ceng"Li, Xuechao"https://zbmath.org/authors/?q=ai:li.xuechao"Wang, Xing"https://zbmath.org/authors/?q=ai:wang.xing.1|wang.xing"Zhang, Chaojin"https://zbmath.org/authors/?q=ai:zhang.chaojinSummary: The effects of several key factors on the nonlinear optical properties of spherical quantum dots (QDs) with like-Deng-Fan-Eckart potential are studied theoretically, aiming at nonlinear optical absorption coefficients (OACs) and refractive index changes (RICs). By using the Factorization method to solve the Schrödinger equation, the dependent wave functions and energy levels are determined, and the analytical expressions of OACs and RICs are obtained by making use of the iterative method and density-matrix theory. It is worth noting that the numerical results show that the radius of QDs, the depth of limiting potential and the tuned effective mass have an enormous impact on the peaks and formants of OACs and RICs.New classes of quadratically integrable systems in magnetic fields: the generalized cylindrical and spherical caseshttps://zbmath.org/1521.811062023-11-13T18:48:18.785376Z"Kubů, Ondřej"https://zbmath.org/authors/?q=ai:kubu.ondrej"Marchesiello, Antonella"https://zbmath.org/authors/?q=ai:marchesiello.antonella"Šnobl, Libor"https://zbmath.org/authors/?q=ai:snobl.liborSummary: We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no longer be connected to separation of variables in the Hamilton-Jacobi equation and can have more general leading order terms. We focus on two cases extending the physically relevant cylindrical- and spherical-type integrals. We find three new integrable systems in the generalized cylindrical case but none in the spherical one. We conjecture that this is related to the presence, respectively absence, of maximal abelian Lie subalgebra of the three-dimensional Euclidean algebra generated by first order integrals in the limit of vanishing magnetic field. By investigating superintegrability, we find only one (minimally) superintegrable system among the integrable ones. It does not separate in any orthogonal coordinate system. This system provides a mathematical model of a helical undulator placed in an infinite solenoid.Solution of quantum eigenvalue problems by means of algebraic consistency conditionshttps://zbmath.org/1521.811082023-11-13T18:48:18.785376Z"de la Peña, L."https://zbmath.org/authors/?q=ai:pena.l-de-la"Cetto, A. M."https://zbmath.org/authors/?q=ai:cetto.ana-maria"Valdés-Hernández, A."https://zbmath.org/authors/?q=ai:valdes-hernandez.andrea(no abstract)Algebra of the spinor invariants and the relativistic hydrogen atomhttps://zbmath.org/1521.811112023-11-13T18:48:18.785376Z"Eremko, Alexander"https://zbmath.org/authors/?q=ai:eremko.alexander"Brizhik, Larissa"https://zbmath.org/authors/?q=ai:brizhik.larissa"Loktev, Vadim"https://zbmath.org/authors/?q=ai:loktev.vadimSummary: It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry \(SU(2)\) of the Dirac equation. The total symmetry of the Dirac equation is the symmetry \(SO(3)\otimes SU(2)\). The generator of the \(SO(3)\) symmetry group is given by the total momentum operator, and the generator of \(SU(2)\) group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson-Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers.Semiclassical analysis of quantum asymptotic fields in the Yukawa theoryhttps://zbmath.org/1521.811362023-11-13T18:48:18.785376Z"Ammari, Zied"https://zbmath.org/authors/?q=ai:ammari.zied"Falconi, Marco"https://zbmath.org/authors/?q=ai:falconi.marco"Olivieri, Marco"https://zbmath.org/authors/?q=ai:olivieri.marcoSummary: In this article, we study the asymptotic fields of the Yukawa particle-field model of quantum physics in the semiclassical regime \(\hslash \to 0\), with an interaction subject to an ultraviolet cutoff. We show that the transition amplitudes between final (respectively initial) states converge towards explicit quantities involving the outgoing (respectively incoming) wave operators of the nonlinear Schrödinger-Klein-Gordon (S-KG) equation. Thus, we rigorously link the scattering theory of the Yukawa model to that of the Schrödinger-Klein-Gordon equation. Moreover, we prove that the asymptotic vacuum states of the Yukawa model have a phase space concentration property around classical radiationless solutions. Under further assumptions, we show that the S-KG energy admits a unique minimizer modulo symmetries and we identify exactly the semiclassical measure of Yukawa ground states. Some additional consequences of asymptotic completeness are also discussed, and some further open questions are raised.Fractional skyrmion molecules in a \(\mathbb{C}P^{N- 1}\) modelhttps://zbmath.org/1521.811702023-11-13T18:48:18.785376Z"Akagi, Yutaka"https://zbmath.org/authors/?q=ai:akagi.yutaka"Amari, Yuki"https://zbmath.org/authors/?q=ai:amari.yuki"Gudnason, Sven Bjarke"https://zbmath.org/authors/?q=ai:gudnason.sven-bjarke"Nitta, Muneto"https://zbmath.org/authors/?q=ai:nitta.muneto"Shnir, Yakov"https://zbmath.org/authors/?q=ai:shnir.yakov-mSummary: We study fractional Skyrmions in a \(\mathbb{C}P^2\) baby Skyrme model with a generalization of the easy-plane potential. By numerical methods, we find stable, metastable, and unstable solutions taking the shapes of molecules. Various solutions possess discrete symmetries, and the origin of those symmetries are traced back to congruencies of the fields in homogeneous coordinates on \(\mathbb{C}P^2\).Multi-soliton dynamics of anti-self-dual gauge fieldshttps://zbmath.org/1521.811732023-11-13T18:48:18.785376Z"Hamanaka, Masashi"https://zbmath.org/authors/?q=ai:hamanaka.masashi"Huang, Shan-Chi"https://zbmath.org/authors/?q=ai:huang.shan-chiSummary: We study dynamics of multi-soliton solutions of anti-self-dual Yang-Mills equations for \(G = \mathrm{GL}(2, \mathbb{C})\) in four-dimensional spaces. The one-soliton solution can be interpreted as a codimension-one soliton in four-dimensional spaces because the principal peak of action density localizes on a three-dimensional hyperplane. We call it the soliton wall. We prove that in the asymptotic region, the \(n\)-soliton solution possesses \(n\) isolated localized lumps of action density, and interpret it as \(n\) intersecting soliton walls. More precisely, each action density lump is essentially the same as a soliton wall because it preserves its shape and ``velocity'' except for a position shift of principal peak in the scattering process. The position shift results from the nonlinear interactions of the multi-solitons and is called the phase shift. We calculate the phase shift factors explicitly and find that the action densities can be real-valued in three kind of signatures. Finally, we show that the gauge group can be \(G = \mathrm{SU}(2)\) in the Ultrahyperbolic space \(\mathbb{U}\) (the split signature \((+, +, -, -)\)). This implies that the intersecting soliton walls could be realized in all region in \(N = 2\) string theories. It is remarkable that quasideterminants dramatically simplify the calculations and proofs.Topological recursion and uncoupled BPS structures. II: Voros symbols and the \(\tau\)-functionhttps://zbmath.org/1521.813942023-11-13T18:48:18.785376Z"Iwaki, Kohei"https://zbmath.org/authors/?q=ai:iwaki.kohei"Kidwai, Omar"https://zbmath.org/authors/?q=ai:kidwai.omarSummary: We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's ``BPS Riemann-Hilbert problem''. In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices \(\Omega\) in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's \(\tau\)-function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function \(Z_\mathrm{TR}\), up to a simple factor.
For Part I, see [the authors, Adv. Math. 398, Article ID 108191, 54 p. (2022; Zbl 1486.81157)].Solution of the quantum three-body problem in a neighborhood of the three-particle forward scattering directionhttps://zbmath.org/1521.814142023-11-13T18:48:18.785376Z"Budylin, A. M."https://zbmath.org/authors/?q=ai:budylin.a-m"Levin, S. B."https://zbmath.org/authors/?q=ai:levin.s-bSummary: The asymptotic behavior of the solution of the scattering problem for three three- dimensional Coulomb quantum particles in a neighborhood of the three-particle forward scattering direction is considered.High-energy two-electron transfer in ion-atom collisionshttps://zbmath.org/1521.814212023-11-13T18:48:18.785376Z"Belkić, Dževad"https://zbmath.org/authors/?q=ai:belkic.dzevadSummary: Two-electron transfer by fast heavy nuclei from heliumlike targets is studied. A detailed sequence of comprehensive computations is carried out in a large keV-MeV range of the projectile energies. This set is illustrated with total cross sections for double capture by alpha particles from helium atoms using several frequently applied four-body quantum-mechanical distorted wave models with the correct boundary conditions. The sensitivity of the obtained total cross sections is examined for different choices of the bound and continuum states. Especially at high energies, the influence of the compactness of the bound states is investigated by reference to the mechanism of the velocity matching kinematic double electron capture. Also considered is the dependence of these cross sections on the electronic screening of the projectile and the target nuclear charges in the bound and continuum states. The impact of this electronic shielding on total cross sections is assessed by reference to the corresponding bare nuclear charges in the bound and continuum states. Relative to all the available experimental data (100--6000 keV), the found striking model-dependence implies that two-electron transfer is sharply different from the associated one-electron transfer involving the same colliding particles.Photodetachment of the outer-most electron(s) in few-electron atomic systems. Variational principle for the cross sectionshttps://zbmath.org/1521.814242023-11-13T18:48:18.785376Z"Frolov, Alexei M."https://zbmath.org/authors/?q=ai:frolov.alexei-mSummary: Photodetachment of the outer-most electron(s) in a few-electron atomic systems is investigated. In particular, photodetachment of the outer-most electron is considered in the neutral atoms and positively charged atomic ions which contain more than one electron. We also consider photodetachment of the outer-most electron in the negatively charged atomic ions, including the negatively charged hydrogen \(\mathrm{H}^-\) ion. In all these cases we have derived the closed analytical formulas for the differential and total cross section(s) of photodetachment of the outer-most electron(s). For one-electron atoms and ions our method allows one to derive the formulas for the photoionization cross sections which exactly coincide with the expressions obtained in earlier studies. A rigorous variational approach, which can be applied to determine atomic photoionization and photodetachment cross sections, is also developed. This our approach is based on the principle of optimal projection which is the fundamental principle used the physics of transition processes and reactions.Vacuum instability due to the creation of neutral fermion with anomalous magnetic moment by magnetic-field inhomogeneitieshttps://zbmath.org/1521.814552023-11-13T18:48:18.785376Z"Adorno, T. C."https://zbmath.org/authors/?q=ai:adorno.tiago-c"He, Zi-Wang"https://zbmath.org/authors/?q=ai:he.zi-wang"Gavrilov, S. P."https://zbmath.org/authors/?q=ai:gavrilov.s-p"Gitman, D. M."https://zbmath.org/authors/?q=ai:gitman.dmitri-mSummary: We study neutral fermions pair creation with anomalous magnetic moment from the vacuum by time-independent magnetic-field inhomogeneity as an external background. We show that the problem is technically reduced to the problem of charged-particle creation by an electric step, for which the nonperturbative formulation of strong-field QED is used. We consider a magnetic step given by an analytic function and whose inhomogeneity may vary from a ``gradual'' to a ``sharp'' field configuration. We obtain corresponding exact solutions of the Dirac-Pauli equation with this field and calculate pertinent quantities characterizing vacuum instability, such as the differential mean number and flux density of pairs created from the vacuum, vacuum fluxes of energy and magnetic moment. We show that the vacuum flux in one direction is formed from fluxes of particles and antiparticles of equal intensity and with the same magnetic moments parallel to the external field. Backreaction to the vacuum fluxes leads to a smoothing of the magnetic-field inhomogeneity. We also estimate critical magnetic field intensities, near which the phenomenon could be observed.Compatibility of DFT+U with non-collinear magnetism and spin-orbit coupling within a framework of numerical atomic orbitalshttps://zbmath.org/1521.814912023-11-13T18:48:18.785376Z"Gómez-Ortiz, Fernando"https://zbmath.org/authors/?q=ai:gomez-ortiz.fernando"Carral-Sainz, Nayara"https://zbmath.org/authors/?q=ai:carral-sainz.nayara"Sifuna, James"https://zbmath.org/authors/?q=ai:sifuna.james"Monteseguro, Virginia"https://zbmath.org/authors/?q=ai:monteseguro.virginia"Cuadrado, Ramón"https://zbmath.org/authors/?q=ai:cuadrado.ramon"García-Fernández, Pablo"https://zbmath.org/authors/?q=ai:fernandez.pablo-garcia"Junquera, Javier"https://zbmath.org/authors/?q=ai:junquera.javierSummary: We report the extension of the density-functional theory plus Hubbard U (DFT+U) method to the case of non-collinear magnetism and spin-orbit coupling in a framework of numerical atomic orbitals. Both the Hubbard repulsion term \(U\), and the exchange \(J\) parameters are explicitly included and treated separately. The occupation numbers of the localized orbitals belonging to the correlated shell are computed from the projections of the Kohn-Sham eigenfunctions onto a set of non-overlapping, orthogonal, localized projectors. We provide the detailed expressions for the total energy, forces and stresses including the Pulay corrections. Our implementation on the version 5.0 of the \textsc{siesta} package has been validated with simulations carried out in isolated atoms and bulk solids including atoms with a strong spin-orbit coupling.Two-body Coulomb problem and \(g^{(2)}\) algebra (once again about the hydrogen atom)https://zbmath.org/1521.814942023-11-13T18:48:18.785376Z"Turbiner, Alexander V."https://zbmath.org/authors/?q=ai:turbiner.alexander"Escobar Ruiz, Adrian M."https://zbmath.org/authors/?q=ai:escobar-ruiz.mauricio-aSummary: Taking the Hydrogen atom as an example it is shown that if the symmetry of a three-dimensional system is \(O(2) \oplus Z_2\), the variables \((r, \rho, \varphi)\) allow a separation of the variable \(\varphi\), and the eigenfunctions define a new family of orthogonal polynomials in two variables, \((r, \rho^2)\). These polynomials are related to the finite-dimensional representations of the algebra \(gl(2) \ltimes R^3\in g^{(2)}\) (discovered by S Lie around 1880 which went almost unnoticed), which occurs as the hidden algebra of the \(G_2\) rational integrable system of 3 bodies on the line with 2- and 3-body interactions (the Wolfes model). Namely, those polynomials occur intrinsically in the study of the Zeeman effect on Hydrogen atom. It is shown that in the variables \((r, \rho, \varphi)\) in the quasi-exactly-solvable generalized Coulomb problem new polynomial eigenfunctions in \((r, \rho^2)\)-variables are found.Spinsim: a GPU optimized Python package for simulating spin-half and spin-one quantum systemshttps://zbmath.org/1521.815002023-11-13T18:48:18.785376Z"Tritt, Alex"https://zbmath.org/authors/?q=ai:tritt.alex"Morris, Joshua"https://zbmath.org/authors/?q=ai:morris.joshua"Hochstetter, Joel"https://zbmath.org/authors/?q=ai:hochstetter.joel"Anderson, R. P."https://zbmath.org/authors/?q=ai:anderson.ross-p"Saunderson, James"https://zbmath.org/authors/?q=ai:saunderson.james"Turner, L. D."https://zbmath.org/authors/?q=ai:turner.l-dSummary: \textit{Spinsim} simulates the quantum dynamics of unentangled spin-1/2 and spin-1 systems evolving under time-dependent control. While other solvers for the time-dependent Schrödinger equation optimize for larger state spaces but less temporally-rich control, \textit{spinsim} is optimized for intricate time evolution of a minimalist system. Efficient simulation of individual or ensemble quanta driven by adiabatic sweeps, elaborate pulse sequences, complex signals and non-Gaussian noise is the primary target application. We achieve fast and robust evolution using a geometric integrator to bound errors over many steps, and split the calculation parallel-in-time on a GPU using the \textit{numba} just-in-time compiler. Speed-up is three orders of magnitude over \textit{QuTip}'s \texttt{sesolve} and \textit{Mathematica}'s \texttt{NDSolve}, and four orders over \textit{SciPy}'s \texttt{ivp\_solve} for equal accuracy. Interfaced through python, \textit{spinsim} should be useful for simulating robust state preparation, inversion and dynamical decoupling sequences in NMR and MRI, and in quantum control, memory and sensing applications with two- and three-level quanta.On a magnetic Lieb-Thirring-type estimate and the stability of bipolarons in graphenehttps://zbmath.org/1521.815022023-11-13T18:48:18.785376Z"Alves, Magno B."https://zbmath.org/authors/?q=ai:alves.magno-branco"Del Cima, Oswaldo M."https://zbmath.org/authors/?q=ai:del-cima.oswaldo-m"Franco, Daniel H. T."https://zbmath.org/authors/?q=ai:franco.daniel-h-t"Pereira, Emmanuel"https://zbmath.org/authors/?q=ai:pereira.emmanuel-aSummary: Two-dimensional Weyl-Dirac relativistic fermions have attracted tremendous interest in condensed matter as they mimic relativistic high-energy physics. This paper concerns two-dimensional Weyl-Dirac operators in the presence of magnetic fields, in addition to a short-range scalar electric potential of the Bessel-Macdonald-type, restricted to its positive spectral subspace. This operator emerges from the action of a pristine graphene-like \(\mathrm{QED}_3\) model recently proposed by \textit{W. B. De Lima}, \textit{O. M. Del Cima} and \textit{E. S. Miranda} [Eur. Phys. J. B, Condens. Matter Complex Syst. B 93, No. 10, Paper No. 187, (2020; \url{doi:10.1140/epjb/e2020-100594-7})]. A magnetic Lieb-Thirring-type inequality \textit{à la} Shen is derived for the sum of the negative eigenvalues of the magnetic Weyl-Dirac operators restricted to their positive spectral subspace. An application to the stability of bipolarons in graphene in the presence of magnetic fields is given.
{\copyright 2023 American Institute of Physics}Retraction note to: ``Algebraic approach to the Tavis-Cummings model with three modes of oscillation''https://zbmath.org/1521.815032023-11-13T18:48:18.785376Z"Choreño, E."https://zbmath.org/authors/?q=ai:choreno.e"Ojeda-Guillén, D."https://zbmath.org/authors/?q=ai:ojeda-guillen.d"Granados, V. D."https://zbmath.org/authors/?q=ai:granados.victor-dRetraction note to the authors' paper [J. Math. Phys. 59, No. 7, 073506, 14 p. (2018; Zbl 1394.81181)].Heat kernel for the quantum Rabi modelhttps://zbmath.org/1521.815082023-11-13T18:48:18.785376Z"Reyes-Bustos, Cid"https://zbmath.org/authors/?q=ai:reyes-bustos.cid"Wakayama, Masato"https://zbmath.org/authors/?q=ai:wakayama.masatoSummary: The quantum Rabi model (QRM) is widely recognized as a particularly important model in quantum optics and beyond. It is considered to be the simplest and most fundamental system describing quantum light-matter interaction. The objective of the paper is to give an analytical formula of the heat kernel of the Hamiltonian explicitly by infinite series of iterated integrals. The derivation of the formula is based on the direct evaluation of the Trotter-Kato product formula without the use of Feynman-Kac path integrals. More precisely, the infinite sum in the expression of the heat kernel arises from the reduction of the Trotter-Kato product formula into sums over the orbits of the action of the infinite symmetric group \(\mathfrak{S}_\infty\) on the group \(\mathbb{Z}^\infty_2\), and the iterated integrals are then considered as the orbital integral for each orbit. Here, the groups \(\mathbb{Z}^\infty_2\) and \(\mathfrak{S}_\infty\) are the inductive limit of the families \(\{\mathbb{Z}^n_2 \}_{n \geq 0}\) and \(\{\mathfrak{S}_n\}_{n \geq 0}\), respectively. In order to complete the reduction, an extensive study of harmonic (Fourier) analysis on the inductive family of abelian groups \(\mathbb{Z}^n_2\) (\(n \geq 0\)) together with a graph theoretical investigation is crucial. To the best knowledge of the authors, this is the first explicit computation for obtaining a closed formula of the heat kernel for a non-trivial realistic interacting quantum system. The heat kernel of this model is further given by a two-by-two matrix valued function and is expressed as a direct sum of two respective heat kernels representing the parity (\(\mathbb{Z}_2\)-symmetry) decomposition of the Hamiltonian by parity.Dyson diffusion on a curved contourhttps://zbmath.org/1521.820072023-11-13T18:48:18.785376Z"Zabrodin, A. V."https://zbmath.org/authors/?q=ai:zabrodin.anton-vSummary: We define the Dyson diffusion process on a curved smooth closed contour in the plane and derive the Fokker-Planck equation for the probability density. Its stationary solution is shown to be the Boltzmann weight for the logarithmic gas confined on the contour.Maxwell's equations and Lorentz transformationshttps://zbmath.org/1521.830022023-11-13T18:48:18.785376Z"Aguirregabiria, J. M."https://zbmath.org/authors/?q=ai:aguirregabiria.juan-maria"Hernández, A."https://zbmath.org/authors/?q=ai:hernandez.amanda|hernandez.alexis-r|hernandez.anderson-melchor|hernandez.alejandro-lopez|hernandez.alan-e-lopez|hernandez.ankai|hernandez.alfonso|hernandez.angelica|hernandez.alejandro-s|hernandez.andres-f|hernandez.a-calvo|hernandez.aracelis|hernandez.arezky-h|hernandez.a-i|melle-hernandez.alejandro|hernandez.adrian|hernandez.adolfo|hernandez.alejo|hernandez.arturo|hernandez.alberto-j|hernandez.aurelio|hernandez.alberto-m|hernandez.alejandro-mario|hernandez.antonio-j|hernandez.alvaro|hernandez.a-e-carcamo|hernandez.araceli|hernandez.anabel"Rivas, M."https://zbmath.org/authors/?q=ai:rivas.miriam|rivas.manuel-a|rivas.mauricio-a|rivas.mariolys|rivas-rodriguez.maria-teresa|rivas.m-teresa|rivas.matthew-l|rivas.martin|rivas.maria-jesus|rivas.migdalia(no abstract)Exact analytical vacuum solutions of \(R^n\)-gravity model depending on two variableshttps://zbmath.org/1521.830122023-11-13T18:48:18.785376Z"Shubina, Maria"https://zbmath.org/authors/?q=ai:shubina.mariaSummary: In this paper we consider the metric power-law \(f(R)\sim R^n\)-gravity model for the four-dimensional metric tensor depending on two coordinates. We obtain exact analytical vacuum solutions for different values of \(n\). These solutions contain both non-stationary configurations of the travelling wave type and stationary ones, in particular, depending on one radial variable.The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild backgroundhttps://zbmath.org/1521.830212023-11-13T18:48:18.785376Z"Hennig, Jörg"https://zbmath.org/authors/?q=ai:hennig.jorg-dieterSummary: We resume former discussions of the conformally invariant wave equation on a Schwarzschild background, with a particular focus on the behaviour of solutions near the `cylinder', i.e. Friedrich's representation of spacelike infinity. This analysis can be considered a toy model for the behaviour of the full Einstein equations and the resulting logarithmic singularities that appear to be characteristic for massive spacetimes. The investigation of the \textit{Cauchy} problem for the conformally invariant wave equation \textit{J. Frauendiener} and \textit{J. Hennig} [Classical Quantum Gravity 35, No. 6, Article ID 065015, 19 p. (2018; Zbl 1386.83080)] showed that solutions generically develop logarithmic singularities at infinitely many expansion orders at the cylinder, but an arbitrary finite number of these singularities can be removed by appropriately restricting the initial data prescribed at \(t = 0\). From a physical point of view, any data at \(t = 0\) are determined from the earlier history of the system and hence not exactly `free data'. Therefore, it is appropriate to ask what happens if we `go further back in time' and prescribe initial data as early as possible, namely at a portion of past null infinity, and on a second past null hypersurface to complete the initial value problem. Will regular data at past null infinity automatically lead to a regular evolution up to future null infinity? Or does past regularity restrict the solutions too much, and regularity at both null infinities is mutually exclusive? Or do we still have suitable degrees of freedom for the data that can be chosen to influence regularity of the solutions to any desired degree? In order to answer these questions, we study the corresponding \textit{characteristic} initial value problem. In particular, we investigate in detail the appearance of singularities at expansion orders \(n = 0, \dots, 4\) for angular modes \(\ell = 0, \dots, 4\).Weyl curvature evolution system for GRhttps://zbmath.org/1521.830332023-11-13T18:48:18.785376Z"Krasnov, Kirill"https://zbmath.org/authors/?q=ai:krasnov.kirill-v"Shaw, Adam"https://zbmath.org/authors/?q=ai:shaw.adamSummary: Starting from the chiral first-order pure connection formulation of General Relativity, we put the field equations of GR in a strikingly simple evolution system form. The two dynamical fields are a complex symmetric tracefree \(3 \times 3\) matrix \(\Psi^{ij}\), which encodes the self-dual part of the Weyl curvature tensor, as well as a spatial \(\mathrm{SO}(3, \mathbb{C})\) connection \(A^i_a\). The right-hand sides of the evolution equations also contain the triad for the spatial metric, and this is constructed non-linearly from the field \(\Psi^{ij}\) and the curvature of the spatial connection \(A^i_a\). The evolution equations for this pair are first order in both time and spatial derivatives, and so simple that they could have been guessed without a computation. They are the most natural spin two generalisations of Maxwell's spin one equations. We also determine the modifications of the evolution system needed to enforce the `constraint sweeping', so that any possible numerical violation of the constraints present becomes propagating and gets removed from the computational grid.Hamilton-Jacobi and Klein-Gordon-Fock equations for a charged test particle in space-time with simply transitive four-parameter groups of motionshttps://zbmath.org/1521.830352023-11-13T18:48:18.785376Z"Obukhov, V. V."https://zbmath.org/authors/?q=ai:obukhov.valery-vSummary: Metric components of potentials of admissible electromagnetic fields in spaces with simply transitive motion group \(G_4\) are found. The components of vector tetrads corresponding to the components of the metric tensors found by \textit{A. Z. Petrov} [Einstein spaces, Oxford: Pergamon Press (1969; Zbl 0174.28305)] are given. The results obtained complement the coordinate-free classification given in [\textit{A. A. Magazev} et al., Theor. Math. Phys. 156, No. 2, 1127--1141 (2008; Zbl 1151.37049); translation from Teor. Mat. Fiz. 156, No. 2, 189--206 (2008)]. Previously, admissible electromagnetic fields were found for the case when three- and four-parameter groups of motions act on hypersurfaces of spacetime. Thus, non-equivalent sets of potentials for all electromagnetic fields that admit three- and four-parameter groups of motions are known now.
{\copyright 2023 American Institute of Physics}Friedmann equations and cosmic bounce in a modified cosmological scenariohttps://zbmath.org/1521.830382023-11-13T18:48:18.785376Z"Alonso-Serrano, Ana"https://zbmath.org/authors/?q=ai:alonso-serrano.ana"Liška, Marek"https://zbmath.org/authors/?q=ai:liska.marek"Vicente-Becerril, Antonio"https://zbmath.org/authors/?q=ai:vicente-becerril.antonioSummary: In this work we present a derivation of modified Raychaudhuri and Friedmann equations from a phenomenological model of quantum gravity based on the thermodynamics of spacetime. Starting from general gravitational equations of motion which encode low-energy quantum gravity effects, we found its particular solution for homogeneous and isotropic universes with standard matter content, obtaining a modified Raychaudhuri equation. Then, we imposed local energy conservation and used a perturbative treatment to derive a modified Friedmann equation. The modified evolution in the early universe we obtained suggests a replacement of the Big Bang singularity by a regular bounce. Lastly, we also briefly discuss the range of validity of the perturbative approach and its results.A scattering theory approach to Cauchy horizon instability and applications to mass inflationhttps://zbmath.org/1521.831362023-11-13T18:48:18.785376Z"Luk, Jonathan"https://zbmath.org/authors/?q=ai:luk.jonathan"Oh, Sung-Jin"https://zbmath.org/authors/?q=ai:oh.sung-jin"Shlapentokh-Rothman, Yakov"https://zbmath.org/authors/?q=ai:shlapentokh-rothman.yakovSummary: Motivated by the strong cosmic censorship conjecture, we study the linear scalar wave equation in the interior of subextremal strictly charged Reissner-Nordström black holes by analyzing a suitably defined ``scattering map'' at 0 frequency. The method can already be demonstrated in the case of spherically symmetric scalar waves on Reissner-Nordström: we show that assuming suitable \((L^2\)-averaged) upper and lower bounds on the event horizon, one can prove \((L^2\)-averaged) polynomial lower bound for the solution
\begin{itemize}
\item[(1)] on any radial null hypersurface transversally intersecting the Cauchy horizon, and
\item[(2)] along the Cauchy horizon toward timelike infinity.
\end{itemize}
Taken together with known results regarding solutions to the wave equation in the exterior, (1) above in particular provides yet another proof of the linear instability of the Reissner-Nordström Cauchy horizon. As an application of (2) above, we prove a conditional mass inflation result for a nonlinear system, namely the Einstein-Maxwell-(real)-scalar field system in spherical symmetry. For this model, it is known that for a generic class of Cauchy data \(\mathcal{G}\), the maximal globally hyperbolic future developments are \(C^2\)-future-inextendible. We prove that if a (conjectural) improved decay result holds in the exterior region, then for the maximal globally hyperbolic developments arising from initial data in \(\mathcal{G}\), the Hawking mass blows up identically on the Cauchy horizon.Kähler-Einstein metrics near an isolated log-canonical singularityhttps://zbmath.org/1521.831642023-11-13T18:48:18.785376Z"Datar, Ved"https://zbmath.org/authors/?q=ai:datar.ved-v"Fu, Xin"https://zbmath.org/authors/?q=ai:fu.xin.1"Song, Jian"https://zbmath.org/authors/?q=ai:song.jianSummary: We construct Kähler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler-Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler-Einstein metrics from hyperbolic geometry.Static neutron stars perspective of quadratic and induced inflationary attractor scalar-tensor theorieshttps://zbmath.org/1521.831742023-11-13T18:48:18.785376Z"Oikonomou, V. K."https://zbmath.org/authors/?q=ai:oikonomou.vasilis-kSummary: This study focuses on the static neutron star perspective for two types of cosmological inflationary attractor theories, namely the induced inflationary attractors and the quadratic inflationary attractors. The two cosmological models can be discriminated cosmologically, since one of the two does not provide a viable inflationary phenomenology, thus in this paper we investigate the predictions of these theories for static neutron stars, mainly focusing on the mass and radii of neutron stars. We aim to demonstrate that although the models have different inflationary phenomenology, the neutron star phenomenology predictions of the two models are quite similar. We solve numerically the Tolman-Oppenheimer-Volkoff equations in the Einstein frame using a powerful double shooting numerical technique, and after deriving the mass-radius graphs for three types of polytropic equations of state, we derive the Jordan frame mass and radii. With regard the equations of state we use polytropic equation of state with the small density part being either the Wiringa-Fiks-Fabrocini, the Akmal-Pandharipande-Ravenhall or the intermediate stiffness equation of state Skyrme-Lyon (SLy). The results of our models will be confronted with quite stringent recently developed constraints on the radius of neutron stars with specific mass.
As we show, the only equation of state which provides results compatible with the constraints is the SLy, for both the quadratic and induced inflation attractors. Thus nowadays, scalar-tensor descriptions of neutron stars are quite scrutinized due to the growing number of constraining observations, which eventually may also constrain theories of inflation.Spin-orbit gravitational locking -- an effective potential approachhttps://zbmath.org/1521.850022023-11-13T18:48:18.785376Z"Clouse, Christopher"https://zbmath.org/authors/?q=ai:clouse.christopher"Ferroglia, Andrea"https://zbmath.org/authors/?q=ai:ferroglia.andrea"Fiolhais, Miguel C. N."https://zbmath.org/authors/?q=ai:fiolhais.miguel-c-nSummary: The objective of this paper is to study the tidally locked 3:2 spin-orbit resonance of Mercury around the Sun. In order to achieve this goal, the effective potential energy that determines the spinning motion of an ellipsoidal planet around its axis is considered. By studying the rotational potential energy of an ellipsoidal planet orbiting a spherical star on an elliptic orbit with fixed eccentricity and semi-major axis, it is shown that the system presents an infinite number of metastable equilibrium configurations. These states correspond to local minima of the rotational potential energy averaged over an orbit, where the ratio between the rotational period of the planet around its axis and the revolution period around the star is fixed. The configurations in which this ratio is an integer or an half integer are of particular interest. Among these configurations, the deepest minimum in the average potential energy corresponds to a situation where the rotational and orbital motion of the planet are synchronous, and the system is tidally locked. The next-to-the deepest minimum corresponds to the case in which the planet rotates three times around its axis in the time that it needs to complete two orbits around the Sun. The latter is indeed the case that describes Mercury's motion. The method discussed in this work allows one to identify the integer and half-integer ratios that correspond to spin-orbit resonances and to describe the motion of the planet in the resonant orbit.A steady azimuthal stratified flow modelling the antarctic circumpolar currenthttps://zbmath.org/1521.860022023-11-13T18:48:18.785376Z"Abrashkin, A. A."https://zbmath.org/authors/?q=ai:abrashkin.a-a"Constantin, A."https://zbmath.org/authors/?q=ai:constantin.adrian|constantin.alexandre|constantin.andreiSummary: We investigate steady flow moving purely in the azimuthal direction on a rotating sphere and having a meridionally localized jet structure. An exact solution for a stratified inviscid fluid, which admits a depth-dependent velocity profile below the surface, is constructed in spherical coordinates. This solution is relevant to the modelling of the Antarctic Circumpolar Current. We show that the stratification enables us to dispense with the nonconservative body force that was invoked in recent spherical-coordinate models to produce realistic flow profiles.A note on the derivation of the quasi-geostrophic potential vorticity equationhttps://zbmath.org/1521.860212023-11-13T18:48:18.785376Z"Fowler, A. C."https://zbmath.org/authors/?q=ai:fowler.andrew-cSummary: The derivation of the quasi-geostrophic potential vorticity equation of mathematical meteorology is usually done using fairly sophisticated techniques of perturbation theory, but stops short of deriving self-consistently the stratification parameter of the mean atmospheric state. In this note we suggest how this should be done within the confines of the theory, and as a consequence we raise the possibility that the atmosphere could become globally unstable, with dramatic consequences.Reflection of an internal wave at an interface representing a rapid increase in viscosityhttps://zbmath.org/1521.860242023-11-13T18:48:18.785376Z"Mchugh, John"https://zbmath.org/authors/?q=ai:mchugh.john-revere|mchugh.john-philip"Grimshaw, Roger"https://zbmath.org/authors/?q=ai:grimshaw.roger-h-jSummary: Internal waves at high altitudes are greatly damped by the drastic increase in molecular viscosity and thermal diffusivity, resulting in important heating and other effects at those altitudes. Here we consider the case where this increase in viscosity is very rapid, idealized as an interface with inviscid flow in the lower layer and constant viscosity in the upper layer. The results show that waves are partially reflected by this interface, with a reflection coefficient that increases monotonically with an increase in the viscosity of the upper layer. This mechanism would have a significant impact on the vertical distribution of thermal energy at high altitudes.Laterally loaded piles and pile groups partially embedded in transversely isotropic fractional viscoelastic saturated soilshttps://zbmath.org/1521.860362023-11-13T18:48:18.785376Z"Ai, Zhi Yong"https://zbmath.org/authors/?q=ai:ai.zhiyong"Wang, Da Shan"https://zbmath.org/authors/?q=ai:wang.da-shan"Zhao, Yong Zhi"https://zbmath.org/authors/?q=ai:zhao.yongzhi"Li, Pan Cong"https://zbmath.org/authors/?q=ai:li.pan-cong(no abstract)Lying with heterogeneous image concernshttps://zbmath.org/1521.910472023-11-13T18:48:18.785376Z"Zakharov, Alexei"https://zbmath.org/authors/?q=ai:zakharov.aleksei-olegovich|zakharov.alexei-vSummary: We study reporting in a lying game where the agents differ in two characteristics: the intrinsic cost of lying, and the social cost due to one's report being perceived as a lie. We show that making lying more socially costly can lead to making lying more frequent. This happens because increasing the social cost for a subset of agents will make them choose high-value reports less frequently, making such reports less socially costly for the rest of the agents, and inviting them to lie more.Identifying an earnings process with dependent contemporaneous income shockshttps://zbmath.org/1521.911602023-11-13T18:48:18.785376Z"Ben-Moshe, Dan"https://zbmath.org/authors/?q=ai:ben-moshe.danSummary: This paper proposes a novel approach for identifying coefficients in an earnings dynamics model with arbitrarily dependent contemporaneous income shocks. Traditional methods relying on second moments fail to identify these coefficients, emphasizing the need for non-Gaussianity assumptions that capture information from higher moments. Our results contribute to the literature on earnings dynamics by allowing models of earnings to have, for example, the permanent income shock of a job change to be linked to the contemporaneous transitory income shock of a relocation bonus.Inter-league competition and the optimal broadcasting revenue-sharing rulehttps://zbmath.org/1521.912642023-11-13T18:48:18.785376Z"Rocaboy, Yvon"https://zbmath.org/authors/?q=ai:rocaboy.yvonSummary: We propose a model where two sports leagues compete for sporting talent, and at the same time consider the competitive balance in their domestic championships. The allocation of broadcasting revenues by the league-governing body acts as an incentive for teams to invest in talent. We derive a strategic league authority's optimal sharing rule of broadcasting revenues across teams in the league. While a weighted form of performance-based sharing is the best way of attracting talent, cross-subsidization from high- to low-payroll teams is required to improve competitive balance. The optimal sharing rule is then a combination of these two ``sub-rules''. We show that the distribution of broadcasting revenues in two first divisions in European men's football, the English Premier League (EPL) and the French Ligue 1 (L1), corresponds to the optimal sharing rule we discuss. We propose a new method to assess empirically the cross-subsidization impact of the sharing formula. As the impact of cross-subsidization is greater in the EPL than L1, we conclude that ensuring domestic competitive balance seems to be a more important target for the EPL than for L1.Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: a mesh-free approachhttps://zbmath.org/1521.913042023-11-13T18:48:18.785376Z"Rasanan, Amir Hosein Hadian"https://zbmath.org/authors/?q=ai:rasanan.amir-hosein-hadian"Evans, Nathan J."https://zbmath.org/authors/?q=ai:evans.nathan-j"Rieskamp, Jörg"https://zbmath.org/authors/?q=ai:rieskamp.jorg"Rad, Jamal Amani"https://zbmath.org/authors/?q=ai:rad.jamal-amani(no abstract)Optimal investment and reinsurance strategies under 4/2 stochastic volatility modelhttps://zbmath.org/1521.913222023-11-13T18:48:18.785376Z"Wang, Wenyuan"https://zbmath.org/authors/?q=ai:wang.wenyuan"Muravey, Dmitry"https://zbmath.org/authors/?q=ai:muravey.dmitry"Shen, Yang"https://zbmath.org/authors/?q=ai:shen.yang"Zeng, Yan"https://zbmath.org/authors/?q=ai:zeng.yanIn this paper, the authors study a mean-variance reinsurance-investment problem under a class of 4/2 stochastic volatility models. Using the parametrix method coupled with the integral transform method, they obtain explicit solutions to parabolic partial differential equations arising from the problem in certain particular cases. By invoking the use of the Lie symmetry analysis, the authors obtain a four-parameter family of the 4/2 stochastic volatility models. The advantage of this family of stochastic volatility models is that the respective parabolic partial differential equations admit closed-form solutions. Using the closed-form solutions to the parabolic partial differential equations, representations for the efficient strategy and the efficient frontier are obtained.
Based on the Monte-Carlo simulation method, numerical examples illustrating the efficient frontier are provided. The model dynamics and underlying assumptions are presented in Section 2. The insurer's optimization problem based on the mean-variance criterion and the respective parabolic partial differential equation are also presented in this section. Section 3 is concerned with using the parametrix method to solve the parabolic partial differential equation. The key idea of the parametrix method is to expand the fundamental solution of a parabolic partial differential equation using a basic function, which is called the parametrix. The integral method is also discussed. The basic idea of the integral method is to replace a differential operator with an algebraic operator. Theorem 3.1 establishes the existence and uniqueness results for the solution to the parabolic partial differential equation. It also gives the unique solution using the parametrix method. Specifically, the unique solution is characterised by considering two cases. By means of special functions, an explicit solution to a parabolic partial differential equation for the case of a classical Heston's stochastic volatility model is provided in Theorem 3.2. The exponential affine representation for the solution to a parabolic partial differential equation is obtained in Theorem 3.3. The coefficients of the exponential affine representation are also explicitly determined in Theorem 3.3.
Section 4 is devoted to the Lie symmetry analysis. Specifically, the Lie symmetry analysis is adopted to derive a closed-form solution to a parabolic partial differential equation. The key theoretical result for the Lie symmetry analysis is presented in Proposition 4.1, where the necessary and sufficient conditions to ensure the existence of non-trivial symmetries are provided. The solution to the mean-variance optimization problem is derived in Section 5. Specifically, the main theoretical result giving the solution is presented in Theorem 5.1, where the efficient strategy and the efficient frontier are provided. The numerical examples illustrating the theoretical results including the efficient strategy and the efficient frontier are provided in Section 6.
Reviewer: Tak Kuen Siu (Sydney)Symmetries of the Black-Scholes-Merton equation for European optionshttps://zbmath.org/1521.913552023-11-13T18:48:18.785376Z"Bakirova, L. N."https://zbmath.org/authors/?q=ai:bakirova.l-n"Shurygina, M. A."https://zbmath.org/authors/?q=ai:shurygina.m-a"Shurygin, V. V. jun."https://zbmath.org/authors/?q=ai:shurygin.vadim-v-junIn the present paper the authors add some corrections to the result by \textit{A. Paliathanasis} et al. [Mathematics 4, No. 2, Paper No. 28, 14 p. (2016; Zbl 1358.91101)], on the symmetry of Lie algebra of the Black-Scholes-Merton partial differential equation (PDE) for European option pricing. They shown that this PDE admits additional symmetries.
Reviewer: Anatoliy Swishchuk (Calgary)Pricing options under time-fractional model using Adomian decompositionhttps://zbmath.org/1521.913872023-11-13T18:48:18.785376Z"Kharrat, Mohamed"https://zbmath.org/authors/?q=ai:kharrat.mohamedSummary: In this chapter, we define and describe some models to price European and American options. As the dynamics of volatility is intrinsic in terms of the hedging and the pricing of options, we shall present in this chapter both cases: the constant volatility [\textit{F. Black} and \textit{M. Scholes}, J. Polit. Econ. 81, No. 3, 637--654 (1973; Zbl 1092.91524)] and the stochastic volatility model [\textit{S. L. Heston}, Rev. Financ. Stud. 6, No. 2, 327--343 (1993; Zbl 1384.35131)]. At the beginning, we display a closed-form solution of a European option generated by the fractional Heston stochastic volatility model. Subsequently, we provide the analytical solution to the fractional linear complement problem related to the evaluation of American put option generated by the fractional Black and Scholes model. In the closing of this chapter, we attempt to set forward the solution of the fractional linear complementarity problem related to the evaluation of American put option generated by the fractional Heston stochastic volatility model. Investing the Adomian decomposition, a numerical investigation is undertaken to corroborate the theoretical results. The results of this chapter are published in [\textit{M. Kharrat}, Nonlinear Dyn. Syst. Theory 18, No. 2, 191--195 (2018; Zbl 1416.91377); ``Pricing American put option under fractional Heston model'', Pramana J. Phys. 95, No. 1, Article No. 3, 7 p. (2021; \url{doi:10.1007/s12043-020-02039-z}); ``Pricing American put option under fractional model'', Filomat 35, No. 10, 3433--3441 (2021; \url{doi:10.2298/FIL2110433K})].
For the entire collection see [Zbl 1497.37001].Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation datahttps://zbmath.org/1521.920072023-11-13T18:48:18.785376Z"Lee, Seungjoon"https://zbmath.org/authors/?q=ai:lee.seungjoon"Psarellis, Yorgos M."https://zbmath.org/authors/?q=ai:psarellis.yorgos-m"Siettos, Constantinos I."https://zbmath.org/authors/?q=ai:siettos.constantinos-i"Kevrekidis, Ioannis G."https://zbmath.org/authors/?q=ai:kevrekidis.ioannis-gSummary: We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs) -- and the closures that lead to them- from high-fidelity, individual-based stochastic simulations of \textit{Escherichia coli} bacterial motility. The fine scale, chemomechanical, hybrid (continuum -- Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. Using a parsimonious set of collective observables, we learn effective, coarse-grained ``Keller-Segel class'' chemotactic PDEs using machine learning regressors: (a) (shallow) feedforward neural networks and (b) Gaussian Processes. The learned laws can be \textit{black-box} (when no prior knowledge about the PDE law structure is assumed) or \textit{gray-box} when parts of the equation (e.g. the pure diffusion part) is known and ``hardwired'' in the regression process. More importantly, we discuss data-driven \textit{corrections} (both additive and functional), to analytically known, \textit{approximate} closures.A POD-RBF-FD scheme for simulating chemotaxis models on surfaceshttps://zbmath.org/1521.920082023-11-13T18:48:18.785376Z"Mohammadi, Vahid"https://zbmath.org/authors/?q=ai:mohammadi.vahid"Dehghan, Mehdi"https://zbmath.org/authors/?q=ai:dehghan.mehdi(no abstract)The role of \(\mathrm{A}\beta\) and tau proteins in Alzheimer's disease: a mathematical model on graphshttps://zbmath.org/1521.920342023-11-13T18:48:18.785376Z"Bertsch, Michiel"https://zbmath.org/authors/?q=ai:bertsch.michiel"Franchi, Bruno"https://zbmath.org/authors/?q=ai:franchi.bruno"Tesi, Maria Carla"https://zbmath.org/authors/?q=ai:tesi.maria-carla"Tora, Veronica"https://zbmath.org/authors/?q=ai:tora.veronicaSummary: In this note we study a mathematical model for the progression of Alzheimer's disease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as \(\beta\) amyloid and tau play in fact a crucial role in the development of Alzheimer's disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy.Astrocytic clearance and fragmentation of toxic proteins in Alzheimer's disease on large-scale brain networkshttps://zbmath.org/1521.920422023-11-13T18:48:18.785376Z"Shaheen, Hina"https://zbmath.org/authors/?q=ai:shaheen.hina"Pal, Swadesh"https://zbmath.org/authors/?q=ai:pal.swadesh"Melnik, Roderick"https://zbmath.org/authors/?q=ai:melnik.roderick-v-nicholasSummary: The human brain is the most complicated biological structure on the planet. A major challenge of brain network modelling lies in its multi-scale spatio-temporal nature, covering scales from synapses to the whole brain. The coupled multiphysics and biochemical activities which spread through such a complex system shape brain capacity inside a structure-function relationship that requires a particular mathematical framework. Next-generation coupled-based mathematical modelling approaches to brain networks and the analysis of data-driven dynamical systems are needed to advance state-of-the-art therapeutic strategies for treating neurodegenerative diseases (NDDs) that affect millions of people worldwide, such as Alzheimer's disease (AD) and Parkinson's disease (PD). Importantly, AD is marked by the presence of amyloid-beta (A\(\beta\)) plaques and tau (\(\tau\)) proteins. Some disease-specific misfolded proteins can interact with healthy proteins to form long chains and aggregates of different sizes that have different transport properties and toxicity. An improved large-scale brain network model is proposed here to understand the pathogenesis of AD, especially the role of astrocytes in the presence of misfolded proteins (A\(\beta\) and \(\tau\)). The idea involves astrocytic clearance, which assists in eliminating toxic A\(\beta\) via fragmentation. We use the general Smoluchowski theory of nucleation, aggregation, and fragmentation to predict the development and propagation of aggregates of misfolded proteins in the brain. It has been shown that the developed model leads to different size distributions and propagation along the network. We predicted that astrocytic clearance varies with the aggregate size, which is key to slowing down AD progression. The clearance and fragmentation of toxic proteins span several spatial and temporal scales, and this research will potentially yield new insight into the associated processes and brain networks in health and disease. Detailed multi-scale brain modelling provides a promising approach for consolidating, organizing, and bridging the data sets of data-driven brain network models.Choice of spatial discretisation influences the progression of viral infection within multicellular tissueshttps://zbmath.org/1521.920442023-11-13T18:48:18.785376Z"Williams, Thomas"https://zbmath.org/authors/?q=ai:williams.thomas-r|williams.thomas-c|williams.thomas-a|williams.thomas-e"McCaw, James M."https://zbmath.org/authors/?q=ai:mccaw.james-m"Osborne, James M."https://zbmath.org/authors/?q=ai:osborne.james-mSummary: There has been an increasing recognition of the utility of models of the spatial dynamics of viral spread within tissues. Multicellular models, where cells are represented as discrete regions of space coupled to a virus density surface, are a popular approach to capture these dynamics. Conventionally, such models are simulated by discretising the viral surface and depending on the rate of viral diffusion and other considerations, a finer or coarser discretisation may be used. The impact that this choice may have on the behaviour of the system has not been studied. Here we demonstrate that under realistic parameter regimes -- where viral diffusion is small enough to support the formation of familiar ring-shaped infection plaques -- the choice of spatial discretisation of the viral surface can qualitatively change key model outcomes including the time scale of infection. Importantly, we show that the choice between implementing viral spread as a cell-scale process, or as a high-resolution converged PDE can generate distinct model outcomes, which raises important conceptual questions about the strength of assumptions underpinning the spatial structure of the model. We investigate the mechanisms driving these discretisation artefacts, the impacts they may have on model predictions, and provide guidance on the design and implementation of spatial and especially multicellular models of viral dynamics. We obtain our results using the simplest TIV construct for the viral dynamics, and therefore anticipate that the important effects we describe will also influence model predictions in more complex models of virus-cell-immune system interactions. This analysis will aid in the construction of models for robust and biologically realistic modelling and inference.Cell orientation under stretch: a review of experimental findings and mathematical modellinghttps://zbmath.org/1521.920512023-11-13T18:48:18.785376Z"Giverso, Chiara"https://zbmath.org/authors/?q=ai:giverso.chiara"Loy, Nadia"https://zbmath.org/authors/?q=ai:loy.nadia"Lucci, Giulio"https://zbmath.org/authors/?q=ai:lucci.giulio"Preziosi, Luigi"https://zbmath.org/authors/?q=ai:preziosi.luigiSummary: The key role of electro-chemical signals in cellular processes had been known for many years, but more recently the interplay with mechanics has been put in evidence and attracted substantial research interests. Indeed, the sensitivity of cells to mechanical stimuli coming from the microenvironment turns out to be relevant in many biological and physiological circumstances. In particular, experimental evidence demonstrated that cells on elastic planar substrates undergoing periodic stretches, mimicking native cyclic strains in the tissue where they reside, actively reorient their cytoskeletal stress fibres. At the end of the realignment process, the cell axis forms a certain angle with the main stretching direction. Due to the importance of a deeper understanding of mechanotransduction, such a phenomenon was studied both from the experimental and the mathematical modelling point of view. The aim of this review is to collect and discuss both the experimental results on cell reorientation and the fundamental features of the mathematical models that have been proposed in the literature.Mathematical model of dynamics of a cell cycle based on the allometric theory of growthhttps://zbmath.org/1521.920522023-11-13T18:48:18.785376Z"Petelin, Dmitriĭ V."https://zbmath.org/authors/?q=ai:petelin.dmitrii-v"Sadovskiĭ, Mikhaĭl G."https://zbmath.org/authors/?q=ai:sadovskii.mikhail-gSummary: The mathematical model of dynamics of a cell cycle on the basis of the theory allometric growth is developed. The way of a finding of stationary and periodic points on the basis of the geometrical description of model has been found.Noise-induced dynamics in a single species model with Allee effect driven by correlated colored noiseshttps://zbmath.org/1521.920792023-11-13T18:48:18.785376Z"Yu, Xingwang"https://zbmath.org/authors/?q=ai:yu.xingwang"Ma, Yuanlin"https://zbmath.org/authors/?q=ai:ma.yuanlinSummary: In this paper, a single species model with Allee effect driven by correlated colored noises is proposed and investigated. The stationary probability density of the model is obtained using the approximative Fokker-Planck equation, and its shape is discussed in detail. P-bifurcation and noise-induced bistability are explored, followed by the observation of noise-enhanced stability through mean first passage time analysis. Our findings demonstrate that: (i) noise can induce P-bifurcation, causing the structure of a stationary probability distribution to shift from unimodal to monotonic under positive correlation and switch from unimodal to bimodal under negative correlation; (ii) correlation time promotes population growth, leading to a higher probability of large population size and delaying the extinction process; (iii) noise-enhanced stability induced by multiplicative noise depends on both additive noise and correlation time, while it always exists for additive noise.SARS-CoV-2 rate of spread in and across tissue, groundwater and soil: a meshless algorithm for the fractional diffusion equationhttps://zbmath.org/1521.920822023-11-13T18:48:18.785376Z"Bavi, O."https://zbmath.org/authors/?q=ai:bavi.o"Hosseininia, M."https://zbmath.org/authors/?q=ai:hosseininia.masoumeh"Heydari, M. H."https://zbmath.org/authors/?q=ai:heydari.mohammad-hossien"Bavi, N."https://zbmath.org/authors/?q=ai:bavi.n(no abstract)Exact controllability of wave equations with interior degeneracy and one-sided boundary controlhttps://zbmath.org/1521.930112023-11-13T18:48:18.785376Z"Bai, Jinyan"https://zbmath.org/authors/?q=ai:bai.jinyan"Chai, Shugen"https://zbmath.org/authors/?q=ai:chai.shugenSummary: In this paper, the authors mainly consider the exact controllability for degenerate wave equation, which degenerates at the interior point, and boundary controls acting at only one of the boundary points. The main results are that, it is possible to control both the position and the velocity at every point of the body and at a certain time \(T\) for the wave equation with interior weakly degeneracy. Moreover, it is shown that the exact controllability fails for the wave equation with interior strongly degeneracy. In order to steer the system to a certain state, one needs controls to act on both boundary points for the wave equation with interior strongly degeneracy. The difficulties are addressed by means of spectral analysis.Approximate controllability results in \(\alpha \)-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaceshttps://zbmath.org/1521.930172023-11-13T18:48:18.785376Z"Ndambomve, Patrice"https://zbmath.org/authors/?q=ai:ndambomve.patrice"Kpoumie, Moussa El-Khalil"https://zbmath.org/authors/?q=ai:kpoumie.moussa-el-khalil"Ezzinbi, Khalil"https://zbmath.org/authors/?q=ai:ezzinbi.khalilIn this work the authors have discussed the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces under the assumption that the corresponding linear part is approximately controllable. The results are obtained by using the fractional power theory and \(\alpha\)-norm, the measure of noncompactness and the Mönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. This paper is a generalization of the work of \textit{N. I. Mahmudov} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 3, 536--546 (2008; Zbl 1129.93004)] and established without the assumption of compactness of the resolvent operator. An example is provided to illustrate the main results.
Reviewer: Krishnan Balachandran (Coimbatore)Controllability of the Schrödinger equation on unbounded domains without geometric control conditionhttps://zbmath.org/1521.930202023-11-13T18:48:18.785376Z"Täufer, Matthias"https://zbmath.org/authors/?q=ai:taufer.matthiasSummary: We prove controllability of the Schrödinger equation in \(\mathbb{R}^d\) in any time \(T > 0\) with internal control supported on nonempty, periodic, open sets. This demonstrates in particular that controllability of the Schrödinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schrödinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation. Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.Observability of a string-beams network with many beamshttps://zbmath.org/1521.930232023-11-13T18:48:18.785376Z"Lai, Anna Chiara"https://zbmath.org/authors/?q=ai:lai.anna-chiara"Loreti, Paola"https://zbmath.org/authors/?q=ai:loreti.paola"Mehrenberger, Michel"https://zbmath.org/authors/?q=ai:mehrenberger.michelSummary: We prove the direct and inverse observability inequality for a network connecting one string with infinitely many beams, at a common point, in the case where the lengths of the beams are all equal. The observation is at the exterior node of the string and at the exterior nodes of all the beams except one. The proof is based on a careful analysis of the asymptotic behavior of the underlying eigenvalues and eigenfunctions, and on the use of a Ingham type theorem with weakened gap condition
[C. Baiocchi, V. Komornik and P. Loreti, \textit{Acta Math. Hung.} 97 (2002) 55-95.].
On the one hand, the proof of the crucial gap condition already observed in the case where there is only one beam
[K. Ammari, M. Jellouli and M. Mehrenberger, \textit{Networks Heterogeneous Media} 4 (2009) 2009.]
is new and based on elementary monotonicity arguments. On the other hand, we are able to handle both the complication arising with the appearance of eigenvalues with unbounded multiplicity, due to the many beams case, and the terms coming from the weakened gap condition, arising when at least 2 beams are present.Boundary control of some distributed heterogeneous vibrating system with given states at intermediate time instantshttps://zbmath.org/1521.930832023-11-13T18:48:18.785376Z"Barseghyan, V. R."https://zbmath.org/authors/?q=ai:barseghyan.vanya-rafaelovichSummary: This paper considers the boundary control problem for a distributed heterogeneous vibrating system described by a one-dimensional wave equation with piecewise constant characteristics. The travel time of a wave through each homogeneous section is assumed the same. The control is implemented by displacement at the two ends. A constructive control design approach is proposed to transfer the vibrations on a given time interval from the initial state through the multipoint intermediate states to the terminal state. The control design scheme is as follows: the original problem is reduced to a control problem with distributed actions and zero boundary conditions. Then the variable separation method and control methods for finite-dimensional systems with multipoint intermediate conditions are used. The results are illustrated by an example.Extremal problems for second order hyperbolic systems involving multiple time delayshttps://zbmath.org/1521.930842023-11-13T18:48:18.785376Z"Kowalewski, Adam"https://zbmath.org/authors/?q=ai:kowalewski.adamSummary: Extremal problems for multiple time delay hyperbolic systems are presented. The optimal boundary control problems for hyperbolic systems in which multiple time delays appear both in the state equations and in the Neumann boundary conditions are solved. The time horizon is fixed. Making use of Dubovicki-Milutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.Boundary output feedback stabilisation of a class of reaction-diffusion PDEs with delayed boundary measurementhttps://zbmath.org/1521.931402023-11-13T18:48:18.785376Z"Lhachemi, Hugo"https://zbmath.org/authors/?q=ai:lhachemi.hugo"Prieur, Christophe"https://zbmath.org/authors/?q=ai:prieur.christopheIn this paper, the authors investigate on boundary output feedback stabilisation of a class of reaction-diffusion PDEs with delayed boundary measurement. Concretely, they address the boundary output feedback stabilisation of a general class of 1-D reaction-diffusion PDEs with delayed boundary measurement.
Reviewer: Savin Treanţă (Bucureşti)Prescribed-time stabilisation for uncertain reaction-diffusion equations with Neumann boundary controlhttps://zbmath.org/1521.931422023-11-13T18:48:18.785376Z"Wei, Chengzhou"https://zbmath.org/authors/?q=ai:wei.chengzhou"Li, Junmin"https://zbmath.org/authors/?q=ai:li.junmin"He, Chao"https://zbmath.org/authors/?q=ai:he.chaoIn this paper, the authors investigate on prescribed-time stabilisation for uncertain reaction-diffusion equations with Neumann boundary control. More precisely, they estimate the uncertainties within the prescribed time by constructing an estimator and then propose the prescribed-time boundary control law by utilising backstepping transformation with a time-varying kernel.
Reviewer: Savin Treanţă (Bucureşti)Lyapunov stability analysis for incommensurate nabla fractional order systemshttps://zbmath.org/1521.931492023-11-13T18:48:18.785376Z"Wei, Yiheng"https://zbmath.org/authors/?q=ai:wei.yiheng"Zhao, Xuan"https://zbmath.org/authors/?q=ai:zhao.xuan|zhao.xu-an"Wei, Yingdong"https://zbmath.org/authors/?q=ai:wei.yingdong"Chen, Yangquan"https://zbmath.org/authors/?q=ai:chen.yangquanSummary: This paper investigates the problem of stability analysis for a class of incommensurate nabla fractional order systems. In particular, both Caputo definition and Riemann-Liouville definition are under consideration. With the convex assumption, several elementary fractional difference inequalities on Lyapunov functions are developed. According to the essential features of nabla fractional calculus, the sufficient conditions are given first to guarantee the asymptotic stability for the incommensurate system by using the direct Lyapunov method. To substantiate the efficacy and effectiveness of the theoretical results, four examples are elaborated.Mixed Poisson-Gaussian noise reduction using a time-space fractional differential equationshttps://zbmath.org/1521.940072023-11-13T18:48:18.785376Z"Gholami Bahador, F."https://zbmath.org/authors/?q=ai:gholami-bahador.f"Mokhtary, P."https://zbmath.org/authors/?q=ai:mokhtary.payam"Lakestani, M."https://zbmath.org/authors/?q=ai:lakestani.mehrdadSummary: Images are frequently corrupted by various sorts of mixed or unrecognized noise, including mixed Poisson-Gaussian noise, rather than just a single kind of noise. In this work, we propose a time-space fractional differential equation to remove mixed Poisson-Gaussian noise. Combining fixed- and variable-order fractional derivatives allows us to maintain an image's high- and low-frequency components while eliminating noise. The current model, although primarily intended for mixed noise reduction, can indeed be utilized with great efficacy on images that have been solely degraded by Gaussian noise. In addition to this, a stable discretization strategy is presented. The illustrative results demonstrate that our scheme performs better than earlier models, reduces the staircase effect, and is applicable to electron microscopy and CT images.