Recent zbMATH articles in MSC 35Ahttps://zbmath.org/atom/cc/35A2024-04-15T15:10:58.286558ZWerkzeugNonstationary radiative-conductive heat transfer problem in an absolutely black body with semitransparent inclusionshttps://zbmath.org/1530.350052024-04-15T15:10:58.286558Z"Amosov, Andrey"https://zbmath.org/authors/?q=ai:amosov.andrei(no abstract)A uniqueness result for the two-vortex traveling wave in the nonlinear Schrödinger equationhttps://zbmath.org/1530.350062024-04-15T15:10:58.286558Z"Chiron, David"https://zbmath.org/authors/?q=ai:chiron.david"Pacherie, Eliot"https://zbmath.org/authors/?q=ai:pacherie.eliotSummary: For the nonlinear Schrödinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by \textit{F. Béthuel} et al. [Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math. Laurent Schwartz, Palaiseau 2007--2008, Exp. No. XV, 28 p. (2008; Zbl 1176.35154); Commun. Math. Phys. 285, No. 2, 567--651 (2009; Zbl 1190.35196)] (see also work of \textit{D. Chiron} and \textit{M. Mariş} [Arch. Ration. Mech. Anal. 226, No. 1, 143--242 (2017; Zbl 1391.35351)]). This minimizer is a traveling wave for the nonlinear Schrödinger equation. For large momenta, the propagation speed is small and the minimizer behaves like two well-separated vortices. In that limit, we show the uniqueness of this minimizer, up to the invariances of the problem, hence proving the orbital stability of this traveling wave. This work is a follow up to two previous papers, where we constructed and studied a particular traveling wave of the equation. We show a uniqueness result on this traveling wave in a class of functions that contains in particular all possible minimizers of the energy.Nonuniqueness of a self-similar solution to the Riemann problem for elastic waves in media with a negative nonlinearity parameterhttps://zbmath.org/1530.350072024-04-15T15:10:58.286558Z"Chugainova, A. P."https://zbmath.org/authors/?q=ai:chugainova.anna-p"Polekhina, R. R."https://zbmath.org/authors/?q=ai:polekhina.r-rSummary: We study self-similar solutions of the Riemann problem in the nonuniqueness region for weakly anisotropic elastic media with a negative nonlinearity parameter. We show that all discontinuities contained in the solutions in the nonuniqueness region have a stationary structure. We also show that in the nonuniqueness region one can construct two types of self-similar solutions.On nonuniqueness and nonregularity for gradient flows of polyconvex functionalshttps://zbmath.org/1530.350082024-04-15T15:10:58.286558Z"Yan, Baisheng"https://zbmath.org/authors/?q=ai:yan.baishengSummary: We provide some counterexamples concerning the uniqueness and regularity of weak solutions to the initial-boundary value problem for the gradient flows of certain strongly polyconvex functionals. We show that such a problem can possess a trivial classical solution as well as infinitely many weak solutions that are nowhere smooth. Such polyconvex functions are constructed from some previous examples, and the nonuniqueness and nonregularity results are proved by reformulating the gradient flow as a partial differential relation and then using the convex integration method to construct certain strongly convergent sequences of subsolutions that have a uniform control on local essential oscillations of the spatial gradients.Local and global analyticity for a generalized Camassa-Holm systemhttps://zbmath.org/1530.350092024-04-15T15:10:58.286558Z"Yamane, Hideshi"https://zbmath.org/authors/?q=ai:yamane.hideshiSummary: We solve the analytic Cauchy problem for the generalized two-component Camassa-Holm system introduced by R. M. Chen and Y. Liu. We show the existence of a unique local/global-in-time analytic solution under certain conditions. This is the first result about global analyticity for a Camassa-Holm-like system. The method of proof is basically that developed by Barostichi, Himonas and Petronilho. The main differences between their proof and ours are twofold: (i) the system of Chen and Liu is not symmetric in the two unknowns and our estimates are not trivial generalization of those in their articles, (ii) we have simplified their argument by using fewer function spaces and the main result is stated in a simple and natural way.Multiplicity and concentration of solutions to fractional anisotropic Schrödinger equations with exponential growthhttps://zbmath.org/1530.350102024-04-15T15:10:58.286558Z"Nguyen, Thin Van"https://zbmath.org/authors/?q=ai:van-nguyen.thin"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-dSummary: In this paper, we consider the Schrödinger equation involving the fractional \((p,p_1,\dots ,p_m)\)-Laplacian as follows
\[
(-\Delta )_p^su+\sum_{i=1}^m(-\Delta )_{p_i}^su+V(\varepsilon x)(|u|^{(N-2s)/2s}u+\sum_{i=1}^m|u|^{p_i-2}u)=f(u) \text{ in } {\mathbb{R}}^N, \]
where \(\varepsilon\) is a positive parameter, \(N=ps\), \(s\in (0,1)\), \(2\le p<p_1< \dots< p_m<+\infty\), \(m\ge 1\). The nonlinear function \(f\) has the exponential growth and potential function \(V\) is continuous function satisfying some suitable conditions. Using the penalization method and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter. In our best knowledge, it is the first time that the above problem is studied.He's variational method for the time-space fractional nonlinear Drinfeld-Sokolov-Wilson systemhttps://zbmath.org/1530.350112024-04-15T15:10:58.286558Z"Wang, Kang-Jia"https://zbmath.org/authors/?q=ai:wang.kang-jia"Wang, Guo-Dong"https://zbmath.org/authors/?q=ai:wang.guodong(no abstract)Frölicher structures, diffieties, and a formal KP hierarchyhttps://zbmath.org/1530.350122024-04-15T15:10:58.286558Z"Magnot, Jean-Pierre"https://zbmath.org/authors/?q=ai:magnot.jean-pierre"Reyes, Enrique G."https://zbmath.org/authors/?q=ai:reyes.enrique-g"Rubtsov, Vladimir"https://zbmath.org/authors/?q=ai:rubtsov.vladimir-nSummary: We propose a definition of a diffiety based on the theory of Frölicher structures. As a consequence, we obtain a natural Vinogradov sequence and, under the assumption of the existence of a suitable derivation on a given diffiety, we can form on it a Kadomtsev-Petviashvili hierarchy which is well-posed.
For the entire collection see [Zbl 1519.35008].The Cauchy problem and multi-peakons for the mCH-Novikov-CH equation with quadratic and cubic nonlinearitieshttps://zbmath.org/1530.350192024-04-15T15:10:58.286558Z"Qin, Guoquan"https://zbmath.org/authors/?q=ai:qin.guoquan"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenya"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.bolingSummary: This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation (alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Hölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored.Estimation of conditional stability of the boundary-value problem for the system of parabolic equations with changing direction of timehttps://zbmath.org/1530.350282024-04-15T15:10:58.286558Z"Fayazov, K. S."https://zbmath.org/authors/?q=ai:fayazov.kudratillo-sadridinovich"Khajiev, I. O."https://zbmath.org/authors/?q=ai:khajiev.ikrombek-oSummary: In this article we investigate an ill-posed problem for the system of inhomogeneous equations of parabolic type with changing the direction of time. We obtained an a priori estimate based on the given data. The theorems of uniqueness and conditional stability are proved on the set of correctness of the solution. An approximate solution to the problem is constructed by the regularization method. We calculate an estimate for efficiency of the norm of the difference between exact and approximate solutions.Asymptotic behaviour of the eenergy to the viscoelastic wave equation with localized hereditary memory and supercritical source termhttps://zbmath.org/1530.350352024-04-15T15:10:58.286558Z"Cavalcanti, V. N. Domingos"https://zbmath.org/authors/?q=ai:domingos-cavalcanti.valeria-neves"Cavalcanti, M. M."https://zbmath.org/authors/?q=ai:cavalcanti.marcelo-moreira"Marchiori, T. D."https://zbmath.org/authors/?q=ai:marchiori.talita-druziani"Webler, C. M."https://zbmath.org/authors/?q=ai:webler.claudete-mSummary: We are concerned with the well-posedness of solutions as well as the asymptotic behaviour of the energy related to the viscoelastic wave equation with localized memory with past history and supercritical source and damping terms, posed on a bounded domain \(\Omega \subset{\mathbb{R}}^3\). Avoiding any relation between the damping and source terms, we obtain the global existence of solutions and the uniform energy decay rates, introducing the notion of a potential well and defining the expanded total energy functional. We also prove that weak solutions blow-up in finite time.On a nonlinear boundary problem for thermoelastic coupled beam equations with memory termhttps://zbmath.org/1530.350502024-04-15T15:10:58.286558Z"Le Thi Phuong Ngoc"https://zbmath.org/authors/?q=ai:le-thi-phuong-ngoc."Pham Nguyen Nhat Khanh"https://zbmath.org/authors/?q=ai:pham-nguyen-nhat-khanh."Nguyen Huu Nhan"https://zbmath.org/authors/?q=ai:nguyen-huu-nhan."Nguyen Thanh Long"https://zbmath.org/authors/?q=ai:nguyen-thanh-long.(no abstract)Blow-up versus global well-posedness for the focusing INLS with inverse-square potentialhttps://zbmath.org/1530.350752024-04-15T15:10:58.286558Z"Deng, Mingming"https://zbmath.org/authors/?q=ai:deng.mingming"Lu, Jing"https://zbmath.org/authors/?q=ai:lu.jing"Meng, Fanfei"https://zbmath.org/authors/?q=ai:meng.fanfei(no abstract)Liouville-type theorem for finite Morse index solutions to the Choquard equation involving \(\Delta_\lambda\)-Laplacianhttps://zbmath.org/1530.350892024-04-15T15:10:58.286558Z"Dao Trong Quyet"https://zbmath.org/authors/?q=ai:dao-trong-quyet.(no abstract)\(C^{2,\alpha}\) regularity of free boundaries in optimal transportationhttps://zbmath.org/1530.350922024-04-15T15:10:58.286558Z"Chen, Shibing"https://zbmath.org/authors/?q=ai:chen.shibing"Liu, Jiakun"https://zbmath.org/authors/?q=ai:liu.jiakun"Wang, Xu-Jia"https://zbmath.org/authors/?q=ai:wang.xu-jiaSummary: The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the \(C^{2,\alpha}\) regularity of the potential function as well as that of the free boundary, thereby resolve an open problem raised by Caffarelli and McCann.
{\copyright} 2023 Wiley Periodicals LLC.The soliton solutions for the (4 + 1)-dimensional stochastic Fokas equationhttps://zbmath.org/1530.351052024-04-15T15:10:58.286558Z"Mohammed, Wael W."https://zbmath.org/authors/?q=ai:mohammed.wael-w"Cesarano, Clemente"https://zbmath.org/authors/?q=ai:cesarano.clemente(no abstract)A problem with periodic boundary conditions for the non-Fourier heat equationhttps://zbmath.org/1530.351112024-04-15T15:10:58.286558Z"Takhirov, Jozil"https://zbmath.org/authors/?q=ai:takhirov.jozil-ostanovich|takhirov.jozil-oSummary: All diffusion equations are based on the infinite velocity of potential fields, which leads to well-known paradoxes. Consequently, in non-stationary processes, the evolution of these quantities do not completely obey the above equations due to the lack of parameters in them that take into account the finite rate of potential growth. \par In the heat conduction theory, numerous generalizations of the Fourier law are used as a remedy for these issues. The article gives a brief overview of generalizations of the Fourier law. Some mathematical issues of well-posed boundary value problems for the Guyer-Krumhansl model are discussed. As an application, a boundary value problem for a general quasilinear equation with periodic boundary conditions is considered. Schauder-type a priori estimates are established and the uniqueness of the solution is proved.Resolvents and complex powers of semiclassical cone operatorshttps://zbmath.org/1530.351152024-04-15T15:10:58.286558Z"Hintz, Peter"https://zbmath.org/authors/?q=ai:hintz.peterAuthor's abstract: We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter \(h\) tends to 0. An example of such an operator is the shifted semiclassical Laplacian \(h^2 \Delta_g +1\) on a manifold \((X,g)\) of dimension \(n\geq 3\) with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space \([0,1)_h \times X\times X\) of \(h\)-dependent integral kernels; the construction of complex powers relies on a calculus with a second semiclassical parameter. As an application, we characterize the domains of \(\big(h^2 \Delta_g +1\big)^{w/2}\) for \(\operatorname{Re}w\in \left( -\tfrac{n}{2},\tfrac{n}{2}\right)\) and use this to prove the propagation of semiclassical regularity through a cone point on a range of weighted semiclassical function spaces.
Reviewer: David Kapanadze (Tbilisi)Bounded weak solutions to superlinear Dirichlet double phase problemshttps://zbmath.org/1530.351172024-04-15T15:10:58.286558Z"Sciammetta, Angela"https://zbmath.org/authors/?q=ai:sciammetta.angela"Tornatore, Elisabetta"https://zbmath.org/authors/?q=ai:tornatore.elisabetta"Winkert, Patrick"https://zbmath.org/authors/?q=ai:winkert.patrickThe authors introduce a Dirichlet double phase problem with a parametric superlinear right-hand side that has subcritical growth. They apply variational methods and critical point theory to prove the existence of at least two nontrivial bounded weak solutions to the problem under consideration, in which they do not need to suppose the Ambrosetti-Rabinowitz condition for the perturbation term.
Reviewer: Shengda Zeng (Yulin)Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flowshttps://zbmath.org/1530.351182024-04-15T15:10:58.286558Z"Cheng, Li-Juan"https://zbmath.org/authors/?q=ai:cheng.lijuan"Thalmaier, Anton"https://zbmath.org/authors/?q=ai:thalmaier.antonSummary: Let \(M\) be a differentiable manifold endowed with a family of complete Riemannian metrics \(g(t)\) evolving under a geometric flow over the time interval \([0,T[\). We give a probabilistic representation for the derivative of the corresponding conjugate semigroup on \(M\) which is generated by a Schrödinger-type operator. With the help of this derivative formula, we derive fundamental Harnack-type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.Energy classification in a nonstandard fourth-order parabolic equation with a Navier boundary conditionhttps://zbmath.org/1530.351232024-04-15T15:10:58.286558Z"Liu, Bingchen"https://zbmath.org/authors/?q=ai:liu.bingchen"Sun, Xizheng"https://zbmath.org/authors/?q=ai:sun.xizheng"Wang, Yiming"https://zbmath.org/authors/?q=ai:wang.yiming(no abstract)Density-constrained chemotaxis and Hele-Shaw flowhttps://zbmath.org/1530.351312024-04-15T15:10:58.286558Z"Kim, Inwon"https://zbmath.org/authors/?q=ai:kim.inwon-christina"Mellet, Antoine"https://zbmath.org/authors/?q=ai:mellet.antoine"Wu, Yijing"https://zbmath.org/authors/?q=ai:wu.yijingSummary: We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint (incompressible parabolic-elliptic Patlak-Keller-Segel model). We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we rigorously establish the convergence of the solution to the characteristic function of a set whose evolution is determined by the classical Hele-Shaw free boundary problem with surface tension. The problem is set in a bounded domain, which leads to an interesting analysis on the limiting boundary conditions. Namely, we prove that the assumption of Robin boundary conditions for the chemical potential leads to a contact angle condition for the free interface (in particular Neumann boundary conditions lead to an orthogonal contact angle condition, while Dirichlet boundary conditions lead to a tangential contact angle condition).Generalized parabolic problems having a gradient term with natural growth and a measure initial datahttps://zbmath.org/1530.351322024-04-15T15:10:58.286558Z"Abdellaoui, Mohammed"https://zbmath.org/authors/?q=ai:abdellaoui.mohammed-amin"Redwane, Hicham"https://zbmath.org/authors/?q=ai:redwane.hichamSummary: We study a class of generalized nonlinear parabolic porous medium equations whose model
\[
({\mathcal{P}}_b)
\begin{cases}
b(x, u)_t - \Delta_p u + g(u) |\nabla u|^p = f \text{ in } Q := (0, T) \times \Omega,\\
u(t, x)=0 \text{ on } \Sigma := (0, T) \times \partial \Omega, \; b(x, u)(0) = b(x, u_0) \text{ in } \Omega,
\end{cases}
\]
where \(\Omega\) is a bounded open set of \(\mathbb{R}^N\) \((N \geq 2)\), \(T>0\), \(\Delta_p\) denotes the so-called \(p\)-Laplace operator \((p>1)\) and \(g\) is a continuous real function satisfying a sign condition. Given \(f \in L^1(Q)\), \(u_0 \in \mathcal{M}(\Omega)\) (the space of initial \textit{Radon} measures on \(\Omega\)), we study under which conditions on \(g\) problem \((\mathcal{P}_b)\) admits a generalized (renormalized) solution. Our methods rely on a-priori estimates and compactness arguments applied to a large class of equations involving \textit{Leray-Lions} operators and general class of lower order terms with natural \(p\)-growth.Boundary value problems for a class of linear homogeneous first-order hyperbolic systemshttps://zbmath.org/1530.351452024-04-15T15:10:58.286558Z"Rashoian, Mari"https://zbmath.org/authors/?q=ai:rashoian.mari"Sigua, Irine"https://zbmath.org/authors/?q=ai:sigua.irineSummary: Boundary value problems for a class of linear homogeneous first-order hyperbolic systems are considered. Necessary and sufficient conditions, imposed on the boundary coefficients that ensure the correctness of the problem are found. It is shown what type of violation of the correctness of the problem occurs when these conditions are not fulfilled. It is also shown what changes in the initial conditions should be made to make the problem correct. In the case of a correctly posed problem, the solution is written out explicitly.Quantitative symmetry in a mixed Serrin-type problem for a constrained torsional rigidityhttps://zbmath.org/1530.351622024-04-15T15:10:58.286558Z"Magnanini, Rolando"https://zbmath.org/authors/?q=ai:magnanini.rolando"Poggesi, Giorgio"https://zbmath.org/authors/?q=ai:poggesi.giorgioSummary: We consider a mixed boundary value problem in a domain \(\Omega\) contained in a half-ball \(B_+\) and having a portion \(\overline{T}\) of its boundary in common with the curved part of \(\partial B_+\). The problem has to do with some sort of constrained torsional rigidity. In this situation, the relevant solution \(u\) satisfies a Steklov condition on \(T\) and a homogeneous Dirichlet condition on \(\Sigma = \partial\Omega\setminus\overline{T}\subset B_+\). We provide an integral identity that relates (a symmetric function of) the second derivatives of the solution in \(\Omega\) to its normal derivative \(u_\nu\) on \(\Sigma\). A first significant consequence of this identity is a rigidity result under a quite weak overdetermining integral condition for \(u_\nu\) on \(\Sigma \): in fact, it turns out that \(\Sigma\) must be a spherical cap that meets \(T\) orthogonally. This result returns the one obtained by Guo and Xia under the stronger pointwise condition that the values of \(u_\nu\) be constant on \(\Sigma\). A second important consequence is a set of stability bounds, which quantitatively measure how \(\Sigma\) is far uniformly from being a spherical cap, if \(u_\nu\) deviates from a constant in the norm \(L^1(\Sigma)\).The time-domain scattering by the elastic shell in a two-layered unbounded structurehttps://zbmath.org/1530.351682024-04-15T15:10:58.286558Z"Zhang, Lei"https://zbmath.org/authors/?q=ai:zhang.lei.14(no abstract)Linear half-space problems in kinetic theory: abstract formulation and regime transitionshttps://zbmath.org/1530.351742024-04-15T15:10:58.286558Z"Bernhoff, Niclas"https://zbmath.org/authors/?q=ai:bernhoff.niclasSummary: In this work, a general formulation, which is based on steady boundary layer problems for the Boltzmann equation, of a half-space problem is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. The solutions will converge exponentially fast ``far away'' from the interface. For linearized kinetic half-space problems similar to the one of evaporation and condensation in kinetic theory, slowly varying modes might occur near regime transitions where the number of conditions needed to obtain well-posedness changes (corresponding to transition between evaporation and condensation, or subsonic and supersonic evaporation/condensation), preventing uniform exponential speed of convergence. However, those modes might be eliminated by imposing extra conditions on the indata at the interface. Flow velocities at the far end for which regime transitions occur are presented for Boltzmann equations: for monatomic and polyatomic single species and mixtures; as well as bosons and fermions.On electroconvection in porous mediahttps://zbmath.org/1530.351762024-04-15T15:10:58.286558Z"Abdo, Elie"https://zbmath.org/authors/?q=ai:abdo.elie"Ignatova, Mihaela"https://zbmath.org/authors/?q=ai:ignatova.mihaelaSummary: We consider the evolution of a surface charge density interacting with a two-dimensional fluid in a porous medium. In the momentum equation, Stokes's law is replaced by Darcy's law balanced by the electrical forces. This results in an active scalar equation, in which the transport velocity is computed from the scalar charge density via a nonlinear and nonlocal relation. We address the model in the whole space \(\mathbb{R}^2\) and in the periodic setting on \(\mathbb{T}^2\). We prove the global existence and uniqueness of solutions in Besov spaces \(\dot{B}_{p,1}^{2/p}\) for small initial data. We also obtain the analyticity, regularity, and long-time behavior of solutions.Existence, uniqueness, and asymptotic stability results for the 3-D steady and unsteady Navier-Stokes equations on multi-connected domains with inhomogeneous boundary conditionshttps://zbmath.org/1530.351772024-04-15T15:10:58.286558Z"Avrin, Joel"https://zbmath.org/authors/?q=ai:avrin.joel-dThis paper discusses solutions of Navier-Stokes equations on three-dimensional domains with finite number of disjoint surfaces as their boundaries. The boundary conditions are inhomogeneous and satisfy an obvious compatibility condition. Extending and generalizing analysis in [\textit{H. Kozono} and \textit{T. Yanagisawa}, Morningside Lect. Math. 3, 237--290 (2013; Zbl 1348.35172)] existence, uniqueness and asymptotic stability of weak solutions are shown. Moreover, regularity properties of those solutions are studied.
Reviewer: Piotr Biler (Wrocław)Initial-boundary value problem for flows of a fluid with memory in a 3D network-like domainhttps://zbmath.org/1530.351782024-04-15T15:10:58.286558Z"Baranovskii, E. S."https://zbmath.org/authors/?q=ai:baranovskii.evgeni-sergeevich|baranovskii.evgenii-sergeevichSummary: We consider an initial-boundary value problem for an integro-differential system that describes 3D flows of a non-Newtonian fluid with memory in a network-like domain. The problem statement uses the Dirichlet boundary conditions for the velocity and pressure fields as well as Kirchhoff-type transmission conditions at the internal nodes of the network. A theorem on the existence and uniqueness of a time-continuous weak solution is proved. In addition, an energy equality for this solution is derived.Anisotropic Liouville type theorem for the stationary Navier-Stokes equations in \(\mathbb{R}^3\)https://zbmath.org/1530.351812024-04-15T15:10:58.286558Z"Chae, Dongho"https://zbmath.org/authors/?q=ai:chae.donghoSummary: In this brief note we study the Liouville type problem for the stationary Navier-Stokes equations in \(\mathbb{R}^3\). We show that under certain anisotropic integrability conditions on the components of the velocity implies that the solution is trivial.Global solutions to coupled (Navier-)Stokes Newton systems in \(\mathbb{R}^3\)https://zbmath.org/1530.351882024-04-15T15:10:58.286558Z"Hillairet, M."https://zbmath.org/authors/?q=ai:hillairet.matthieu"Sabbagh, L."https://zbmath.org/authors/?q=ai:sabbagh.lamis|sabbagh.laraSummary: We consider the motion of spherical particles in the whole space \(\mathbb{R}^3\) filled with a viscous fluid. We show that, when modelling the fluid behavior with an incompressible Stokes system, solutions are global and no collision occurs between the spheres in finite time.Ricci curvature and the size of initial data for the Navier-Stokes equations on Einstein manifoldshttps://zbmath.org/1530.351902024-04-15T15:10:58.286558Z"Nguyen, Thieu Huy"https://zbmath.org/authors/?q=ai:nguyen-thieu-huy."Vu, Thi Ngoc Ha"https://zbmath.org/authors/?q=ai:vu-thi-ngoc-ha.Summary: Consider a noncompact Einstein manifold \((M, g)\) with negative Ricci curvature tensor (\({\mathrm{Ric}}_{ij}=rg_{ij}\) for a curvature constant \(r<0\)). Denoting by \(\Gamma (TM)\) the set of all vector fields on \(M\), we study the Navier-Stokes equations
\[
\begin{cases}
\partial_t u + \nabla_u u + \operatorname{grad}\pi = \operatorname{div}(\nabla u + \nabla u^t)^{\sharp },\;\; \mathrm{div}\,u=0,\\
u|_{t=0}= u_0 \in \Gamma (TM),\; \mathrm{div}\,u_0=0,
\end{cases}
\]
for the vector field \(u\in \Gamma (TM)\). Given any initial datum \(u_0\in \Gamma (TM)\), we prove that if the curvature constant \(r\) is large enough, then the Navier-Stokes equations on the Einstein manifold \((M, g)\) always have a unique solution \(u(\cdot ,t)\in \Gamma (TM)\) which is defined for all \(t\ge 0\) with \(u(\cdot ,0)=u_0\). We also prove the exponential decay of solutions under appropriate conditions.Optimal decay for the 2D anisotropic Navier-Stokes equations with mixed partial dissipationhttps://zbmath.org/1530.351922024-04-15T15:10:58.286558Z"Shang, Haifeng"https://zbmath.org/authors/?q=ai:shang.haifeng"Zhou, Daoguo"https://zbmath.org/authors/?q=ai:zhou.daoguoSummary: This paper studies the stability and large time behavior to the 2D anisotropic Navier-Stokes equations with mixed partial dissipation. We establish the uniform upper bounds and the global stability of solutions, and obtain the optimal decay properties to these global solutions and their higher order derivatives without any small assumptions on the initial data.Gromov-Hausdorff stability of global attractors for the 3D Navier-Stokes equations with dampinghttps://zbmath.org/1530.351932024-04-15T15:10:58.286558Z"Tao, Zhengwang"https://zbmath.org/authors/?q=ai:tao.zhengwang"Yang, Xin-Guang"https://zbmath.org/authors/?q=ai:yang.xinguang"Miranville, Alain"https://zbmath.org/authors/?q=ai:miranville.alain-m"Li, Desheng"https://zbmath.org/authors/?q=ai:li.deshengSummary: This paper is concerned with the Gromov-Hausdorff stability of global attractors for the 3D Navier-Stokes equations with damping under variations of the domain, which describes the complexity of the dynamics of the motion of a fluid flow. The Gromov-Hausdorff stability accounts for the Gromov-Hausdorff distance between two global attractors which may lie in disjoint phase spaces, as well as the stability of global attractors under perturbations of the domain. The same phase space cannot be used for the convergence via the Gromov-Hausdorff distance, which can be overcome, following \textit{J. Lee} et al. [J. Differ. Equations 269, No. 1, 125--147 (2020; Zbl 1436.35049)], by introducing a Banach space defined on a variable domain without ``pull-backing'' the perturbed system onto the original domain.Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignmenthttps://zbmath.org/1530.351942024-04-15T15:10:58.286558Z"Bai, Xiang"https://zbmath.org/authors/?q=ai:bai.xiang"Miao, Qianyun"https://zbmath.org/authors/?q=ai:miao.qianyun"Tan, Changhui"https://zbmath.org/authors/?q=ai:tan.changhui"Xue, Liutang"https://zbmath.org/authors/?q=ai:xue.liutangSummary: In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behaviour and optimal decay estimates of the solutions as \(t\to\infty\).
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}The regularity of the boundary of vortex patches for some nonlinear transport equationshttps://zbmath.org/1530.351952024-04-15T15:10:58.286558Z"Cantero, Juan Carlos"https://zbmath.org/authors/?q=ai:cantero.juan-carlos"Mateu, Joan"https://zbmath.org/authors/?q=ai:mateu.joan"Orobitg, Joan"https://zbmath.org/authors/?q=ai:orobitg.joan"Verdera, Joan"https://zbmath.org/authors/?q=ai:verdera.joanSummary: We prove the persistence of boundary smoothness of vortex patches for a nonlinear transport equation in \(\mathbb{R}^n\) with velocity field given by convolution of the density with an odd kernel, homogeneous of degree \(-(n-1)\) and of class \(C^2(\mathbb{R}^n\setminus\{0\}, \mathbb{R}^n)\). This allows the velocity field to have nontrivial divergence. The quasigeostrophic equation in \(\mathbb{R}^3\) and the Cauchy transport equation in the plane are examples.The expansion of a non-ideal gas around a sharp corner for 2-D compressible Euler systemhttps://zbmath.org/1530.351962024-04-15T15:10:58.286558Z"Chen, Jianjun"https://zbmath.org/authors/?q=ai:chen.jianjun.1"Shen, Zhimin"https://zbmath.org/authors/?q=ai:shen.zhimin"Yin, Gan"https://zbmath.org/authors/?q=ai:yin.gan(no abstract)Bounded solutions in incompressible hydrodynamicshttps://zbmath.org/1530.351972024-04-15T15:10:58.286558Z"Cobb, Dimitri"https://zbmath.org/authors/?q=ai:cobb.dimitriSummary: In this article, we study bounded solutions of Euler-type equations on \(\mathbb{R}^d\) which have no integrability at \(|x| \to +\infty\). As has been previously noted, such solutions fail to achieve uniqueness in an initial value problem, even under strong smoothness conditions. This contrasts with well-posedness results that have been obtained by using the Leray projection operator in these equations. This apparent paradox is solved by noting that using the Leray projector requires an extra condition the solutions must fulfill at \(|x| \to + \infty\). Our goal is to find one such condition which is sharp. We then apply the methods we develop to prove a full uniqueness result for Besov-Lipschitz solutions, as to the theory of Serfati solutions. In the last Section, we see how these techniques also apply to the Elsässer variables used in ideal MHD.An inviscid free boundary fluid-wave modelhttps://zbmath.org/1530.351992024-04-15T15:10:58.286558Z"Kukavica, Igor"https://zbmath.org/authors/?q=ai:kukavica.igor"Tuffaha, Amjad"https://zbmath.org/authors/?q=ai:tuffaha.amjad-mSummary: We consider the local existence and uniqueness of solutions for a system consisting of an inviscid fluid with a free boundary, modeled by the Euler equations, in a domain enclosed by an elastic boundary, which evolves according to the wave equation. We derive a priori estimates for the local existence of solutions and also conclude the uniqueness. Both, existence and uniqueness are obtained under the assumption that the Euler data belongs to \(H^r\), where \(r>2.5\), which is known to be the borderline exponent for the Euler equations. Unlike the setting of the Euler equations with vacuum, the membrane is shown to stabilize the system in the sense that the Rayleigh-Taylor condition does not need to be assumed.Well-posedness of Whitham-Broer-Kaup equation with negative dispersionhttps://zbmath.org/1530.352062024-04-15T15:10:58.286558Z"Bedjaoui, Nabil"https://zbmath.org/authors/?q=ai:bedjaoui.nabil"Mammeri, Youcef"https://zbmath.org/authors/?q=ai:mammeri.youcefSummary: In this work, we discuss the well-posedness of Whitham-Broer-Kaup equation with negative dispersion term. A symmetrizer is built, then we prove the existence and uniqueness of a solution using the vanishing viscosity method.Extension of the Hoff solutions framework to cover Navier-Stokes equations for a compressible fluid with anisotropic viscous-stress tensorhttps://zbmath.org/1530.352072024-04-15T15:10:58.286558Z"Bresch, Didier"https://zbmath.org/authors/?q=ai:bresch.didier"Burtea, Cosmin"https://zbmath.org/authors/?q=ai:burtea.cosminSummary: This paper deals with the Navier-Stokes system governing the evolution of a compressible barotropic fluid. We extend Hoff's intermediate regularity solutions framework
[\textit{D. Hoff}, J. Differ. Equations 120, No. 1, 215--254 (1995; Zbl 0836.35120); Arch. Ration. Mech. Anal. 132, No. 1, 1--14 (1995; Zbl 0836.76082)]
by relaxing the integrability needed for the initial density which is usually assumed to be \(L^{\infty} \). By achieving this, we are able to take into account general fourth-order symmetric viscous-stress tensors with coefficients depending smoothly on the time-space variables. More precisely, in space dimensions \(d=2,3\), under periodic boundary conditions, considering a pressure law \(p(\rho)=a\rho^{\gamma}\) with \(a>0\) (respectively, \( \gamma\geq d/(4-d)\)), and under the assumption that the norms of the initial data \((\rho_0-M,u_0)\in L^{2\gamma}(\mathbb{T}^d)\times(H^1(\mathbb{T}^d))^d\) are sufficiently small, we are able to construct global weak solutions. Above, \(M\) denotes the total mass of the fluid while \(\mathbb{T}\) with \(d=2,3\) stands for the periodic box. When comparing to the results known for the global weak solutions à la Leray (i.e., constructed assuming only the basic energy bounds), we obtain a relaxed condition on the range of admissible adiabatic coefficients \(\gamma \).Analysis of inhomogeneous boundary value problems for generalized Boussinesq model of mass transferhttps://zbmath.org/1530.352082024-04-15T15:10:58.286558Z"Brizitskii, R. V."https://zbmath.org/authors/?q=ai:brizitskii.r-v|brizitskii.roman-victorovich|brizitskii.roman-viktorovich"Zh. Yu., Saritskaia"https://zbmath.org/authors/?q=ai:zh-yu.saritskaiaSummary: The global solvability of the boundary value problem for the nonlinear mass transfer equations is proved under inhomogeneous Dirichlet boundary conditions for the velocity given on the entire boundary, and for the substance's concentration, given on the part of the boundary. It is assumed that the reaction coefficient in one of the equations of the model depends nonlinearly on the substance's concentration, and also depends on spatial variables. The local uniqueness of a weak solution of the boundary value problem is proved, the principle of maximum and minimum for the substance's concentration is established. Several types of conditions for the reaction coefficient are considered, each of them has its own mathematical apparatus.On uniqueness results for solutions of the Benjamin equationhttps://zbmath.org/1530.352112024-04-15T15:10:58.286558Z"Cunha, Alysson"https://zbmath.org/authors/?q=ai:cunha.alyssonSummary: We prove that the uniqueness results obtained in [\textit{J. Jiménez Urrea}, J. Differ. Equations 254, No. 4, 1863--1892 (2013; Zbl 1259.35217)] for the Benjamin equation, cannot be extended for any pair of non-vanishing solutions. On the other hand, we study uniqueness results of solutions of the Benjamin equation. With this purpose, we showed that for any solutions \(u\) and \(v\) defined in \(\mathbb{R}\times[0,T]\), if there exists an open set \(I\subset\mathbb{R}\) such that \(u(\cdot, 0)\) and \(v(\cdot, 0)\) agree in \(I,\partial_t u(\cdot,0)\) and \(\partial_tv(\cdot,0)\) agree in \(I\), then \(u\equiv v\). A better version of this uniqueness result is also established. To finish, this type of uniqueness results were also proved for the nonlocal perturbation of the Benjamin-Ono equation (npBO) and for the regularized Benjamin-Ono equation (rBO).Optimal shape design for a time-dependent Brinkman flow using asymptotic analysis techniqueshttps://zbmath.org/1530.352142024-04-15T15:10:58.286558Z"Dhif, R."https://zbmath.org/authors/?q=ai:dhif.rabeb"Meftahi, H."https://zbmath.org/authors/?q=ai:meftahi.houcine.1"Rjaibi, B."https://zbmath.org/authors/?q=ai:rjaibi.badreddineSummary: In this paper, we consider the geometric inverse problem of recovering an obstacle \(\omega\) immersed in a bounded fluid flow \(\Omega\) governed by the time-dependent Brinkman model. We reformulate the inverse problem into an optimization problem using a least squares functional. We prove the existence of an optimal solution to the optimization problem. Then, we perform the asymptotic expansion of the cost function in a simple way using a penalty method. An important advantage of this method is that it avoids the truncation method used in the literature. To reconstruct the obstacle, we propose a fast algorithm based on the topological derivative. Finally, we present some numerical experiments in two- and three-dimensional cases showing the efficiency of the proposed method.On the global well-posedness for a multi-dimensional compressible Navier-Stokes-Poisson systemhttps://zbmath.org/1530.352152024-04-15T15:10:58.286558Z"Dong, Junting"https://zbmath.org/authors/?q=ai:dong.junting"Wang, Zheng"https://zbmath.org/authors/?q=ai:wang.zheng"Xu, Fuyi"https://zbmath.org/authors/?q=ai:xu.fuyiSummary: This article is dedicated to the study of the Cauchy problem for compressible Navier-Stokes-Poisson system in spatial dimensions two and higher. We prove the global well-posedness when the initial data are close to a stable equilibrium state in critical \(L^p\) framework.Existence and blow up criterion for strong solutions to the compressible biaxial nematic liquid crystal flowhttps://zbmath.org/1530.352172024-04-15T15:10:58.286558Z"Fang, Jiahui"https://zbmath.org/authors/?q=ai:fang.jiahui"Lin, Junyu"https://zbmath.org/authors/?q=ai:lin.junyu(no abstract)On the motion of a nearly incompressible viscous fluid containing a small rigid bodyhttps://zbmath.org/1530.352182024-04-15T15:10:58.286558Z"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-daniIn this interesting paper, the authors consider the motion of an isentropic compressible viscous fluid containing a moving rigid body confined to a planar domain \(\Omega\subset \mathbb{R}^2\). The main result states that the influence of the immersed body on the fluid is negligible, provided that (i) the diameter of the body is small and (ii) the fluid is nearly incompressible (in the low Mach number regime). The specific shape of the body as well as the boundary conditions on the fluid-body interface are irrelevant and collisions with the boundary \(\partial \Omega\) are allowed. The rigid body motion may be enforced externally or governed solely by its interaction with the fluid. This paper is a first good attempt to study the negligibility of a small rigid body immersed in a planar viscous \textit{compressible} fluid.
Reviewer: Liutang Xue (Beijing)Wong-Zakai approximations and attractors for stochastic three-dimensional globally modified Navier-Stokes equations driven by nonlinear noisehttps://zbmath.org/1530.352192024-04-15T15:10:58.286558Z"Ho Thi Hang"https://zbmath.org/authors/?q=ai:ho-thi-hang."Bui Kim My"https://zbmath.org/authors/?q=ai:bui-kim-my."Pham Tri Nguyen"https://zbmath.org/authors/?q=ai:pham-tri-nguyen.Summary: We analyze the asymptotic behavior of the stochastic three dimensional globally modified Navier-Stokes equations with general Lipschitz nonlinear noise. By using the Wong-Zakai approximation, we first show that the approximate equation has a unique random attractor, and then when the stochastic equation is driven by a linear multiplicative noise or additive white noise, we show the convergence of solutions and attractors of Wong-Zakai approximations of the approximate random systems as the size of the approximation tends to zero.Global existence of strong solutions to the kinetic Cucker-Smale model coupled with the two dimensional incompressible Navier-Stokes equationshttps://zbmath.org/1530.352212024-04-15T15:10:58.286558Z"Jin, Chunyin"https://zbmath.org/authors/?q=ai:jin.chunyinSummary: In this paper, we investigate existence of global-in-time strong solutions to the Cauchy problem of the kinetic Cucker-Smale model coupled with the incompressible Navier-Stokes equations in the two dimensional space. By introducing a weighted Sobolev space and using the maximal regularity estimate on the linear non-stationary Stokes equations, we present a complete analysis on existence of global-in-time strong solutions to the coupled model, without any smallness assumptions on initial data.The slightly compressible Brinkman-Forcheimer equations and its incompressible approximationhttps://zbmath.org/1530.352262024-04-15T15:10:58.286558Z"Li, Xinhua"https://zbmath.org/authors/?q=ai:li.xinhua"Sun, Chunyou"https://zbmath.org/authors/?q=ai:sun.chunyouSummary: This paper investigates the slightly compressible Brinkman-Forchheimer equations (BFEs): \(\partial_t u - \varDelta_x u + \nabla_x p + f(u) = g\) with \(D^{-1}(t) \partial_t p + \operatorname{div}u = 0\) in a bounded 3D domain with Dirichlet boundary conditions. The features of this problem is that, formally, this system is partially dissipative, and will recover to incompressible BFEs when the time-dependent coefficient \(D(t)\) goes to infinity as \(t\to\infty\). The well-posedness and dissipation are obtained in the natural energy space. In addition, our result reveals that the velocity field \(u\) can be approximated by the solution \(u^{in}\) of incompressible BFEs in \(L^2\) provided that \(\partial_t p^{in}\) is uniform bounded in \(L^2\), here \(p^{in}\) is the pressure of incompressible BFEs.Uniqueness and regularity of weak solutions of a fluid-rigid body interaction system under the Prodi-Serrin conditionhttps://zbmath.org/1530.352282024-04-15T15:10:58.286558Z"Maity, Debayan"https://zbmath.org/authors/?q=ai:maity.debayan"Takahashi, Takéo"https://zbmath.org/authors/?q=ai:takahashi.takeoSummary: In this article, we study the weak uniqueness and the regularity of the weak solutions of a fluid-structure interaction system. More precisely, we consider the motion of a rigid ball in a viscous incompressible fluid and we assume that the fluid-rigid body system fills the entire space \(\mathbb{R}^3.\) We prove that the corresponding weak solutions that additionally satisfy a classical Prodi-Serrin condition, including a critical one, are unique. We also show that the weak solutions are regular under the Prodi-Serrin conditions, with a smallness condition in the critical case.On the blow-up phenomena of the compressible Navier-Stokes-Korteweg system with degenerate viscosityhttps://zbmath.org/1530.352292024-04-15T15:10:58.286558Z"Tang, Tong"https://zbmath.org/authors/?q=ai:tang.tong"Wei, Xu"https://zbmath.org/authors/?q=ai:wei.xu"Ling, Zhi"https://zbmath.org/authors/?q=ai:ling.zhi(no abstract)Long-time existence for a Whitham-Boussinesq system in two dimensionshttps://zbmath.org/1530.352302024-04-15T15:10:58.286558Z"Tesfahun, Achenef"https://zbmath.org/authors/?q=ai:tesfahun.achenefSummary: This paper is concerned with a two-dimensional Whitham-Boussinesq system modeling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with existence time of scale \(\mathcal{O}(\mu^{3/2-}\epsilon^{-2+})\), where \(\mu\) and \(\epsilon\) are small parameters related to the level of dispersion and nonlinearity, respectively. In particular, in the KdV regime \(\{\mu\sim\epsilon\}\), the existence time is of order \(\epsilon^{-1/2}\). The main ingredients in the proof are frequency loacalized dispersive estimates and bilinear Strichartz estimates that depend on the parameter \(\mu\).Optimal decay rate of solutions to the two-phase flow modelhttps://zbmath.org/1530.352322024-04-15T15:10:58.286558Z"Wu, Yakui"https://zbmath.org/authors/?q=ai:wu.yakui"Zhang, Yue"https://zbmath.org/authors/?q=ai:zhang.yue.2"Tang, Houzhi"https://zbmath.org/authors/?q=ai:tang.houzhi(no abstract)An initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equationshttps://zbmath.org/1530.352352024-04-15T15:10:58.286558Z"Ye, Jueling"https://zbmath.org/authors/?q=ai:ye.jueling"Guo, Houbin"https://zbmath.org/authors/?q=ai:guo.houbin"Hu, Yanbo"https://zbmath.org/authors/?q=ai:hu.yanboSummary: We investigate an initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations. The Dirichlet boundary conditions are imposed only on the velocity, while no boundary condition is imposed on the height of the fluid or the magnetic field. We derive a series of a priori estimates for the approximate solution sequences to show that they are Cauchy in a suitable Sobolev space. The local well-posedness in time of strong solutions for the initial-boundary value problem is established by the strong convergence of the approximate solution sequences.Local well-posedness of the plasma-vacuum interface problem for the ideal incompressible MHDhttps://zbmath.org/1530.352382024-04-15T15:10:58.286558Z"Zhao, Wenbin"https://zbmath.org/authors/?q=ai:zhao.wenbinSummary: In this article, we consider the plasma-vacuum interface problem for the incompressible ideal MHD. The plasma magnetic field is tangential to the interface while the vacuum magnetic field vanishes. We shall prove the stability of the interface under the Taylor sign condition. By deriving the evolution equation of the interface in the Eulerian coordinates, we are able to identify different stability mechanisms which correspond to the hyperboliticity of this evolution equation. Once the optimal regularity of the interface is obtained, all the other quantities can be estimated in the Eulerian coordinates. Therefore, we do not need the change of coordinates or the use of the Alinhac's good unknowns.Well-posedness for the one-dimensional inviscid Cattaneo-Christov systemhttps://zbmath.org/1530.352392024-04-15T15:10:58.286558Z"Zhu, Limin"https://zbmath.org/authors/?q=ai:zhu.liminSummary: This paper is devoted to studying the inviscid compressible Cattaneo-Christov system in one-dimensional space. The iterative method will be used to establish the local well-posedness of this system for large data in critical Besov spaces based on the \(L^2\) framework. Moreover, by using the renormalized energy method, one can prove the global existence of a strong solution when the initial perturbation around a constant state is sufficiently small.Solvability of the initial-boundary value problem for the Kelvin-Voigt fluid motion model with variable densityhttps://zbmath.org/1530.352412024-04-15T15:10:58.286558Z"Zvyagin, V. G."https://zbmath.org/authors/?q=ai:zvyagin.viktor-grigorevich"Turbin, M. V."https://zbmath.org/authors/?q=ai:turbin.mikhail-vSummary: The solvability of the initial-boundary value problem for the Kelvin-Voigt fluid motion model with a variable density is investigated. First, using the Laplace transform, from the rheological relation for the Kelvin-Voigt fluid motion model and the fluid motion equation in the Cauchy form, we derive a system of equations that describes the fluid motion in the Kelvin-Voigt model with a variable density. For the resulting system of equations, an initial-boundary value problem is posed, a definition of its weak solution is given, and its existence is proved. The proof is based on an approximation-topological approach to the study of fluid dynamic problems. Namely, the original problem is approximated by another one, whose solvability is proved using a version of the Leray-Schauder theorem. Then, on the basis of a priori estimates, it is proved that from the sequence of solutions of the approximation problem, it is possible to extract a subsequence that weakly converges to the solution of the original problem.Existence in the nonlinear Schrödinger equation with bounded magnetic fieldhttps://zbmath.org/1530.352442024-04-15T15:10:58.286558Z"Schindler, Ian"https://zbmath.org/authors/?q=ai:schindler.ian"Tintarev, Cyril"https://zbmath.org/authors/?q=ai:tintarev.kyrilThis paper investigates the existence of ground states for the nonlinear Schrödinger equation with a bounded external magnetic field. The equation includes an external magnetic field represented by a real-valued covector field \(A\), and an external potential \(V\). The study focuses on the existence of solutions without requiring lattice periodicity or symmetry of the magnetic field, or the presence of an external electric field. The paper builds on previous research on the existence of solutions for the nonlinear magnetic Schrödinger equation, extending the analysis to more general cases of bounded external magnetic fields. The authors provide new results and theorems related to the existence of ground states for this equation, considering critical exponents and concentration-compactness principles. The paper is structured into sections covering preliminary concepts, profile decomposition, and critical exponent problems. The results presented in the paper contribute to the understanding of the behavior of solutions to the nonlinear Schrödinger equation in the presence of bounded external magnetic fields.
\begin{itemize}
\item [1)] The existence of ground states for the nonlinear Schrödinger equation with a general external magnetic field, without requiring lattice periodicity, symmetry of the magnetic field, or the presence of an external electric field.
\item [2)] The existence conditions for ground states are refined, using a concentration-compactness argument that overcomes the lack of compactness of Sobolev embeddings in the whole space.
\item [3)] The study introduces the concept of energy at infinity, which is evaluated via lattice shifts, and provides a refined existence condition based on comparing the magnetic field and electric potential with their respective limits at infinity.
\item [4)] The paper builds upon previous research on the existence of solutions to the nonlinear magnetic Schrödinger equation, extending the analysis to cases with bounded magnetic fields and without relying on strong electric fields dominating the entire space.
\end{itemize}
The theorems and lemmas presented in the paper.
\begin{itemize}
\item [Theorem 4.5: ] Addresses the existence of minimizers in the constraint problem under a penalty condition, providing insights into the minimum in the problem and the convergence of minimizing sequences.
\item [Theorem 4.2: ] Focuses on the existence of minimizers in a model minimization problem involving the Aharonov-Bohm magnetic potential, a singular electric potential, and critical Sobolev nonlinearity.
\item [Theorem 5.3: ] Explores the critical exponent problem, specifically addressing the minimum in the problem and the convergence of minimizing sequences under certain conditions.
\item [Lemma 4.1: ] Provides a proof of the existence of minimizers in the constraint problem, demonstrating the convergence of minimizing sequences to a minimizer
\item [Lemma 4.3: ] Offers insights into the relaxation of conditions, allowing for a broader range of physical scenarios in the analysis of the nonlinear Schrödinger equation.
\item [Lemma 5.1: ] Addresses the minimum in the critical exponent problem, providing a proof of the attainment of the minimum under specific conditions.
\end{itemize}
These theorems and lemmas collectively contribute to the understanding of the behavior of solutions to the nonlinear Schrödinger equation in the presence of bounded external magnetic fields, offering valuable insights and implications for further research in this area.
Previous research mentioned in references.
The paper mentions previous research on the existence of solutions for the nonlinear magnetic Schrödinger equation
\begin{itemize}
\item [Existence Results:] [\textit{P.-L. Lions}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109--145 (1984; Zbl 0541.49009)]: This paper is one of the earliest existing results for nonlinear magnetic Schrödinger equations. It considers the case where the magnetic field is assumed to be constant.
\item [Generalization:] [\textit{G. Arioli} and \textit{A. Szulkin}, Arch. Ration. Mech. Anal. 170, No. 4, 277--295 (2003; Zbl 1051.35082)]: This paper generalizes the existence result of Esteban and Lions to the case of periodic magnetic fields. It introduces the concept of energy-preserving operators called ''magnetic shifts'' to control the loss of compactness in problems with periodic magnetic fields.
\item [Quasiclassical Asymptotics:] The paper also mentions the study of quasiclassical asymptotics in the context of the magnetic Schrödinger equation. This line of research explores the behavior of solutions in the limit of large quantum numbers, providing insights into the semiclassical behavior of the system.
\item [Properties of Solutions:] The paper refers to studies of the properties of solutions to the magnetic Schrödinger equation.
\end{itemize}
Conclusion.
In this paper, the authors contribute to the existing literature on the nonlinear Schrödinger equation with a bounded external magnetic field by providing new results and theorems related to the existence of ground states. Specifically, the paper extends the analysis to more general cases of bounded external magnetic fields, without requiring lattice periodicity or symmetry of the magnetic field, or the presence of an external electric field. The new contributions in this paper include the development of theorems and results that address critical exponents and concentration-compactness principles in the context of the nonlinear Schrödinger equation with a bounded external magnetic field. These contributions expand the understanding of the behavior of solutions to the nonlinear Schrödinger equation in the presence of bounded external magnetic fields, providing valuable insights into this area of study.
Reviewer: Mustafa Moumni (Batna)Efficient analytical algorithms to study Fokas dynamical models involving M-truncated derivativehttps://zbmath.org/1530.352502024-04-15T15:10:58.286558Z"Ehsan, Haiqa"https://zbmath.org/authors/?q=ai:ehsan.haiqa"Abbas, Muhammad"https://zbmath.org/authors/?q=ai:abbas.muhammad-mohsin"Nazir, Tahir"https://zbmath.org/authors/?q=ai:nazir.tahir"Mohammed, Pshtiwan Othman"https://zbmath.org/authors/?q=ai:mohammed.pshtiwan-othman"Chorfi, Nejmeddine"https://zbmath.org/authors/?q=ai:chorfi.nejmeddine"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: The dynamical behaviour of the (4+1)-dimensional fractional Fokas equation is investigated in this paper. The modified auxiliary equation method and extended \((\frac{G^\prime}{G^2})\)-expansion method, two reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions used the definition of the truncated M-derivative (TMD) to understand its dynamical behaviour. The hyperbolic, periodic, and trigonometric function solutions are used to derive the analytical solutions for the given model. As a result, dark, bright, and singular solitary wave solitons are obtained. We observe the fractional parameter impact of the above derivative on the physical phenomena. Each set of travelling wave solutions have a symmetrical mathematical form. Last but not least, we employ Mathematica to produce 2D and 3D figures of the analytical soliton solutions to emphasize the influence of TMD on the behaviour and symmetry of the solutions for the proposed problem. The physical importance of the solutions found for particular values of the combination of parameters during the representation of graphs as well as understanding of physical incidents.Orbital stability of pseudo-peakons for the fifth-order Camassa-Holm type equationhttps://zbmath.org/1530.352522024-04-15T15:10:58.286558Z"Hu, Qinghua"https://zbmath.org/authors/?q=ai:hu.qinghua"Zhu, Mingxuan"https://zbmath.org/authors/?q=ai:zhu.mingxuanSummary: In this paper, we consider the fifth-order Camassa-Holm type equation which is integrable and admits the single pseudo-peakons and multi-pseudo-peakons. We discuss the orbital stability of single pseudo-peakons.Initial-boundary value problem for the Hirota equation posed on a finite intervalhttps://zbmath.org/1530.352622024-04-15T15:10:58.286558Z"Wu, Jun"https://zbmath.org/authors/?q=ai:wu.jun.5"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.bolingSummary: In this paper, we turn our attention to a nonhomogeneous initial boundary value problem for Hirota equation posed on a bounded interval \((0,L)\). In particular, the explicit solution formula of linear nonhomogeneous boundary value problem is established by Laplace transform. Using space \(L^2(0,T;H_0^{s-1}(0,1))\), which is preparing for trilinear estimates and Lions-Magenes interpolation theorem, we prove the local existence, uniqueness, and Lipschitz continuous in \(C(0,T;H^s(0,1))\cap L^2(0,T;H^{s+1} (0,1))\) corresponding to the initial and boundary data. Moreover, the local solution extends to a global one by a priori bound.A new approach to the connections between moving curves and a family of complex KdV type systemshttps://zbmath.org/1530.352672024-04-15T15:10:58.286558Z"Yüzbaşı, Zühal Küçükarslan"https://zbmath.org/authors/?q=ai:yuzbasi.zuhal-kucukarslan(no abstract)Lower bounds on the radius of spatial analyticity for the higher order nonlinear dispersive equation on the real linehttps://zbmath.org/1530.352702024-04-15T15:10:58.286558Z"Zhang, Zaiyun"https://zbmath.org/authors/?q=ai:zhang.zaiyun"Liu, Zhenhai"https://zbmath.org/authors/?q=ai:liu.zhenhai"Deng, Youjun"https://zbmath.org/authors/?q=ai:deng.youjunSummary: In this paper, we consider the Cauchy problem for the higher order nonlinear dispersive equation with the initial data in Gevrey space \(G^{\sigma, s}\). First, using Tao's \([k, Z]\)-multiplier method, we establish the basic estimate on dyadic blocks. Also, using the Fourier restriction norm method, we establish the bilinear estimate and approximate conservation law. Then, using the contraction mapping principle, iteration technique as well as the bilinear estimate, we prove the local well-posedness for the initial data \(u_0\in G^{\sigma, s}\) with \(s\geq -\frac{11}{4}\). Finally, based on the local well-poseness and the approximate conservation law, we obtain that the analyticity radium does not decay faster than \(t^{-\frac{4}{11}}\) as time \(t\) goes to infinity. This result improves earlier ones in the literatures, such as
[\textit{J. Ahn} et al., Anal. Math. Phys. 11, No. 1, Paper No. 28, 22 p. (2021; Zbl 1457.32021)],
[\textit{A. Boukarou} et al., Math. Bohem. 147, No. 1, 19--32 (2022; Zbl 1513.35119)] and
[\textit{G. Petronilho} and \textit{P. L. da Silva}, Math. Nachr. 292, No. 9, 2032--2047 (2019; Zbl 1427.35220)].Some connections between stochastic mechanics, optimal control, and nonlinear Schrödinger equationshttps://zbmath.org/1530.352722024-04-15T15:10:58.286558Z"Albeverio, Sergio"https://zbmath.org/authors/?q=ai:albeverio.sergio-a"De Vecchi, Francesco Carlo"https://zbmath.org/authors/?q=ai:de-vecchi.francesco-carlo"Ugolini, Stefania"https://zbmath.org/authors/?q=ai:ugolini.stefaniaSummary: We first recall how the quantum mechanics of \(N\) particles is related, in the limit of large \(N\), to certain nonlinear Schrödinger equations, used also to describe the physical effect of Bose-Einstein condensation. We then discuss how, under the influence of Nelson's stochastic mechanics, a stochastic variational approach to both quantum mechanics and heat diffusion has been developed. We present such topics together with a newer stochastic optimal control approach to Bose-Einstein condensation. Future lines of research in the different areas of mathematics involved in these studies are mentioned.
For the entire collection see [Zbl 1515.01005].Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equationhttps://zbmath.org/1530.352732024-04-15T15:10:58.286558Z"Brzeźniak, Zdzisław"https://zbmath.org/authors/?q=ai:brzezniak.zdzislaw"Ferrario, Benedetta"https://zbmath.org/authors/?q=ai:ferrario.benedetta"Zanella, Margherita"https://zbmath.org/authors/?q=ai:zanella.margheritaThe authors consider the nonlinear damped stochastic Schrödinger equation
\begin{align*}
\mathrm{d}u(t) &= -\left( iAu(t)+i |u(t)|^{\alpha-1}u(t)+\beta u(t)\right)\mathrm{d}t-iBu(t)\circ \mathrm{d}W(t)\\
&\quad -iG(u(t))\mathrm{d}\mathbf W(t),\qquad t>0,
\end{align*}
where the space variable belong to a bounded two-dimensional domain, any power \(\alpha\in (1,\infty)\) of the defocusing nonlinearity is allowed, \(\beta>0\) is a damping constant, \(W\) and \(\mathbf W\) are two independent Wiener processes, the first stochastic differential being in the Stratonovich form, and the other one in the Itô form. The operator \(-A\) is, according to the three settings considered here:
\begin{itemize}
\item \(-A=\Delta_g\), the Laplace-Beltrami operator, on a 2D Riemannian manifold \((M,g)\) without boundary,
\item \(-A=\Delta_D\), the Laplacian with Dirichlet boundary condition on a smooth, relatively compact domain \(\mathcal{O}\subset \mathbb{R}^2\),
\item \(-A=\Delta_N\), the Laplacian with Neumann boundary condition on a smooth, relatively compact domain \(\mathcal{O}\subset \mathbb{R}^2\).
\end{itemize}
Initial data are allowed to be random. In the stochastic terms, \(B\) is a linear operator, \(G\) a Lipschitz continuous nonlinearity; precise assumptions can be found in Section~2.3. The equation is analyzed in the Itô form, by considering the Stratonovich correction term.
The authors construct a martingale solution using a modified Faedo-Galerkin method, and suitable compactness arguments. They also show pathwise uniqueness of solutions, thanks to Strichartz estimates adapted from the deterministic setting. Finally, the existence of at least one invariant measure is established by means of a version of the Krylov-Bogoliubov method, provided that the damping term \(\beta\) is sufficiently large. Extra attention is paid to the special case of a purely multiplicative noise, where, under a weaker constraint on \(\beta\), existence and uniqueness of the invariant measure are proved.
Reviewer: Rémi Carles (Rennes)On elliptic problems with Choquard term and singular nonlinearityhttps://zbmath.org/1530.352752024-04-15T15:10:58.286558Z"Choudhuri, Debajyoti"https://zbmath.org/authors/?q=ai:choudhuri.debajyoti"Repovš, Dušan D."https://zbmath.org/authors/?q=ai:repovs.dusan-d"Saoudi, Kamel"https://zbmath.org/authors/?q=ai:saoudi.kamelSummary: Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded a.e. in the domain \(\Omega\) and is Hölder continuous.Existence and uniqueness of constraint minimizers for the planar Schrödinger-Poisson system with logarithmic potentialshttps://zbmath.org/1530.352782024-04-15T15:10:58.286558Z"Guo, Yujin"https://zbmath.org/authors/?q=ai:guo.yujin"Liang, Wenning"https://zbmath.org/authors/?q=ai:liang.wenning"Li, Yan"https://zbmath.org/authors/?q=ai:li.yan.67Summary: In this paper, we study constraint minimizers \(u\) of the planar Schrödinger-Poisson system with a logarithmic convolution potential \(\ln | x | \ast u^2\) and a logarithmic external potential \(V(x) = \ln(1 + | x |^2)\), which can be described by the \(L^2\)-critical constraint minimization problem with a subcritical perturbation. We prove that there is a threshold \(\rho^\ast \in(0, \infty)\) such that constraint minimizers exist if and only if \(0 < \rho < \rho^\ast \). In particular, the local uniqueness of positive constraint minimizers as \(\rho \nearrow \rho^\ast\) is analyzed by overcoming the sign-changing property of the logarithmic convolution potential and the non-invariance under translations of the logarithmic external potential.Local well-posedness of a system describing laser-plasma interactionshttps://zbmath.org/1530.352792024-04-15T15:10:58.286558Z"Herr, Sebastian"https://zbmath.org/authors/?q=ai:herr.sebastian"Kato, Isao"https://zbmath.org/authors/?q=ai:kato.isao"Kinoshita, Shinya"https://zbmath.org/authors/?q=ai:kinoshita.shinya"Spitz, Martin"https://zbmath.org/authors/?q=ai:spitz.martinSummary: A degenerate Zakharov system arises as a model for the description of laser-plasma interactions. It is a coupled system of a Schrödinger and a wave equation with a non-dispersive direction. In this paper, a new local well-posedness result for rough initial data is established. The proof is based on an efficient use of local smoothing and maximal function norms.On well-posedness of nonlocal evolution equationshttps://zbmath.org/1530.352802024-04-15T15:10:58.286558Z"Himonas, A. Alexandrou"https://zbmath.org/authors/?q=ai:himonas.a-alexandrou"Yan, Fangchi"https://zbmath.org/authors/?q=ai:yan.fangchiSummary: This work studies questions of existence, uniqueness, dependence on initial data, and regularity of solutions to the Cauchy problem for nonlocal evolution equations with data in Sobolev spaces. The focus is on integrable Camassa-Holm type equations and in particular the Novikov equation and its dispersive modification. These equations apart from being interesting on their own right, also they can serve as ``toy'' models for the Euler equations.Probabilistic local wellposedness of 1D quintic NLS below \(L^2(\mathbb{R})\)https://zbmath.org/1530.352822024-04-15T15:10:58.286558Z"Hwang, Gyeongha"https://zbmath.org/authors/?q=ai:hwang.gyeongha"Yoon, Haewon"https://zbmath.org/authors/?q=ai:yoon.haewonSummary: We consider the Cauchy problem of the nonlinear Schrödinger equation \(i\partial_tu+\partial_x^2u\pm u^5=0\) on the real line, which is \(L^2\)-critical. We prove the local well-posedness of the initial value problem (IVP) for the scaling supercritical regularity regime \(-\frac{1}{10}<s<0\) in probabilistic manner. One of the main ingredient is to establish the probabilistic bilinear Strichartz estimate.Global solutions for 1D cubic defocusing dispersive equations: Part Ihttps://zbmath.org/1530.352832024-04-15T15:10:58.286558Z"Ifrim, Mihaela"https://zbmath.org/authors/?q=ai:ifrim.mihaela"Tataru, Daniel"https://zbmath.org/authors/?q=ai:tataru.danielSummary: This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are both \textit{small} and \textit{localized}. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data.
In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for \(L^2\) initial data which are \textit{small} and \textit{nonlocalized}. Our main structural assumption is that our nonlinearity is \textit{defocusing}. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team.
In terms of scattering, we prove that our global solutions satisfy both global \(L^6\) Strichartz estimates and bilinear \(L^2\) bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS. There, by scaling, our result also admits a large data counterpart.Combined effects in planar quasilinear Schrödinger equations with superlinear reactionhttps://zbmath.org/1530.352892024-04-15T15:10:58.286558Z"Zhang, Ning"https://zbmath.org/authors/?q=ai:zhang.ning.4"Tang, Xianhua"https://zbmath.org/authors/?q=ai:tang.xian-hua"Chen, Sitong"https://zbmath.org/authors/?q=ai:chen.sitongSummary: In this paper, we prove the existence of nontrivial solutions for the following planar quasilinear Schrödinger equation:
\[
- \Delta u + V (x) u - \Delta (u^2) u = g (u), \quad x \in \mathbb{R}^2,
\]
where \(V \in \mathcal{C} (\mathbb{R}^2, [0, \infty))\) and \(g \in \mathcal{C} (\mathbb{R}, \mathbb{R})\) is of subcritical exponential growth satisfying some mild conditions. In particular, by means of the Trudinger-Moser inequality, we give a different method from the one of the polynomial growth nonlinearities to prove the Brézis-Lieb split property when \(f\) has subcritical exponential growth. Our result extends and complements the one of \textit{S. Chen} et al. [Rev. Mat. Iberoam. 36, No. 5, 1549--1570 (2020; Zbl 1460.35100)] dealing with the higher dimensions \(N \geqslant 3\) to the dimension \(N = 2\).Boundary localization of transmission eigenfunctions in spherically stratified mediahttps://zbmath.org/1530.352932024-04-15T15:10:58.286558Z"Jiang, Yan"https://zbmath.org/authors/?q=ai:jiang.yan"Liu, Hongyu"https://zbmath.org/authors/?q=ai:liu.hongyu"Zhang, Jiachuan"https://zbmath.org/authors/?q=ai:zhang.jiachuan"Zhang, Kai"https://zbmath.org/authors/?q=ai:zhang.kaiSummary: Consider the transmission eigenvalue problem for \(u \in H^1 (\Omega)\) and \(v \in H^1 (\Omega)\):
\[
\begin{aligned}
\begin{cases}
\nabla \cdot (\sigma \nabla u) + k^2 \mathbf{n}^2 u = 0 \quad &\text{in } \Omega,\\
\Delta v + k^2 v = 0 \quad &\text{in } \Omega,\\
u = v, \sigma \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} &\text{on } \partial \Omega,
\end{cases}
\end{aligned}
\]
where \(\Omega\) is a ball in \(\mathbb{R}^N\), \(N = 2, 3\). If \(\sigma\) and \(\mathbf{n}\) are both radially symmetric, namely they are functions of the radial parameter \(r\) only, we show that there exists a sequence of transmission eigenfunctions \(\{u_m, v_m\}_{m \in \mathbb{N}}\) associated with \(k_m \rightarrow + \infty\) as \(m \rightarrow + \infty\) such that the \(L^2\)-energies of \(v_m\)'s are concentrated around \(\partial \Omega\). If \(\sigma\) and \(\mathbf{n}\) are both constant, we show the existence of transmission eigenfunctions \(\{u_j, v_j\}_{j \in \mathbb{N}}\) such that both \(u_j\) and \(v_j\) are localized around \(\partial \Omega\). Our results extend the recent studies in [\textit{Y. T. Chow} et al., SIAM J. Imaging Sci. 14, No. 3, 946--975 (2021; Zbl 1478.35159)]. Through numerics, we also discuss the effects of the medium parameters, namely \(\sigma\) and \(\mathbf{n}\), on the geometric patterns of the transmission eigenfunctions.Normal form and dynamics of the Kirchhoff equationhttps://zbmath.org/1530.352992024-04-15T15:10:58.286558Z"Baldi, Pietro"https://zbmath.org/authors/?q=ai:baldi.pietro"Haus, Emanuele"https://zbmath.org/authors/?q=ai:haus.emanueleSummary: We summarize some recent results on the Cauchy problem for the Kirchhoff equation
\[
\partial_{tt} u - \Delta u \Bigg (1 + \int_{\mathbb{T}^d} |\nabla u|^2 \Bigg) = 0
\]
on the \(d\)-dimensional torus \(\mathbb{T}^d\), with initial data \(u(0, x)\), \(\partial_t u(0,x)\) of size \(\varepsilon\) in Sobolev class. While the standard local theory gives an existence time of order \(\varepsilon^{-2}\), a quasilinear normal form allows to give a lower bound on the existence time of the order of \(\varepsilon^{-4}\) for all initial data, improved to \(\varepsilon^{-6}\) for initial data satisfying a suitable nonresonance condition. We also use such a normal form in an ongoing work with \textit{F. Giuliani} and \textit{M. Guardia} [``Effective chaos for the Kirchhoff equation on tori'', Preprint, \url{arXiv:2303.00688}] to prove existence of chaotic-like motions for the Kirchhoff equation.Exponential stability for a Timoshenko thermoelastic system with second sound and a time-varying delay term in the internal feedbackhttps://zbmath.org/1530.353042024-04-15T15:10:58.286558Z"Khalili, Zineb"https://zbmath.org/authors/?q=ai:khalili.zineb"Ouchenane, Djamel"https://zbmath.org/authors/?q=ai:ouchenane.djamelSummary: The main goal of this paper is to investigate the exponential stability of the Timoshenko system in thermoelasticity of second sound with a time-varying delay term in the internal feedback. The well-posedness of the problem is assured by using the variable norm technique of Kato. Furthermore the stability of the system is shown by applying the energy method.Global well-posedness of Vlasov-Poisson-type systems in bounded domainshttps://zbmath.org/1530.353122024-04-15T15:10:58.286558Z"Cesbron, Ludovic"https://zbmath.org/authors/?q=ai:cesbron.ludovic"Iacobelli, Mikaela"https://zbmath.org/authors/?q=ai:iacobelli.mikaelaSummary: In this paper we prove global existence of classical solutions to the Vlasov-Poisson and ionic Vlasov-Poisson models in bounded domains. On the boundary, we consider the specular reflection boundary condition for the Vlasov equation and either homogeneous Dirichlet or Neumann conditions for the Poisson equations.Propagation of velocity moments and uniqueness for the magnetized Vlasov-Poisson systemhttps://zbmath.org/1530.353132024-04-15T15:10:58.286558Z"Rege, Alexandre"https://zbmath.org/authors/?q=ai:rege.alexandreSummary: We present two results regarding the three-dimensional Vlasov-Poisson system in the full space with an external magnetic field. First, we investigate the propagation of velocity moments for solutions to the system when the magnetic field is uniform and time-dependent. We combine the classical moment approach with an induction procedure depending on the cyclotron period \(T_c = \|B\|_\infty^{-1}\) This allows us to obtain, like in the unmagnetized case, the propagation of velocity moments of order \(k > 2\) in the full space case and of order \(k > 3\) in the periodic case. Second, this time taking a general magnetic field that depends on both time and position, we manage to extend a result by \textit{E. Miot} [Commun. Math. Phys. 346, No. 2, 469--482 (2016; Zbl 1357.82041)] regarding uniqueness for Vlasov-Poisson to the magnetized framework.Global well-posedness for eddy-mean vorticity equations on \(\mathbb{T}^2\)https://zbmath.org/1530.353152024-04-15T15:10:58.286558Z"Cacchio', Yuri"https://zbmath.org/authors/?q=ai:cacchio.yuriSummary: We consider the two-dimensional, \(\beta\)-plane, eddy-mean vorticity equations for an incompressible flow, where the zonally averaged flow varies on scales much larger than the perturbation. We prove global existence and uniqueness of the solution to the equations on periodic settings.Improved local existence result of the Green-Naghdi equations with the Coriolis effecthttps://zbmath.org/1530.353162024-04-15T15:10:58.286558Z"Khorbatly, Bashar"https://zbmath.org/authors/?q=ai:khorbatly.basharSummary: The aim of this paper is to provide an alternative proof of the well-posedness of the Green-Naghdi equations with the Coriolis effect established by \textit{R. M. Chen} et al. [Adv. Math. 340, 106--137 (2018; Zbl 1403.35230)]. We showed that an additional assumption on the initial horizontal velocity is not necessary to obtain well-posedness. Indeed, with a refined symmetrizer and appropriately scaling the rotation parameter, we can derive a prior energy estimate based solely on the physically relevant depth-condition.Stability of a vector-borne disease model with a delayed nonlinear incidencehttps://zbmath.org/1530.353262024-04-15T15:10:58.286558Z"Traoré, Ali"https://zbmath.org/authors/?q=ai:traore.aliSummary: A vector-borne disease model with spatial diffusion with time delays and a general incidence function is studied. We derived conditions under which the system exhibits threshold behavior. The stability of the disease-free equilibrium and the endemic equilibrium are analyzed by using the linearization method and constructing appropriate Lyapunov functionals. It is shown that the given conditions are satisfied by at least two common forms of the incidence function.Global dynamics and traveling waves for a diffusive SEIVS epidemic model with distributed delayshttps://zbmath.org/1530.353272024-04-15T15:10:58.286558Z"Wang, Lianwen"https://zbmath.org/authors/?q=ai:wang.lianwen.1"Wang, Xingyu"https://zbmath.org/authors/?q=ai:wang.xingyu"Liu, Zhijun"https://zbmath.org/authors/?q=ai:liu.zhijun.1"Wang, Yating"https://zbmath.org/authors/?q=ai:wang.yatingSummary: This contribution develops a delayed diffusive SEIVS epidemic model for predicting and quantifying transmission dynamics for some slowly progressive diseases with long-term latent stage, governed by reaction-diffusion integro-differential equations taking distributed delays of latency and waning immunity, spatial mobility, vaccination strategies, temporary immunity into account. Necessary and sufficient conditions not only for global asymptotic stability of the disease-free and endemic equilibria are just determined by the basic reproduction number, but also for the existence and nonexistence of traveling wave solution connecting the two equilibria fully depend on the minimal wave velocity and the basic reproduction number. The targeted model with exponential distributions is applied to fit the pulmonary tuberculosis (TB) case data in China, predict its spread trend and provide us with an improving understanding of the effectiveness of a few interventions. Furthermore, our analytical findings are numerically corroborated to characterize the spatiotemporal evolution of pulmonary TB.Analysis of global stability and asymptotic properties of a cholera model with multiple transmission routes, spatial diffusion and incomplete immunityhttps://zbmath.org/1530.353282024-04-15T15:10:58.286558Z"Wang, Shengfu"https://zbmath.org/authors/?q=ai:wang.shengfu"Nie, Linfei"https://zbmath.org/authors/?q=ai:nie.linfeiSummary: A novel reaction diffusion model is proposed, in this paper, to consider the impacts on the spatially heterogeneity, horizontal and environmental transmission, incomplete immunity and exposed individuals on the spread and control of Cholera. Firstly, the existence of the solution of this model, the boundedness and the existence of the global attractor are investigated. The basic reproduction number \(\mathcal{R}_0\) of model is further defined, and then the threshold results of its global dynamics are established based on \(\mathcal{R}_0\). In particular, we analyze the asymptotic behavior of the steady state when the diffusion rate is small or large in susceptible and exposed individuals. With the help of numerical simulations, we found that the spread of susceptible individuals does not change the spatial distribution of the disease and the local epidemic risk level during the spread of the disease in different regions, while the spread of exposed individuals has an important effect on the spatial distribution of the infected population.Global bounded solution of a 3D chemotaxis-Stokes system with nonlinear doubly degenerate diffusionhttps://zbmath.org/1530.353292024-04-15T15:10:58.286558Z"Zhou, Xindan"https://zbmath.org/authors/?q=ai:zhou.xindan"Li, Zhongping"https://zbmath.org/authors/?q=ai:li.zhongpingSummary: The paper considers the following chemotaxis-Stokes system with nonlinear doubly degenerate diffusion
\[
\begin{cases}
n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n^m|^{p-2}\nabla n^m)-\chi\nabla\cdot(n\nabla c),\quad & x\in\Omega,\, t>0,\\
c_t+u\cdot\nabla c=\Delta c-cn, \quad & x\in\Omega,\, t>0,\\
u_t+\nabla P=\Delta u+n\nabla\Phi, \quad & x\in\Omega,\, t>0, \\
\nabla\cdot u=0, \quad & x\in\Omega,\, t>0
\end{cases}
\]
in a bounded domain \(\Omega\subset\mathbb{R}^3\) with zero-flux boundary conditions and no-slip boundary condition. In this paper, we proved that global bounded weak solutions exist whenever \(m>1\) and \(p\geq 2\). It removes the restrict \(8mp-8m+3p>15\) and improves the result of \textit{Q. Lin} [J. Math. Anal. Appl. 506, No. 1, Article ID 125545, 32 p. (2022; Zbl 1475.35054)].Existence and multiplicity of positive solutions of certain nonlocal scalar field equationshttps://zbmath.org/1530.353372024-04-15T15:10:58.286558Z"Bhakta, Mousomi"https://zbmath.org/authors/?q=ai:bhakta.mousomi"Chakraborty, Souptik"https://zbmath.org/authors/?q=ai:chakraborty.souptik"Ganguly, Debdip"https://zbmath.org/authors/?q=ai:ganguly.debdipIn this paper, the authors considered the existence and multiplicity of positive solutions of the following nonlocal subcritical scalar field equations
\[
(-\Delta)^su+u=a(x)|u|^{p-1}u+f(x),\ u\in H^s(\mathbb{R}^N),
\]
where \(s\in(0,1)\), \(N>2s\), \(1<p<2_s^*-1\), \(0<a(x)\in L^{\infty}(\mathbb{R}^N)\) and \(f\in H^{-s}(\mathbb{R}^N)\) is a nongegative functional. \par The main results of this paper consisting of three parts: (1) the above equation has three positive solutions provided \(a(x)\leq 1\) with \(a(x)\to1\) as \(|x|\to\infty\) and \(\|f\|_{H^{-s}}\) is small enough but not zero. (2) The above equation has two positive solutions when \(a(x)\geq 1\) with \(a(x)\to1\) as \(|x|\to\infty\) and \(\|f\|_{H^{-s}}\) is small enough but not zero. (3) The above equation has a positive solution if \(f=0\) and \(a(x)\) satisfies
\[
0<\lim\limits_{|x|\to\infty}a(x)=\inf\limits_{x\in\mathbb{R}^N}a(x).
\]
The arguments are variational and rely on a profile decomposition of a Palais-Smale sequence of the functional associated with the above equation.
Reviewer: Fukun Zhao (Kunming)Solvability of a nonlocal fractional \(p\)-Kirchhoff type problemhttps://zbmath.org/1530.353382024-04-15T15:10:58.286558Z"Bouabdallah, Mohamed"https://zbmath.org/authors/?q=ai:bouabdallah.mohamed"Chakrone, Omar"https://zbmath.org/authors/?q=ai:chakrone.omar"Chehabi, Mohammed"https://zbmath.org/authors/?q=ai:chehabi.mohammed"Zuo, Jiabin"https://zbmath.org/authors/?q=ai:zuo.jiabinSummary: In this work, we solve a Kirchhoff type boundary problem governed by the nonlocal fractional \(p\)-Laplacian operator
\[
\begin{cases}
M\left( \displaystyle\iint_{\mathbb{R}^{2N}}\vert u(x)-u(y)\vert^p K(x-y)\mathrm{d}x\mathrm{d}y \right) \mathcal{L}^p_K u=f(x,u), & \text{in}\, \Omega, \\
u=0 & \text{on} \,\mathbb{R}^N \backslash \Omega,
\end{cases}
\]
where \(\mathcal{L}^p_K\) is a non-local operator with singular kernel \(K\), \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\) with smooth boundary \(\partial \Omega\), \(M\) is a continuous function and the nonlinearity \(f\) is a Caratheodory function which does not verify the Ambrosetti-Rabinowitz type condition. Through some adequate assumptions, we prove the existence of nontrivial weak solutions to our problem by applying new skills and variational methods.A study of extremal parameter for fractional singular Choquard problemhttps://zbmath.org/1530.353512024-04-15T15:10:58.286558Z"Mishra, Pawan Kumar"https://zbmath.org/authors/?q=ai:mishra.pawan-kumar"Tripathi, Vinayak Mani"https://zbmath.org/authors/?q=ai:tripathi.vinayak-maniSummary: In this work, we study the singular problem involving fractional Laplacian operator perturbed with a Choquard nonlinearity using the idea of constrained minimization based on Nehari manifold. Precisely, for some \(\epsilon > 0\), we have proved the existence of two solutions when the parameter \(\lambda \in (0, \lambda^* + \epsilon)\), adding to the existing works dealing with multiplicity of solutions when the parameter \(\lambda\) strictly lies below \(\lambda^*\). We have given a variational characterization of the parametric value \(\lambda^*\), which is an extremal value of the parameter \(\lambda\) involved in the problem up to which the Nehari manifold method can be applied successfully. The paper highlights a fine analysis via fibering maps even for \(\lambda \geq \lambda^*\) to establish an existence of two different positive solutions for the underlying problem.
{\copyright} 2023 Wiley-VCH GmbH.Existence of two solutions of the inverse problem for a mathematical model of sorption dynamicshttps://zbmath.org/1530.353602024-04-15T15:10:58.286558Z"Denisov, A. M."https://zbmath.org/authors/?q=ai:denisov.alexander-m"Zhu Dongqin"https://zbmath.org/authors/?q=ai:zhu-dongqin.Summary: The inverse problem for a nonlinear mathematical model of sorption dynamics with an unknown variable kinetic coefficient is considered. A theorem on the existence of two solutions of the inverse problem is proved, and an iterative method for solving it is justified. An example of the application of the proposed method to the numerical solution of the inverse problem is given.Canonical Fourier-Bessel transform and their applicationshttps://zbmath.org/1530.353792024-04-15T15:10:58.286558Z"Ghazouani, Sami"https://zbmath.org/authors/?q=ai:ghazouani.sami"Sahbani, Jihed"https://zbmath.org/authors/?q=ai:sahbani.jihedSummary: The aim of this paper is to introduce a translation operator associated to the canonical Fourier-Bessel transform \(\mathcal{F}_{\nu}^{\mathbf{m}}\) and study some of its important properties. We derive a convolution product for this transform and as application we study the heat equation related to \(\Delta_{\nu}^{\mathbf{m}^{-1}}\) given by
\[
\Delta_{\nu}^{\mathbf{m}^{-1}}=\dfrac{d^2}{dx^2}+\left( \dfrac{2\nu +1}{x}+2i \dfrac{a}{b} x\right) \dfrac{d}{dx}-\left( \dfrac{a^2}{b^2}x^2 -2i (\nu +1) \dfrac{a}{b}\right).
\]Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivityhttps://zbmath.org/1530.440042024-04-15T15:10:58.286558Z"Webber, James W."https://zbmath.org/authors/?q=ai:webber.james-w"Holman, Sean"https://zbmath.org/authors/?q=ai:holman.sean-f"Quinto, Eric Todd"https://zbmath.org/authors/?q=ai:quinto.eric-toddSummary: We present novel microlocal and injectivity analyses of ellipsoid and hyperboloid Radon transforms. We introduce a new Radon transform, \(R\), which defines the integrals of a compactly supported \(L^2\) function, \(f\), over ellipsoids and hyperboloids with centers on a smooth connected surface, \(S\). Our transform is shown to be a Fourier Integral Operator (FIO) and in our main theorem we prove that \(R\) satisfies the Bolker condition if the support of \(f\) is contained in a connected open set that is not intersected by any plane tangent to \(S\). Under certain conditions, this is an equivalence. We give examples where our theory can be applied. Focusing specifically on a cylindrical geometry of interest in Ultrasound Reflection Tomography (URT), we prove injectivity results and investigate the visible singularities. In addition, we present example reconstructions of image phantoms in two-dimensions and validate our microlocal theory.Remarks on the coarea formula of Fleming and Rishelhttps://zbmath.org/1530.490362024-04-15T15:10:58.286558Z"Dierkes, Ulrich"https://zbmath.org/authors/?q=ai:dierkes.ulrichThe coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function. A special case is Fubini's theorem. It was originally published by \textit{W. H. Fleming} and \textit{R. Rishel} [Arch. Math. 11, 218--222 (1960; Zbl 0094.26301)]. In the present article the author provides a modern proof of the formula and discusses some applications.
Reviewer: Andreas Arvanitoyeorgos (Pátra)Microlocal analysis of singular measureshttps://zbmath.org/1530.580142024-04-15T15:10:58.286558Z"Banica, Valeria"https://zbmath.org/authors/?q=ai:banica.valeria"Burq, Nicolas"https://zbmath.org/authors/?q=ai:burq.nicolasSummary: The purpose of this article is to investigate the structure of singular measures from a microlocal perspective. Motivated by the result of \textit{G. De Philippis} and \textit{F. Rindler} [Ann. Math. (2) 184, No. 3, 1017--1039 (2016; Zbl 1352.49050)], and the notions of wave cones of \textit{F. Murat} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 489--507 (1978; Zbl 0399.46022); Res. Notes Math. 39, 136--212 (1979; Zbl 0437.35004); \textit{J. M. Ball} (ed.), ``Systems of Nonlinear Partial Differential Equations'', NATO ASI series. Series C 111, 263--285 (1983), \url{https://link.springer.com/content/pdf/10.1007/978-94-009-7189-9.pdf}] and of polarisation set of \textit{N. Dencker} [J. Funct. Anal. 46, 351--372 (1982; Zbl 0487.58028)] we introduce a notion of \(L^1\)-regularity wave front set for scalar and vector distributions. Our main result is a proper microlocal characterisation of the support of the singular part of tempered Radon measures and of their polar functions at these points. The proof is based on De Philippis-Rindler's approach reinforced by microlocal analysis techniques and some extra geometric measure theory arguments. We deduce a sharp \(L^1\) elliptic regularity result which appears to be new even for scalar measures and which enlightens the interest of the techniques from geometric measure theory to the study of harmonic analysis questions. For instance we prove that \(\Psi^0 L^1 \cap \mathcal{M}_{loc} \subseteq L^1_{loc}\), and in particular we obtain \(L^1\) elliptic regularity results as \(\Delta u \in L^1_{loc}\), \(D^2 u \in \mathcal{M}_{loc} \Longrightarrow D^2 u \in L^1_{loc}\). We also deduce several consequences including extensions of the results by De Philippis and Rindler [loc. cit.] giving constraints on the polar function at singular points for measures constrained by a PDE, and of Alberti's rank one theorem. Finally, we also illustrate the interest of this microlocal approach with a result of propagation of singularities for constrained measures.Analysis and discretization of a variable-order fractional wave equationhttps://zbmath.org/1530.651272024-04-15T15:10:58.286558Z"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1Summary: We analyze a variable-order time-fractional wave equation, which models, e.g., the vibration of a membrane in a viscoelastic environment. We prove that the solutions to the variable-order ordinary differential equations in the spectral decomposition of the solution to the fractional wave equation exhibit power-law decaying characteristics and overcome the difficulty that its solution operator does not have an exponential decay in contrast to its variable-order fractional diffusion analogue.
We prove an optimal-order error estimate of a numerical discretization of the variable-order fractional wave equation only under regularity assumptions of the data of the model but with no smoothness assumption of its solution. As the solution exhibits initial weak singularity, the local truncation error is suboptimal. A conventional analysis gives a suboptimal-order estimate. We develop a new technique to derive the desired optimal-order convergence rate. We also conduct numerical experiments to substantiate the mathematically proved findings.Analysis of a direct discretization of the Maxwell eigenproblemhttps://zbmath.org/1530.651512024-04-15T15:10:58.286558Z"Du, Zhijie"https://zbmath.org/authors/?q=ai:du.zhijie"Duan, Huoyuan"https://zbmath.org/authors/?q=ai:duan.huoyuanSummary: A direct discretization is analyzed for the computation of the eigenvalues of the Maxwell eigenproblem, where the finite element space \((P_k)^d + \nabla P_{k + 1}\) with the pair of the \(k\)th order \(P_k\) and \((k + 1)\)th order \(P_{k + 1}\) Lagrange element spaces (\(k \geq 1\)) on generic simplexes are used. The finite element space directly mimics the Hodge decomposition of the second-kind \(k\)th order Nédélec \((P_k)^d\) elements while the finite element formulation directly uses the classical variational formulation. We prove the convergence of the resulting finite element solutions.Well-posedness and finite element approximation of mixed dimensional partial differential equationshttps://zbmath.org/1530.651652024-04-15T15:10:58.286558Z"Hellman, Fredrik"https://zbmath.org/authors/?q=ai:hellman.fredrik"Målqvist, Axel"https://zbmath.org/authors/?q=ai:malqvist.axel"Mosquera, Malin"https://zbmath.org/authors/?q=ai:mosquera.malinSummary: In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings.An asymptotic-preserving finite element method for a forth order singular perturbation problem with boundary layershttps://zbmath.org/1530.651692024-04-15T15:10:58.286558Z"Li, Hongliang"https://zbmath.org/authors/?q=ai:li.hongliang"Ming, Pingbing"https://zbmath.org/authors/?q=ai:ming.pingbingSummary: We propose an asymptotic-preserving finite element method for a fourth order singular perturbation problem, which completely preserves the asymptotic transition of the underlying partial differential equation. As a representative, we analyze the NZT element and a linear convergence rate is proved for the solution with sharp boundary layer. Numerical examples in two and three dimensions are consistent with the theoretical prediction.A hybrid high-order method for a class of strongly nonlinear elliptic boundary value problemshttps://zbmath.org/1530.651702024-04-15T15:10:58.286558Z"Mallik, Gouranga"https://zbmath.org/authors/?q=ai:mallik.gouranga"Gudi, Thirupathi"https://zbmath.org/authors/?q=ai:gudi.thirupathiSummary: In this article, we design and analyze a hybrid high-order (HHO) finite element approximation for a class of strongly nonlinear boundary value problems. We consider an HHO discretization for a suitable linearized problem and show its well-posedness using the Gårding type inequality. The essential ingredients for the HHO approximation involve local reconstruction and high-order stabilization. We establish the existence of a unique solution for the HHO approximation using the Brouwer fixed point theorem and contraction principle. We derive an optimal order a priori error estimate in the discrete energy norm. Numerical experiments are performed to illustrate the convergence histories.Iterative two-grid methods for discontinuous Galerkin finite element approximations of semilinear elliptic problemhttps://zbmath.org/1530.651742024-04-15T15:10:58.286558Z"Zhan, Jiajun"https://zbmath.org/authors/?q=ai:zhan.jiajun"Zhong, Liuqiang"https://zbmath.org/authors/?q=ai:zhong.liuqiang"Peng, Jie"https://zbmath.org/authors/?q=ai:peng.jieSummary: In this paper, we design and analyze the iterative two-grid methods for the discontinuous Galerkin (DG) discretization of semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that is just like the classical iterative two-grid methods for nonsymmetric or indefinite linear elliptic PDEs, namely, to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space. Secondly, we designed another iterative two-grid method, which replace the semilinear term by using the corresponding first-order Taylor expansion. Specifically, we need to construct a suitable initial value, which can be sorted out from an auxiliary variational problem, for the second iterative method. We also provide the error estimates for the second iterative algorithm and present numerical experiments to illustrate the theoretical result.Determining carrier-envelope phase of an ultrashort laser pulse in quantum well via tuning inhomogeneityhttps://zbmath.org/1530.810132024-04-15T15:10:58.286558Z"Zhang, Chaojin"https://zbmath.org/authors/?q=ai:zhang.chaojin"Jiang, Yu"https://zbmath.org/authors/?q=ai:jiang.yunping"Du, Henglei"https://zbmath.org/authors/?q=ai:du.henglei"Liu, Chengpu"https://zbmath.org/authors/?q=ai:liu.chengpuSummary: When a few-cycle Inhomogeneous laser pulse propagates in parabolic quantum wells, a soliton pulse can occur after a certain propagation distance. The duration and peak intensity of such a soliton pulse are both sensitively dependent on the carrier-envelope phase (CEP) of the incident few-cycle pulse. This kind of CEP-dependence follows a sinusoidal modulation pattern with the Inhomogeneous degree of the incident laser field, which provides a route to determine the CEP by just shaping the field's inhomogeneity analogous to that by using polar media.Numerical simulation of AES dynamics in Roy's orbital elementshttps://zbmath.org/1530.830042024-04-15T15:10:58.286558Z"Avdyushev, V. A."https://zbmath.org/authors/?q=ai:avdyushev.victor-a"Gontarev, R. A."https://zbmath.org/authors/?q=ai:gontarev.r-a"Mikhaylova, Y. A."https://zbmath.org/authors/?q=ai:mikhaylova.y-a(no abstract)Trautman problem and its solution for plane waves in Riemann and Riemann-Cartan spaceshttps://zbmath.org/1530.830142024-04-15T15:10:58.286558Z"Babourova, O. V."https://zbmath.org/authors/?q=ai:babourova.olga-v"Frolov, B. N."https://zbmath.org/authors/?q=ai:frolov.boris-n"Khetczeva, M. S."https://zbmath.org/authors/?q=ai:khetczeva.m-s"Kushnir, D. V."https://zbmath.org/authors/?q=ai:kushnir.d-vSummary: The Trautman problem determines the conditions under which GWs transfer the information contained in them in an invariant manner. According to the analogy between plane gravitational and electromagnetic waves, the metric tensor of a plane gravitational wave is invariant under the five-dimensional group \(G_5\), which does not change the null hypersurface of the plane wave front. The theorems are proven on the equality to zero for the result of the action of the Lie derivative on the curvature 2-form of a plane GW in Riemann and Riemann-Cartan spaces in the direction determined by the vector generating the group \(G_5\). Thus the curvature tensor of a plane gravitational wave can invariantly transfer the information encoded in the source of the GW.