Recent zbMATH articles in MSC 35Qhttps://zbmath.org/atom/cc/35Q2023-09-22T14:21:46.120933ZWerkzeugNoise-driven bifurcations in a nonlinear Fokker-Planck system describing stochastic neural fieldshttps://zbmath.org/1517.350242023-09-22T14:21:46.120933Z"Carrillo, José A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonio"Roux, Pierre"https://zbmath.org/authors/?q=ai:roux.pierre"Solem, Susanne"https://zbmath.org/authors/?q=ai:solem.susanneSummary: The existence and characterisation of noise-driven bifurcations from the spatially homogeneous stationary states of a nonlinear, non-local Fokker-Planck type partial differential equation describing stochastic neural fields is established. The resulting theory is extended to a system of partial differential equations modelling noisy grid cells. It is shown that as the noise level decreases, multiple bifurcations from the homogeneous steady state occur. Furthermore, the shape of the branches at a bifurcation point is characterised locally. The theory is supported by a set of numerical illustrations of the condition leading to bifurcations, the patterns along the corresponding local bifurcation branches, and the stability of the homogeneous state and the most prevalent pattern: the hexagonal one.Stability and bifurcation of a reaction-diffusion-advection model with nonlinear boundary conditionhttps://zbmath.org/1517.350262023-09-22T14:21:46.120933Z"Li, Zhenzhen"https://zbmath.org/authors/?q=ai:li.zhenzhen"Dai, Binxiang"https://zbmath.org/authors/?q=ai:dai.binxiang"Zou, Xingfu"https://zbmath.org/authors/?q=ai:zou.xingfuThe authors study the dynamics of a reaction-diffusion-advection population model with nonlinear boundary condition, both from a theoretical and numerical point of view.
Trough the Crandall-Rabinowitz bifurcation theorem, a Lyapunov-Schmidt reduction method, and a perturbation method, in which bifurcation from simple eigenvalue and that from degenerate simple eigenvalue are both possible, they analyze the existence and stability of nontrivial steady states.
Next, they study a parabolic equation with monostable nonlinear boundary condition, and a parabolic equation with sublinear growth and super-linear boundary condition. Nonlinear boundary condition can lead to various steady state bifurcations.
Nonlinear boundary condition can induce the multiplicity and growing-up property of positive steady-state solutions for the model with logistic interior growth.
Finally, the numerical results show that the advection term affect to the bifurcation direction of some bifurcation, and also affect to the density distribution of the species.
Reviewer: Rosa Maria Pardo San Gil (Madrid)Small data global regularity and scattering for 3D Ericksen-Leslie compressible hyperbolic liquid crystal modelhttps://zbmath.org/1517.350292023-09-22T14:21:46.120933Z"Huang, Jiaxi"https://zbmath.org/authors/?q=ai:huang.jiaxi"Jiang, Ning"https://zbmath.org/authors/?q=ai:jiang.ning"Luo, Yi-Long"https://zbmath.org/authors/?q=ai:luo.yi-long"Zhao, Lifeng"https://zbmath.org/authors/?q=ai:zhao.lifengSummary: We study the Ericksen-Leslie hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity and scattering for small and smooth initial data near equilibrium are proved for the case that the system is a nonlinear coupling of compressible Navier-Stokes equations with wave map to \(\mathbb{S}^2\). The main strategy relies on an interplay between the control of high order energies and decay estimates, which is based on the idea inspired by the method of space-time resonances. Unlike the incompressible model, the different behaviors of the decay properties of the density and velocity field for compressible fluids at different frequencies play a key role, which is a particular feature of compressible model.Exponential convergence to equilibrium for coupled systems of nonlinear degenerate drift diffusion equationshttps://zbmath.org/1517.350362023-09-22T14:21:46.120933Z"Beck, Lisa"https://zbmath.org/authors/?q=ai:beck.lisa"Matthes, Daniel"https://zbmath.org/authors/?q=ai:matthes.daniel"Zizza, Martina"https://zbmath.org/authors/?q=ai:zizza.martinaSummary: We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a cross-diffusion term that is scaled by a parameter \(\varepsilon \geq 0\). The nonlinearities and potentials are chosen such that in the decoupled system for \(\varepsilon=0\), the evolution is metrically contractive, with a global rate \(\Lambda >0 \Lambda > 0\). The coupling is a singular perturbation in the sense that for any \(\varepsilon > 0\), contractivity of the system is lost. Our main result is that for all sufficiently small \(\varepsilon > 0\), the global attraction to a unique steady state persists, with an exponential rate \(\Lambda_{\varepsilon}=\Lambda - K \varepsilon\) for some \(k > 0\). The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities.Boundedness and asymptotic stabilization in a two-dimensional Keller-Segel-Navier-Stokes system with sub-logistic sourcehttps://zbmath.org/1517.350402023-09-22T14:21:46.120933Z"Dai, Feng"https://zbmath.org/authors/?q=ai:dai.feng"Xiang, Tian"https://zbmath.org/authors/?q=ai:xiang.tianSummary: This paper mainly deals with a Keller-Segel-Navier-Stokes model with sub-logistic source in a two-dimensional bounded and smooth domain. For a large class of cell kinetics including sub-logistic sources, it is shown that under an explicit condition involving the chemotactic strength, asymptotic ``damping'' rate and initial mass of cells, the associated no-flux/no-flux/Dirichlet problem possesses a global and bounded classical solution. Moreover, a systematical treatment has been conducted on convergence of bounded solutions toward constant equilibrium in \(W^{1,\infty}\) for sub- and standard logistic sources. In such chemotaxis-fluid setting, our boundedness improves known blow-up prevention by logistic source to blow-up prevention by sub-logistic source, indicating standard logistic source is not the weakest damping source to prevent blow-up, and our stability improves known algebraic convergence under quadratic degradation to exponential convergence under log-correction of quadratic degradation, implying log-correction of quadratic degradation quickens the decay of bounded solutions. These findings significantly improve and extend previously known ones.Large-time behavior of solutions to the time-dependent damped bipolar Euler-Poisson systemhttps://zbmath.org/1517.350542023-09-22T14:21:46.120933Z"Wu, Qiwei"https://zbmath.org/authors/?q=ai:wu.qiwei"Zheng, Junzhi"https://zbmath.org/authors/?q=ai:zheng.junzhi"Luan, Liping"https://zbmath.org/authors/?q=ai:luan.liping(no abstract)Bi-space global attractors for a class of second-order evolution equations with dispersive and dissipative terms in locally uniform spaceshttps://zbmath.org/1517.350592023-09-22T14:21:46.120933Z"Zhang, Fang-hong"https://zbmath.org/authors/?q=ai:zhang.fanghongSummary: This paper deals with the asymptotic behavior of a class of second-order evolution equations with dispersive and dissipative terms' critical nonlinearity in locally uniform spaces. First of all, we prove the global well-posedness of solutions to the evolution equations in the locally uniform spaces \(H^1_{\mathrm{lu}}(\mathbb{R}^N)\times H^1_{\mathrm{lu}}(\mathbb{R}^N)\) and define a strong continuous analytic semigroup. Secondly, the existence of the \((H^1_{\mathrm{lu}}(\mathbb{R}^N)\times H^1_{\mathrm{lu}}(\mathbb{R}^N), H^1_\rho(\mathbb{R}^N)\times H^1_\rho(\mathbb{R}^N))\)-global attractor is established. Finally, we obtain the asymptotic regularity of solutions which appear to be optimal and the existence of a bounded subset(in \(H^2_{\mathrm{lu}}(\mathbb{R}^N)\times H^2_{\mathrm{lu}}(\mathbb{R}^N)\)), which attracts exponentially every initial \(H^1_{\mathrm{lu}}(\mathbb{R}^N)\times H^1_{\mathrm{lu}}(\mathbb{R}^N)\)-bounded set with respect to the \(H^1_{\mathrm{lu}}(\mathbb{R}^N)\times H^1_{\mathrm{lu}}(\mathbb{R}^N)\)-norm.Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetimehttps://zbmath.org/1517.350702023-09-22T14:21:46.120933Z"Lai, Ning-An"https://zbmath.org/authors/?q=ai:lai.ningan"Zhou, Yi"https://zbmath.org/authors/?q=ai:zhou.yiSummary: We study the semilinear wave equation with power type nonlinearity and small initial data in Schwarzschild spacetime. If the nonlinear exponent \(p\) satisfies \(2\leq p<1+\sqrt{2}\), we establish the sharp upper bound of lifespan estimate, while for the most delicate critical power \(p=1+\sqrt{2}\), we show that the lifespan satisfies
\[
T(\varepsilon)\leq\exp\big(C\varepsilon^{-(2+\sqrt{2})}\big),
\]
the optimality of which remains to be proved. The key novelty is that the compact support of the initial data can be close to the event horizon. By combining the global existence result for \(p>1+\sqrt{2}\) obtained by \textit{H. Lindblad} et al. [Math. Ann. 359, No. 3--4, 637--661 (2014; Zbl 1295.35327)], we then give a positive answer to the interesting question posed by \textit{M. Dafermos} and \textit{I. Rodnianski} [J. Math. Pures Appl. (9) 84, No. 9, 1147--1172 (2005; Zbl 1079.35069), p. 1151]: \(p=1+\sqrt{2}\) is exactly the critical power of \(p\) separating stability and blow-up.Dispersive estimates for kinetic transport equation in Besov spaceshttps://zbmath.org/1517.350752023-09-22T14:21:46.120933Z"He, Cong"https://zbmath.org/authors/?q=ai:he.cong"Chen, Jingchun"https://zbmath.org/authors/?q=ai:chen.jingchun"Fang, Houzhang"https://zbmath.org/authors/?q=ai:fang.houzhang"He, Huan"https://zbmath.org/authors/?q=ai:he.huan(no abstract)New exact solutions of Landau-Ginzburg-Higgs equation using power index methodhttps://zbmath.org/1517.350812023-09-22T14:21:46.120933Z"Ahmad, Khalil"https://zbmath.org/authors/?q=ai:ahmad.khalil"Bibi, Khudija"https://zbmath.org/authors/?q=ai:bibi.khudija"Arif, Muhammad Shoaib"https://zbmath.org/authors/?q=ai:arif.muhammad-shoaib"Abodayeh, Kamaleldin"https://zbmath.org/authors/?q=ai:abodayeh.kamaleldin(no abstract)Multi-mode solitons in a long-short range traffic lattice model with time delayhttps://zbmath.org/1517.350892023-09-22T14:21:46.120933Z"Ren, Xiufang"https://zbmath.org/authors/?q=ai:ren.xiufang"Zhao, Shiji"https://zbmath.org/authors/?q=ai:zhao.shiji(no abstract)Multi solitary waves to stochastic nonlinear Schrödinger equationshttps://zbmath.org/1517.350902023-09-22T14:21:46.120933Z"Röckner, Michael"https://zbmath.org/authors/?q=ai:rockner.michael"Su, Yiming"https://zbmath.org/authors/?q=ai:su.yiming"Zhang, Deng"https://zbmath.org/authors/?q=ai:zhang.dengSummary: In this paper, we present a pathwise construction of multi-soliton solutions for focusing stochastic nonlinear Schrödinger equations with linear multiplicative noise, in both the \(L^2\)-critical and subcritical cases. The constructed multi-solitons behave asymptotically as a sum of \(K\) solitary waves, where \(K\) is any given finite number. Moreover, the convergence rate of the remainders can be of either exponential or polynomial type, which reflects the effects of the noise in the system on the asymptotical behavior of the solutions. The major difficulty in our construction of stochastic multi-solitons is the absence of pseudo-conformal invariance. Unlike in the deterministic case
[\textit{F. Merle}, Commun. Math. Phys. 129, No. 2, 223--240 (1990; Zbl 0707.35021);
\textit{M. Röckner}, \textit{Y. Su} and \textit{D. Zhang}, ``Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equation'', Preprint, \url{arXiv: 2110.04107}],
the existence of stochastic multi-solitons cannot be obtained from that of stochastic multi-bubble blow-up solutions in
[Röckner, Su and Zhang, loc. cit.;
\textit{Y. Su} and \textit{D. Zhang}, ``On the multi-bubble blow-up solutions to rough nonlinear Schrödinger equations'', Preprint, \url{arXiv:2012.14037}].
Our proof is mainly based on the rescaling approach in
[\textit{S. Herr} et al., Commun. Math. Phys. 368, No. 2, 843--884 (2019; Zbl 1416.35239)],
relying on two types of Doss-Sussman transforms, and on the modulation method in
[\textit{R. Côte} and \textit{X. Friederich}, Commun. Partial Differ. Equations 46, No. 12, 2325--2385 (2021; Zbl 1491.35117);
\textit{Y. Martel} and \textit{F. Merle}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 6, 849--864 (2006; Zbl 1133.35093)],
in which the crucial ingredient is the monotonicity of the Lyapunov type functional constructed by
\textit{Y. Martel} et al. [Duke Math. J. 133, No. 3, 405--466 (2006; Zbl 1099.35134)].
In our stochastic case, this functional depends on the Brownian paths in the noise.\(BV\) solutions for a hydrodynamic model of flocking-type with all-to-all interaction kernelhttps://zbmath.org/1517.350932023-09-22T14:21:46.120933Z"Amadori, Debora"https://zbmath.org/authors/?q=ai:amadori.debora"Christoforou, Cleopatra"https://zbmath.org/authors/?q=ai:christoforou.cleopatra-cSummary: We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions with concentration to the Cauchy problem for any \(BV\) initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, under a suitable condition on the initial data, we show that entropy weak solutions with concentration admit time-asymptotic flocking.New type of solutions for the nonlinear Schrödinger equation in \(\mathbb{R}^N\)https://zbmath.org/1517.351042023-09-22T14:21:46.120933Z"Duan, Lipeng"https://zbmath.org/authors/?q=ai:duan.lipeng"Musso, Monica"https://zbmath.org/authors/?q=ai:musso.monicaSolutions \(u \in H^1(\mathbb{R}^N)\) of the nonlinear Schrödinger equation
\[
-\Delta u + V(|y|)u = u^p, \quad u>0\tag{1}
\]
are constructed when \(1<p<\frac{N+2}{N-2}\), \(N \ge 3\), and the radial potential \(V\) is positive, bounded and \[ V(|y|) =V_0 + \frac{a}{|y|^m} + O\left(\frac{1}{|y|^{m+\sigma}}\right) \text{ as }\: |y| \to \infty \] for some constants \(V_0, a,\sigma>0 \) and \(m> \max\{\frac{4}{p-1},2\}\). More precisely, let \(U\) be the radial solution of (1) when \(V =0\) and so that \(U(y) \to 0\) as \(|y|\to 0\). Define \(W_{r,h}(y)= \sum_{j=1}^{2k}U(y-x_j)\) where the \(2k\) points \(x_j\) are symmetrically chosen on the sphere \(y_1^2+y_2^2+y_3^2=r^2\) intersected with the two 2D planes \(y_3=\pm rh\), \(y_j=0\), \(4\le j \le N\). The parameters \(r,h>0\) are chosen in a small range dependent on \(k\). The main result is: For all \(k\) large enough there is a solution \(u_k\) of (1) of the form \(u_k = W_{r_k,h_k}+\omega_k\) where \(\omega_k\in H^1(\mathbb{R}^N)\) has certain symmetry properties and \[ \int_{\mathbb{R}^N}|\nabla \omega_k|^2+V|\omega_k|^2 \to 0 \text{ as } k \to \infty. \] This result is related to that of \textit{J. Wei} and \textit{S. Yan} [Calc. Var. Partial Differ. Equ. 37, No. 3--4, 423--439 (2010; Zbl 1189.35106)]. The latter result is roughly equivalent to taking \(h=0\) above.
Reviewer: Denis A. White (Toledo)Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearityhttps://zbmath.org/1517.351152023-09-22T14:21:46.120933Z"Ma, Shiwang"https://zbmath.org/authors/?q=ai:ma.shiwang"Moroz, Vitaly"https://zbmath.org/authors/?q=ai:moroz.vitalyThe topic of this article is the asymptotic behavior of positive radially symmetric ground state solutions to the nonlinear Schrödinger equation with double power nonlinearity
\[
-\Delta u_\lambda + u_\lambda = u_\lambda^{2^* - 1} + \lambda u_\lambda^{q-1} \quad \text{ in } \mathbb R^N, \quad \lambda > 0,
\]
as \(\lambda \to 0\), where \(N \geq 3\), \(2^* = \frac{2N}{N-2}\) is the critical Sobolev exponent and \(q \in (2, 2^*)\) is subcritical.
In the limit \(\lambda \to 0\), for \(\mu_\lambda = u_\lambda(0)\), the rescaling \(v_\lambda(x) = \mu_\lambda^{-1} u_\lambda( \mu_\lambda^\frac{2}{N-2} x)\) converges to a solution \(U\) of the limit equation \(-\Delta U = U^{2^* - 1}\). In this paper, the authors derive the explicit growth rate of \(\mu_\lambda\) in terms of \(\lambda\). This growth rate is sensitive to the dimension and takes a different expression if \(N \geq 5\), \(N = 4\) and \(N = 3\), respectively. Similar dimension-dependent asymptotics rates are obtained for \(\|u_\lambda\|_2\) and \(\|u_\lambda\|_q\).
Under certain additional assumptions on \(N\) and \(q\), the authors also deduce from this the blow-up asymptotics for ground states of the analogous \(L^2\)-mass-constrained variational problem with double power nonlinearity, where the parameter \(\lambda \to 0\) is replaced by the constraint \(\rho = \|u\|_2 \to 0\).
Reviewer: Tobias König (Frankfurt am Main)Smoothness effects of a quadratic damping term of mixed type on a chemotaxis-type system modeling propagation of urban crimehttps://zbmath.org/1517.351182023-09-22T14:21:46.120933Z"Li, Bin"https://zbmath.org/authors/?q=ai:li.bin.7"Xie, Li"https://zbmath.org/authors/?q=ai:xie.liSummary: This paper focuses on the two-dimensional Neumann initial-boundary value problem of a chemotaxis-type system with a mixed-type quadratic damping term constituting of the product of two unknown functions. It is shown that such quadratic damping term seems sufficient to exclude the possibility of blowup in infinite time. Precisely, the first result indicates that for all reasonably regular initial data and any chemotatic sensitivity, the solution of the initial-boundary value problem is global in time within a suitable generalized framework. Meanwhile, the second result demonstrates that such generalized solution enjoys the eventual boundedness and regularity properties, i.e., it becomes bounded and smooth after some waiting time. Finally, a statement on the asymptotic stability of certain steady states is derived as a by-product.Global dynamics of a diffusive viral infection model with spatial heterogeneityhttps://zbmath.org/1517.351242023-09-22T14:21:46.120933Z"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.21"Feng, Zhaosheng"https://zbmath.org/authors/?q=ai:feng.zhaoshengSummary: To explore the joint impact of cell-free infection and cytokine-enhanced viral infection, we propose a PDE model with spatial heterogeneity by taking into account a general cell reproduction function, free-virus infection function and cytokine-enhanced viral infection function. Mathematical challenges lie in the facts that (i) the solution map of the model system loses its compactness; and (ii) the definition of the basic reproduction number and the global asymptotic stability of the infection-free steady state become challenging since the linear system at the infection-free steady state is constituted by three equations. We define the basic reproduction number \(R_0\) as the spectral radius of the sum of two linear operators corresponding to cell-free infection and cytokine-enhanced viral infection, and prove its threshold role: if \(R_0 < 1\), the infection-free steady state is globally asymptotically stable; if \(R_0 = 1\), the infection-free steady state is locally asymptotically stable; and if \(R_0 > 1\), the model system is uniformly persistent. A special case is given to show the global attractiveness of the infection steady state.On the fundamental solution for degenerate Kolmogorov equations with rough coefficientshttps://zbmath.org/1517.351282023-09-22T14:21:46.120933Z"Anceschi, Francesca"https://zbmath.org/authors/?q=ai:anceschi.francesca"Rebucci, Annalaura"https://zbmath.org/authors/?q=ai:rebucci.annalauraSummary: The aim of this work is to prove the existence of a fundamental solution associated to the Kolmogorov equation \(\mathscr{L}u = f\) in the dilation invariant case, with bounded measurable first order coefficients and bounded diffusion coefficients satisfying a sort of divergence free assumption. Finally, we prove Gaussian upper and lower bounds for the fundamental solution, and other related properties, under less restrictive assumptions on the coefficients.The one-sided Lipschitz condition in the follow-the-leader approximation of scalar conservation lawshttps://zbmath.org/1517.351302023-09-22T14:21:46.120933Z"Francesco, Marco Di"https://zbmath.org/authors/?q=ai:di-francesco.marco"Stivaletta, Graziano"https://zbmath.org/authors/?q=ai:stivaletta.grazianoSummary: We consider the follow-the-leader particle approximation scheme for a \(1d\) scalar conservation law with non-negative compactly supported \(L^\infty\) initial datum and with a \(C^1\) concave flux, which is known to provide convergence towards the entropy solution \(\rho\) to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density \(\rho^n\) is a ``discrete version of an entropy condition''; more precisely, under fairly general assumptions on \(f\) (which imply concavity of \(f)\) we prove that the continuum version \((f(\rho)/\rho)_x\le 1/t\) of said condition allows to select a unique weak solution, despite \((f(\rho)/\rho)_x\le 1/t\) is apparently weaker than the classical Oleinik-Hoff one-sided Lipschitz condition \(f'(\rho)_x\le 1/t\). Said result relies on an improved version of Hoff's uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case \(f(\rho)=\rho(A-\rho^\gamma)\) the one-sided Lipschitz condition can be improved to a discrete version of the classical (and ``sharp'') Oleinik-Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases ``alternative'' ones) of all steps of the convergence of the particle scheme.The Fokas-Lenells equation on the line: global well-posedness with solitonshttps://zbmath.org/1517.351452023-09-22T14:21:46.120933Z"Cheng, Qiaoyuan"https://zbmath.org/authors/?q=ai:cheng.qiaoyuan"Fan, Engui"https://zbmath.org/authors/?q=ai:fan.enguiSummary: In this paper, we prove the existence of global solutions in \(H^3(\mathbb{R}) \cap H^{2 , 1}(\mathbb{R})\) to the Fokas-Lenells (FL) equation on the line when the initial data includes solitons. A key tool in proving this result is a newly modified Darboux transformation, which adds or subtracts a soliton with given spectral and scattering parameters. In this way the inverse scattering transform technique is then applied to establish the global well-posedness of initial value problem with a finite number of solitons based on our previous results on the global well-posedness of the FL equation.Large deviations of Kac's conservative particle system and energy nonconserving solutions to the Boltzmann equation: a counterexample to the predicted rate functionhttps://zbmath.org/1517.351472023-09-22T14:21:46.120933Z"Heydecker, Daniel"https://zbmath.org/authors/?q=ai:heydecker.danielGiven a system with \(N\) indistinguishable particles of mass \(N^{-1}\), of velocities \(V_{t}^{1},\ldots ,V_{t}^{N}\in \mathbb{R}^{d}\), the author introduces the normalized empirical measure \(\mu _{t}^{N}=N^{-1}\sum_{i}\delta _{V_{t}^{i}}\).\ For every (ordered) pair of particles with velocities \(v,v_{\star }\in\operatorname{supp}(\mu _{t}^{N})\), the velocities change according to \(v^{\prime }(v,v_{\star },\sigma )=v-((v-v_{\star })\cdot \sigma )\sigma \), \(v_{\star }^{\prime }(v,v_{\star },\sigma )=v_{\star }+((v-v_{\star })\cdot \sigma )\sigma \), at rate \( 2B(v-v_{\star },\sigma )/N\), where \(B:\mathbb{R}^{d}\times \mathbb{S} ^{d-1}\rightarrow \lbrack 0,\infty )\) is a collision kernel here chosen as \( B(v,\sigma )=1+\left\vert v\right\vert \) (regularized hard spheres), or \( B(v,\sigma )=1\) (Maxwell molecules). The Kac process may be written as a stochastic perturbation of the Boltzmann equation, for any bounded \(f\), through: \(\left\langle f,\mu _{t}^{N}\right\rangle =\left\langle f,\mu _{0}^{N}\right\rangle +\int_{0}^{t}\left\langle f,Q\mu _{s}^{N}\right\rangle ds+M_{t}^{N,f}\), where \(M_{t}^{N,f}\) is a martingale of quadratic variation \( [M^{N,f}]_{t}=O(N^{-1})\). The Gaussian moments are defined as: \(\mathcal{E} _{z}(\mu _{0}^{\star })=\int_{\mathbb{R}^{d}}e^{z\left\vert v\right\vert ^{2}}\mu _{0}^{\star }(dv)\in \lbrack 1,\infty ]\) and they are supposed to satisfy a Gaussian upper bound: there exists \(z_{1}>0\) such that \(\mathcal{E} _{z}(\mu _{0}^{\star })<\infty \), a Gaussian lower bound: there exists \( z_{2}<\infty \) such that \(\mathcal{E}_{z}(\mu _{0}^{\star })<\infty \) for \( z<z_{2}\), and \(\mathcal{E}_{z}(\mu _{0}^{\star })\rightarrow \infty \) as \( z\uparrow z_{2}\). Further \(\mu _{0}^{\star }\) is supposed to have a continuous density \(f_{0}^{\star }\) with respect to the Lebesgue measure, and for some \(z_{3}\in (0,\infty )\) and \(c>0\), \(f_{0}^{\star }\geq ce^{-z_{3}\left\vert v\right\vert ^{2}}\). To the Kac process, the author associates an auxiliary empirical flux \(w_{t}^{N}\) defined on the parameter space of collisions \(E=[0,T]\times \mathbb{R}^{d}\times \mathbb{R}^{d}\times \mathbb{S}^{d-1}\), setting \(w_{t}^{N}=0\) and \(w_{t}^{N}=w_{t-}^{N}+\frac{1}{N }\delta _{(t,v,v_{\star },\sigma )}\) at times \(t\) where there is a collision, choosing one possible assignment \((v,v_{\star },\sigma )\) uniformly at random between the \(4^{2}\) possible choices of collision parameters. The author then proposes a rate function previously identified by \textit{C. Léonard} in [Probab. Theory Relat. Fields 101, No. 1, 1--44 (1995; Zbl 0839.60031)]. For \(\mu _{\bullet }\in \mathcal{D}\), he defines \( \overline{m}_{\mu }\in \mathcal{M}(E)\) by \(\overline{m}_{\mu }(dt,dv,dv_{\star },d\sigma )=B(v-v_{\star })dt\mu _{t}(dv)\mu _{t}(dv_{\star })d\sigma \), where \(\mathcal{M}(E)\) is the space of finite Borel measures on \(E\).\ An element \((\mu _{\bullet },w)\in \mathcal{D}\times \mathcal{M}(E)\) is a measure-flux pair if \(w\ll \overline{m}_{\mu }\) and if they solve the continuity equation: for all \(0\leq t\leq T\), \(\mu _{t}=\mu _{0}+\int_{E}\Delta (v,v_{\star },\sigma )1_{s\leq t}w(ds,dv,dv_{\star },d\sigma )\). Here \(\mathcal{D}\) is the Skorokhod space defined as: \( \mathcal{D}=\{\mu _{\bullet }\in D([0,T],(\mathcal{P}_{2},W):\sup_{t\leq T}\left\vert v\right\vert ^{2},\mu _{t}<\infty \}\), equipped with a metric inducing the Skorokhod \(J_{1}\)-topology, \(\mathcal{P}_{2}\) being the space of probability measures on \(\mathbb{R}^{d}\) with finite second moment, equipped with the Monge-Kantorovich-Wasserstein distance. The dynamic cost of a trajectory \((\mu _{\bullet },w)\in \mathcal{D}\times \mathcal{M}(E)\) is defined as \(\mathcal{J}(\mu _{\bullet },w)=\int_{E}\tau (\frac{dw}{d \overline{m}_{\mu }})\overline{m}_{\mu }(ds,dv,dv_{\star },d\sigma )\) if \( (\mu _{\bullet },w)\) is a measure-flux pair, \(\infty \) else, where \(\tau :[0,\infty ]\rightarrow \lbrack 0,\infty ]\) is the function \(\tau (k)=k\log k-k+1\), and the full rate function is defined as: \(\mathcal{I}(\mu _{\bullet },w)=H(\mu _{0}\mid \mu _{0}^{\ast })+\mathcal{J}(\mu _{\bullet },w)\), with \(H(\mu _{0}\mid \mu _{0}^{\ast })=\int_{\mathbb{R}^{d}}\frac{ d\mu _{0}}{d\mu _{0}^{\ast }}\log(\frac{d\mu _{0}}{d\mu _{0}^{\ast }})\mu _{0}^{\ast }(dv)\) if \(\mu _{0}\ll \mu _{0}^{\ast }\); \(\infty \) else. The first main result proves that for \(N\geq 2\), for every Kac process and its flux \((\mu _{\bullet }^{N},w^{N})\), with particles drawn initially from \(\mu _{0}^{\star }\) satisfying the above hypotheses and \(\mathcal{I}\) the above rate function, then for all \(\mathcal{A}\subset \mathcal{D}\times \mathcal{M} (E)\) closed: \(\limsup_{N}\frac{1}{N}\log\mathbb{P}((\mu _{\bullet }^{N},w^{N})\in \mathcal{A})\leq -\inf\{\mathcal{I}(\mu _{\bullet },w):(\mu _{\bullet },w)\in \mathcal{A}\}\), and for all \(\mathcal{U}\subset \mathcal{D} \times \mathcal{M}(E)\) open: \(\liminf_{N}\frac{1}{N}\log\mathbb{P}((\mu _{\bullet }^{N},w^{N})\in \mathcal{U})\geq -\inf\{\mathcal{I}(\mu _{\bullet },w):(\mu _{\bullet },w)\in \mathcal{U}\cap \mathcal{R}\}\), where \(\mathcal{R }=\{(\mu _{\bullet },w)\in \mathcal{D}\times \mathcal{M}(E):\left\langle 1+\left\vert v\right\vert ^{2}+\left\vert v_{\ast }\right\vert ^{2},w\right\rangle <\infty \}\). The second main result proves estimates in the case where \(B\) is the regularized hard spheres kernel. For the proofs, the author gathers results concerning large deviations and related problems. He proves the upper bound using a variational formulation of the rate function \(\mathcal{I}\). He proves the lower bound using an approximation argument for paths belonging to \(\mathbb{R}\) and a standard ``left tilting'' argument. The proof of the last main result is based on properties of the Kac process and a careful analysis of Cramér bounds.
Reviewer: Alain Brillard (Riedisheim)A survey on some vanishing viscosity limit resultshttps://zbmath.org/1517.351482023-09-22T14:21:46.120933Z"Beirão da Veiga, Hugo"https://zbmath.org/authors/?q=ai:beirao-da-veiga.hugo"Crispo, Francesca"https://zbmath.org/authors/?q=ai:crispo.francescaSummary: We present a survey concerning the convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations. After considering the Cauchy problem, particular attention is given to the convergence under Navier slip-type boundary conditions. We show that, in the presence of flat boundaries (typically, the half-space case), convergence holds, uniformly in time, with respect to the initial data's norm. In spite of this result (and of a similar result for arbitrary two-dimensional domains), strong inviscid limit results are proved to be false in general domains, in correspondence to a very large family of smooth initial data. In Section 6, we present a result in this direction.Existence of controls insensitizing the rotational of the solution of the Navier-Stokes system having a vanishing componenthttps://zbmath.org/1517.351492023-09-22T14:21:46.120933Z"Carreño, N."https://zbmath.org/authors/?q=ai:carreno.nicolas"Prada, J."https://zbmath.org/authors/?q=ai:prada.julia|prada.juan-carlos-garciaSummary: In this paper we study an insensitizing control problem for the Navier-Stokes system. The novelty is that we insensitize the rotational of the solution using controls with one component fixed at zero. This problem can be formulated as a null controllability problem for a nonlinear cascade system for which we follow the usual duality approach. First, we prove a suitable Carleman inequality for a system coupling two Stokes like equations, which leads to the null controllability at any positive time. Finally, we deduce a local null controllability result for the cascade system by a local inverse argument.Weak solutions to the generalized Navier-Stokes equations with mixed boundary conditions and implicit obstacle constraintshttps://zbmath.org/1517.351502023-09-22T14:21:46.120933Z"Cen, Jinxia"https://zbmath.org/authors/?q=ai:cen.jinxia"Nguyen, Van Thien"https://zbmath.org/authors/?q=ai:nguyen-van-thien."Vetro, Calogero"https://zbmath.org/authors/?q=ai:vetro.calogero"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: This paper is concerned with the investigation of a generalized Navier-Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational-hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.The influence of viscous coefficients on the lifespan of 3-D anisotropic Navier-Stokes systemhttps://zbmath.org/1517.351512023-09-22T14:21:46.120933Z"Hao, Tiantian"https://zbmath.org/authors/?q=ai:hao.tiantian"Liu, Yanlin"https://zbmath.org/authors/?q=ai:liu.yanlinSummary: The anisotropic Navier-Stokes system arises in geophysical fluid dynamics, which is derived by changing \(-\nu \Delta\) in the classical Navier-Stokes system to \(-(\nu_1\partial^2_1+\nu_2\partial^2_2+\nu_3\partial^2_3)\). Here \(\nu_1,\nu_2,\nu_3\) are the viscous coefficients, which can be different from each other. This reflects that the fluid can behave differently in each direction. The purpose of this paper is to derive some lower bound estimates on the lifespan to such an anisotropic Navier-Stokes system. We not only investigate the case when \(\nu_1, \nu_2, \nu_3\) are all positive, but also the more sophisticated cases when one or two of them vanish. We find that in these lower bound estimates, the weights of \(\nu_1,\nu_2,\nu_3\) are not equal. A detailed study of this problem can also help us to have a better understanding of the nonlinear structure in the classical Navier-Stokes system.Singularity formation for the multi-dimensional compressible degenerate Navier-Stokes equationshttps://zbmath.org/1517.351522023-09-22T14:21:46.120933Z"Huang, Yucong"https://zbmath.org/authors/?q=ai:huang.yucong"Wang, Qin"https://zbmath.org/authors/?q=ai:wang.qin.3"Zhu, Shengguo"https://zbmath.org/authors/?q=ai:zhu.shengguoSummary: In this paper, the multi-dimensional (M-D) isentropic compressible Navier-Stokes equations with degenerate viscosities (\textbf{ICNS}) is considered in the whole space. We show that for a certain class of initial data with local vacuum, the regular solution of the corresponding Cauchy problem will blow up in finite time, no matter how small and smooth the initial data are. It is worth pointing out that local existence of regular solution considered in this paper has been established.3D anisotropic Navier-Stokes equations in \(\mathbb{R}\): stability and large-time behaviourhttps://zbmath.org/1517.351532023-09-22T14:21:46.120933Z"Ji, Ruihong"https://zbmath.org/authors/?q=ai:ji.ruihong"Tian, Ling"https://zbmath.org/authors/?q=ai:tian.ling"Wu, Jiahong"https://zbmath.org/authors/?q=ai:wu.jiahongSummary: The study on the large-time behaviour of solutions to the 3D incompressible anisotropic Navier-Stokes (ANS) equations is very recent. Powerful tools designed for the Navier-Stokes equations with full Laplacian dissipation such as the Fourier splitting method no longer apply to the case when there is only horizontal dissipation. For the whole space \(\mathbb{R}^3\), as \(t\to\infty\), solutions of the ANS equations converge to the trivial solution and the convergence rate is algebraic. This paper is devoted to the case when the spatial domain \(\Omega\) is \(\mathbb{T}^2\times\mathbb{R}\). Our results reveal that the large-time behaviour for \(\mathbb{T}^2\times\mathbb{R}\) is quite different from that for \(\mathbb{R}^3\). We show that any small initial velocity field \(u_0\in H^2(\Omega)\) leads to a unique global solution \(u\) that remains small in \(H^2(\Omega)\). More importantly, as \(t\to\infty\), the velocity field \(u\) converges to a nontrivial steady state. The first two components of the steady state are given by the horizontal average of the first two components of \(u_0\) while the third component vanishes. In addition, this convergence is exponentially fast.Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equationshttps://zbmath.org/1517.351542023-09-22T14:21:46.120933Z"Koganemaru, Junichi"https://zbmath.org/authors/?q=ai:koganemaru.junichi"Tice, Ian"https://zbmath.org/authors/?q=ai:tice.ianSummary: This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest.Forces for the Navier-Stokes equations and the Koch and Tataru theoremhttps://zbmath.org/1517.351552023-09-22T14:21:46.120933Z"Lemarié-Rieusset, Pierre Gilles"https://zbmath.org/authors/?q=ai:lemarie-rieusset.pierre-gillesSummary: We consider the Cauchy problem for the incompressible Navier-Stokes equations on the whole space \(\mathbb{R}^3\), with initial value \(\vec{u}_0\in \mathrm{BMO}^{-1}\) (as in Koch and Tataru's theorem) and with force \(\vec{f} = \operatorname{div}\mathbb{F}\) where smallness of \(\mathbb{F}\) ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin's solutions (in \(L^2\mathcal{F}^{-1}L^1\)) or solutions in the multiplier space \(\mathcal{M}(\dot{H}^{1/2, 1}_{t, x}\mapsto L^2_{t, x})\).A posteriori error analysis for pressure-robust HDG methods for the stationary incompressible Navier-Stokes equationshttps://zbmath.org/1517.351562023-09-22T14:21:46.120933Z"Leng, Haitao"https://zbmath.org/authors/?q=ai:leng.haitaoSummary: A hybridizable discontinuous Galerkin method with divergence-free and \(H(\mathrm{div})\)-conforming velocity field is considered in this paper for the stationary incompressible Navier-Stokes equations. The pressure-robustness, which means that a priori error estimates for the velocity is independent of the pressure error, is satisfied. As a consequence, an efficient and reliable a posteriori error estimator is proved for the \(L^2\)-errors in the velocity gradient and pressure under a smallness assumption. We conclude by several numerical examples which reveal the pressure-robustness and show the performance of the obtained a posteriori error estimator.Global existence and optimal time-decay rates of the compressible Navier-Stokes-Euler systemhttps://zbmath.org/1517.351572023-09-22T14:21:46.120933Z"Li, Hai-Liang"https://zbmath.org/authors/?q=ai:li.hailiang"Shou, Ling-Yun"https://zbmath.org/authors/?q=ai:shou.lingyunSummary: In this paper, we consider the Cauchy problem of the multidimensional compressible Navier-Stokes-Euler system for two-phase flow motion, which consists of the isentropic compressible Navier-Stokes equations and the isothermal compressible Euler equations coupled with each other through a relaxation drag force. We first establish the local existence and uniqueness of the strong solution for general initial data in a critical homogeneous Besov space, and then prove the global existence of the solution if the initial data are a small perturbation of the equilibrium state. Moreover, under the additional condition that the low-frequency part of the initial perturbation also belongs to another Besov space with lower regularity, we obtain the optimal time-decay rates of the global solution toward the equilibrium state. These results imply that the relaxation drag force and the viscosity dissipation affect the regularity properties and long time behaviors of solutions for the compressible Navier-Stokes-Euler system.Global well-posedness for 2D fractional inhomogeneous Navier-Stokes equations with rough densityhttps://zbmath.org/1517.351582023-09-22T14:21:46.120933Z"Li, Yatao"https://zbmath.org/authors/?q=ai:li.yatao"Miao, Qianyun"https://zbmath.org/authors/?q=ai:miao.qianyun"Xue, Liutang"https://zbmath.org/authors/?q=ai:xue.liutangSummary: This paper is concerned with the global well-posedness issue of the two-dimensional (2D) incompressible inhomogeneous Navier-Stokes equations with fractional dissipation and rough density. By establishing the \(L^q_t(L^p_x)\)-maximal regularity estimate for the generalized Stokes system and using the Lagrangian approach, we prove the global existence and uniqueness of regular solutions for the 2D fractional inhomogeneous Navier-Stokes equations with large velocity field, provided that the initial density is sufficiently close to the constant 1 in \(L^2\cap L^\infty\) and in the norm of some multiplier spaces. Moreover, we also consider the associated density patch problem, and show the global persistence of \(C^{1,\gamma}\)-regularity of the density patch boundary when the piecewise jump of density is small enough.Decay estimates to the inhomogeneous Navier-Stokes equations in \(\mathbb{R}^3\)https://zbmath.org/1517.351592023-09-22T14:21:46.120933Z"Mu, Yanmin"https://zbmath.org/authors/?q=ai:mu.yanminSummary: In this paper, we study the decay estimates to the inhomogeneous incompressible Navier-Stokes Equations in \(\mathbb{R}^3\). The greatest difficulty is that the density \(\rho\) has only \(L^{\infty}\) norm; to overcome, this difficulty we find a new key quantity \(\int^{+\infty}_0\Vert u\Vert^2_{L^{\infty}}\mathrm{d}t< +\infty\).Global wellposedness of the 3D compressible Navier-Stokes equations with free surface in the maximal regularity classhttps://zbmath.org/1517.351602023-09-22T14:21:46.120933Z"Shibata, Yoshihiro"https://zbmath.org/authors/?q=ai:shibata.yoshihiro"Zhang, Xin"https://zbmath.org/authors/?q=ai:zhang.xin.23Summary: This paper concerns the global well posedness issue of the compressible Navier-Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. Combining the maximal \(L_p\)-\(L_q\) estimate and the \(L_p\)-\(L_q\) decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal \(L_p\)-\(L_q\) regularity class for some \(p>2\) and \(q>3\). Namely, the solution is bounded as \(L_p\) in time and \(L_q\) in space. Compared with the previous results of the free boundary value problem of (CNS) in unbounded domains, we relax the regularity assumption on the initial states, which is the advantage by using the maximal \(L_p\)-\(L_q\) regularity framework. On the other hand, the equilibrium state of the moving boundary of the exterior domain is not necessary the sphere. To our knowledge, this paper is the first result on the long time solvability of the free boundary value problem of (CNS) in the exterior domain.Random attractors for non-autonomous stochastic Navier-Stokes-Voigt equations in some unbounded domainshttps://zbmath.org/1517.351612023-09-22T14:21:46.120933Z"Wang, Shu"https://zbmath.org/authors/?q=ai:wang.shu"Si, Mengmeng"https://zbmath.org/authors/?q=ai:si.mengmeng"Yang, Rong"https://zbmath.org/authors/?q=ai:yang.rongSummary: This paper is concerned with the asymptotic behavior for the three dimensional non-autonomous stochastic Navier-Stokes-Voigt equations on unbounded domains. A continuous non-autonomous random dynamical system for the equations is firstly established. We then obtain pullback asymptotic compactness of solutions and prove that the existence of tempered random attractors for the random dynamical system generated by the equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time.Entropy estimates for uniform attractors of 2D Navier-Stokes equations with weakly normal measureshttps://zbmath.org/1517.351622023-09-22T14:21:46.120933Z"Xiong, Yangmin"https://zbmath.org/authors/?q=ai:xiong.yangmin"Song, Xiaoya"https://zbmath.org/authors/?q=ai:song.xiaoya"Sun, Chunyou"https://zbmath.org/authors/?q=ai:sun.chunyouSummary: This paper aims at the long-time behavior of non-autonomous 2D Navier-Stokes equations with a class of external forces which are \(H\)-valued measures in time. We first establish the well-posedness of solutions as well as the existence of a strong uniform attractor, and then pay the main attention on the estimation of \(\varepsilon\)-entropy for such uniform attractor in the standard energy phase space.Global unique solvability of inhomogeneous incompressible Navier-Stokes equations with nonnegative densityhttps://zbmath.org/1517.351632023-09-22T14:21:46.120933Z"Zhang, Jianzhong"https://zbmath.org/authors/?q=ai:zhang.jianzhong.1|zhang.jianzhong"Shi, Weixuan"https://zbmath.org/authors/?q=ai:shi.weixuan"Cao, Hongmei"https://zbmath.org/authors/?q=ai:cao.hongmeiSummary: In this paper, we consider the initial-boundary value problem to the inhomogeneous incompressible Navier-Stokes equations in \(\Omega\subset\mathbb{R}^2\). The initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. The global existence and uniqueness of solutions are proved, for any initial data \((\rho_0,u_0)\in(L^\infty\times H^s_0)\) with \(s>0\), which constitutes a positive answer to the question raised by \textit{R. Danchin} and \textit{P. B. Mucha} [Commun. Pure Appl. Math. 72, No. 7, 1351--1385 (2019; Zbl 1420.35182)], in which the initial velocity \(u_0\in H^1_0\) (see also [\textit{J. Li}, J. Differ. Equations 263, No. 10, 6512--6536 (2017; Zbl 1370.76026)]).On the instantaneous radius of analyticity of \(L^p\) solutions to 3D Navier-Stokes systemhttps://zbmath.org/1517.351642023-09-22T14:21:46.120933Z"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.5|zhang.ping.2|zhang.ping.1|zhang.ping|zhang.ping.3Summary: In this paper, we first investigate the instantaneous radius of space analyticity for the solutions of 3D Navier-Stokes system with initial data in the Besov spaces \(\dot{B}^s_{p,q}({\mathbb{R}}^3)\) for \(p\in ]1,\infty [\), \(q\in [1,\infty ]\) and \(s\in \left [-1+\frac{3}{p},\frac{3}{p}\right [\). Then for initial data \(u_0\in L^p({\mathbb{R}}^3)\) with \(p\) in \(]3, 6[\), we prove that 3D Navier-Stokes system has a unique solution \(u=u_L+v\) with \(u_L \stackrel{\text{def}}= e^{t \Delta} u_0\) and \(v\in{\widetilde{L}^\infty_T\Big (\dot{B}^{1-\frac{3}{p}}_{p,\frac{p}{2}}\Big )}\cap{\widetilde{L}^1_T\Big (\dot{B}^{3-\frac{3}{p}}_{p,\frac{p}{2}}\Big )}\) for some positive time \(T\). Furthermore, we derive an explicit lower bound for the radius of space analyticity of \(v\), which in particular extends the corresponding results in [\textit{R. Hu} and \textit{P. Zhang}, Chin. Ann. Math., Ser. B 43, No. 5, 749--772 (2022; Zbl 1502.35082)] with initial data in \(L^p({\mathbb{R}}^3)\) for \(p\in [3, 18/5[\).Invariant region on a non-isentropic gas dynamics systemhttps://zbmath.org/1517.351652023-09-22T14:21:46.120933Z"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.10|chen.yong.1|chen.yong.3|chen.yong.5|chen.yong.6|chen.yong.8|chen.yong|chen.yong.4|chen.yong.2|chen.yong.9"Klingenberg, Christian"https://zbmath.org/authors/?q=ai:klingenberg.christian"Lu, Yun-guang"https://zbmath.org/authors/?q=ai:lu.yunguang"Wang, Xianting"https://zbmath.org/authors/?q=ai:wang.xianting"You, Guoqiao"https://zbmath.org/authors/?q=ai:you.guoqiaoSummary: In this paper, we found a special non-isentropic gas dynamics system, whose invariant region is the opposite of the corresponding isentropic case. This shows that the powerful invariant region theory introduced by Chueh, Conley and Smoller for general hyperbolic system of two conservation laws cannot be obviously applied to obtain the a priori \(L^\infty\) estimates for systems of more than two equations.Transient gas pipeline flow: analytical examples, numerical simulation and a comparison to the quasi-static approachhttps://zbmath.org/1517.351662023-09-22T14:21:46.120933Z"Gugat, Martin"https://zbmath.org/authors/?q=ai:gugat.martin"Krug, Richard"https://zbmath.org/authors/?q=ai:krug.richard"Martin, Alexander"https://zbmath.org/authors/?q=ai:martin.alexanderSummary: The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a PDE model is necessary to handle certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based on multi-period optimization of steady states does not lead to approximations that converge to the optimal state.The cavitation and concentration of Riemann solutions for the isentropic Euler equations with isothermal dusty gashttps://zbmath.org/1517.351672023-09-22T14:21:46.120933Z"Jiang, Weifeng"https://zbmath.org/authors/?q=ai:jiang.weifeng"Zhang, Yuan"https://zbmath.org/authors/?q=ai:zhang.yuan"Li, Tong"https://zbmath.org/authors/?q=ai:li.tong"Chen, Tingting"https://zbmath.org/authors/?q=ai:chen.tingtingSummary: In this paper, we are mainly concerned with the phenomena of cavitation and concentration to the isentropic Euler equations with isothermal dusty gas as the pressure vanishes with double parameters. Firstly, we solve the Riemann problem by analyzing the properties of the elementary waves due to the existence of the inflection points. Secondly, we investigate the limiting behaviors of the Riemann solutions as the pressure vanishes and observe the cavitation and concentration phenomena. Finally, some numerical simulations are performed and the results are consistent with the theoretical analysis. The highlight of this paper is that we extend the restriction of \(\rho \theta \ll 1\) in the previous works to \(\rho \theta < 1\), which makes the wave curve from convex to non-convex. And we prove that the limit of the Riemann solutions of isothermal dusty gas equations is the Riemann solutions of the limit of that equations as pressure vanishes, while the limiting process to vacuum state is different from the previous works.Incompressible Euler equations with stochastic forcing: a geometric approachhttps://zbmath.org/1517.351682023-09-22T14:21:46.120933Z"Maurelli, Mario"https://zbmath.org/authors/?q=ai:maurelli.mario"Modin, Klas"https://zbmath.org/authors/?q=ai:modin.klas"Schmeding, Alexander"https://zbmath.org/authors/?q=ai:schmeding.alexanderThe authors consider the stochastic Euler equations for an incompressible fluid: \(\frac{\partial u}{\partial t }+\nabla _{u}u+\nabla p=\overset{.}{W}\), \(\operatorname{div}(u)=0\), posed in the fluid domain taken as a compact oriented Riemannian manifold \(K\) of dimension \(d\), possibly with smooth boundary. Here the fluid velocity \(u\) is a vector field on \(K\) of Sobolev regularity \(H^{s}\), \(p\) the pressure, \(\nabla _{u}\) the co-variant derivative along \(u\), and the vector valued noise \(\overset{.} {W}\) a fluctuating external force field. \(W\) is a Wiener process with values in the space of Sobolev, divergence free vector fields. If \(K\) has a boundary, \(u\) and the noise field \(\overset{.}{W}\) are supposed to be tangential to it. The authors reformulate the Euler model using the Lagrangian variable \(\Phi \), with \(\overset{.}{\Phi }=u\circ \Phi \), as: \( \nabla _{\overset{.}{\Phi }}\overset{.}{\Phi }+\nabla p\circ \Phi =\overset{. }{W}\circ \Phi \), \(\operatorname{div}(\overset{.}{\Phi }\circ \Phi ^{-1})=0\). The first main result proves that if \(s>d/2+1\) and if the noise takes values in the space of \(H^{s+2}\) divergence-free (and tangential) vector fields, then the preceding Lagrangian formulation has a local solution which is unique. The second main result proves that if \(s>d/2+4\) and if the noise takes values in the space of \(H^{s+2}\) divergence-free (and tangential) vector fields, then the preceding Euler model has a local solution which is unique. For the proofs, the authors first consider stochastic differential equations on Hilbert manifolds: \(dX_{t}=b(X_{t})dt+\sigma (X_{t})\bullet dW_{t}\), \( X_{0}=\zeta \), where the drift \(b:M\rightarrow TM\) and the diffusion coefficient \(\sigma :M\rightarrow L(E;TM)\) are given sections which are supposed to be continuous and in \(C^{1}\) respectively, and the initial datum \(\zeta :\Omega \rightarrow M\) is a \(\mathcal{F}_{0}\)-measurable random variable. The authors prove properties of such stochastic differential equations, among which Itô's formula for manifold-valued processes, using the definition of Stratonovich differential. They prove an existence result assuming regularity properties of the data. They then consider the geometry of the Hilbert manifold of Sobolev diffeomorphisms preserving a volume form. For the proof of existence and uniqueness results for the above stochastic version of the Euler equation for an incompressible fluid on a manifold, they use the corresponding Ebin-Marsden theory. They apply the preceding local existence result first building an appropriate chart. In the last part of their paper, the authors propose natural extensions of the preceding existence and uniqueness results. They consider the case of fractional order Sobolev spaces for a manifold \(K\) without boundary. They establish the existence of maximal (in time) solutions to the stochastic differential equations and which preserve the regularity of the initial data. They finally consider the case of a multiplicative noise and possible extensions to other Euler-Arnold equations.
Reviewer: Alain Brillard (Riedisheim)Decay rate to contact discontinuities for the one-dimensional compressible Euler-Fourier system with a reacting mixturehttps://zbmath.org/1517.351692023-09-22T14:21:46.120933Z"Peng, Lishuang"https://zbmath.org/authors/?q=ai:peng.lishuang"Li, Yong"https://zbmath.org/authors/?q=ai:li.yong.22|li.yong.13|li.yong.1|li.yong.5|li.yong.17|li.yong.11|li.yong.7|li.yong.10|li.yong.24|li.yong|li.yong.3|li.yong.15|li.yong.12Summary: In this paper, we investigate the nonlinear stability of contact waves to the Cauchy problem of the compressible Euler-Fourier system with a reacting mixture in one dimension under the non-zero mass condition. If the corresponding Riemann problem for the compressible Euler system admits a contact discontinuity solution, it is shown that the contact wave is nonlinearly stable, while the strength of the contact discontinuity and the initial perturbation are suitably small. Especially, we obtain the decay rate of contact waves by using anti-derivative methods and elaborated energy estimates.Onsager's energy conservation of solutions for density-dependent Euler equations in \(\mathbb{T}^d\)https://zbmath.org/1517.351702023-09-22T14:21:46.120933Z"Wu, Xinglong"https://zbmath.org/authors/?q=ai:wu.xinglongSummary: This article is devoted to the study of energy conservation of weak solutions for the incompressible inhomogeneous Euler equations in \(\mathbb{T}^d\), \(d > 1\). We give two sufficient conditions to prove the energy conservation, which only include a regularity of space (and time) on the velocity \(u\). This is different from these results in
[\textit{Q. Chen}, J. Math. Fluid Mech. 22, No. 1, Paper No. 6, 13 p. (2020; Zbl 1429.76036)],
[\textit{R. M. Chen} and \textit{C. Yu}, J. Math. Pures Appl. (9) 131, 1--16 (2019; Zbl 1444.76023)] and
[\textit{Q.-H. Nguyen} et al., J. Differ. Equations 269, No. 9, 7171--7210 (2020; Zbl 1440.35257)],
which contain the regularity of \(\partial_t \rho\) and the pressure \(p\). In particular, our assumption in theorems is weaker than the assumption of Theorem 1.2 in [Chen, loc. cit.] and the assumption of regularity of density of Theorem 1.1 in [Chen and Yu, loc. cit.].Absence of singularities in solutions for the compressible Euler equations with source terms in \(\mathbb{R}^d\)https://zbmath.org/1517.351712023-09-22T14:21:46.120933Z"Wu, Xinglong"https://zbmath.org/authors/?q=ai:wu.xinglongSummary: The present article is devoted to the study of global solution and large time behaviour of solution for the isentropic compressible Euler system with source terms in \(\mathbb{R}^d\), \(d\geqslant 1\), which extends and improves the results obtained by \textit{T. C. Sideris} et al. [Commun. Partial Differ. Equations 28, No. 3--4, 795--816 (2003; Zbl 1048.35051)]. We first establish the existence and uniqueness of global smooth solution provided the initial datum is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude. Next, under the additional smallness assumption, the large time behaviour of solution is investigated, we only obtain the algebra decay of solution besides the \(L^2\)-norm of \(\nabla u\) is exponential decay.Global existence of weak solutions to the Navier-Stokes-Korteweg equationshttps://zbmath.org/1517.351722023-09-22T14:21:46.120933Z"Antonelli, Paolo"https://zbmath.org/authors/?q=ai:antonelli.paolo"Spirito, Stefano"https://zbmath.org/authors/?q=ai:spirito.stefanoIn this article, the authors consider the global existence problem for the Navier-Stokes-Korteweg system on the three-dimensional torus:
\begin{align*}
\partial_t \rho +\operatorname{div}(\rho u)= 0, \ \rho &\geq 0, \\
\partial_t (\rho u) +\operatorname{div}(\rho u \otimes u)+ \nabla \rho^\gamma - \operatorname{div}(\rho D u) - \rho \nabla \Delta \rho &=0, \\
(t,x) \in (0,T) &\times \mathbb{T}^3.
\end{align*}
In particular, the authors allow vacuum and consider possibly large initial data. As their main result they establish the existence of weak solutions under mild integrability and regularity assumptions on the initial density and momentum.
In order to construct these solutions, they introduce an approximate system with additional terms
\[
\epsilon \rho |u|^2 u + \epsilon u - \epsilon \rho \nabla(\frac{\Delta\sqrt{\rho}}{\rho}).
\]
Furthermore, truncations of the velocity field and density and regularizations are used to account for the low regularity of solutions. Following a comprehensive analysis of weak solution of the approximate system, estimates uniform in \(\epsilon>0\) and commutator estimates are established and exploited to pass to the limit \(\epsilon\downarrow 0\).
Reviewer: Christian Zillinger (Karlsruhe)Compactness lemma for sequences of divergence free Bochner measurable functionshttps://zbmath.org/1517.351732023-09-22T14:21:46.120933Z"Bae, Hyeong-Ohk"https://zbmath.org/authors/?q=ai:bae.hyeong-ohk"Wolf, Jörg"https://zbmath.org/authors/?q=ai:wolf.jorgSummary: We provide a local compactness lemma, that can be used for various models of incompressible fluids in a general domain, such as generalized Newtonian fluids. In particular, it helps for the proof of the existence of weak solution to those systems, when it comes to carrying out the passage to limit for sequences of weak solution to a corresponding approximate system. The key argument of our approach is the use of the local pressure decomposition in order to avoid the construction of a global pressure.A shallow water modeling with the Coriolis effect coupled with the surface tensionhttps://zbmath.org/1517.351742023-09-22T14:21:46.120933Z"Berjawi, Marwa"https://zbmath.org/authors/?q=ai:berjawi.marwa"ElArwadi, Toufic"https://zbmath.org/authors/?q=ai:el-arwadi.toufic"Israwi, Samer"https://zbmath.org/authors/?q=ai:israwi.samerSummary: In this paper, we study fluid dynamics taking into account two effects governing fluid's motion, an external one caused by Earth's rotation: Coriolis effect, besides an internal effect, caused by the interaction between fluid's molecules: the surface tension. So a new physical frame is considered here. At first, we will derive formally a new model of equations describing the motion in Camassa-Holm regime. Furthermore, large-time local well-posedness of the main model will be investigated in \({H^s}({\mathbb{R}})\) for \(s > {\frac{3}{2}} \).An exact solution representing equatorial wind-drift currents with depth-dependent continuous stratificationhttps://zbmath.org/1517.351752023-09-22T14:21:46.120933Z"Fan, Lili"https://zbmath.org/authors/?q=ai:fan.lili"Liu, Ruonan"https://zbmath.org/authors/?q=ai:liu.ruonanSummary: In this paper, we aim to derive an exact solution to a linearised version of the geophysical equations in the \(\beta \)-plane setting. The obtained explicit solution represents a steady purely azimuthal stratified flow with a flat surface and an impermeable flat bed that is suitable for describing the Equatorial Current. Moreover, we show that the thermocline exhibits some monotonicity properties.Global well-posedness and asymptotic behavior of the 3D MHD-Boussinesq equationshttps://zbmath.org/1517.351762023-09-22T14:21:46.120933Z"Guo, Zhengguang"https://zbmath.org/authors/?q=ai:guo.zhengguang"Zhang, Zunzun"https://zbmath.org/authors/?q=ai:zhang.zunzun"Skalák, Zdenĕk"https://zbmath.org/authors/?q=ai:skalak.zdenekSummary: In this paper, we study global well-posedness of the three-dimensional MHD-Boussinesq equations. The global existence of axisymmetric strong solutions to the MHD-Boussinesq equations in the presence of magnetic diffusion is shown by providing some smallness conditions only on the swirl component of velocity. As a by-product, long-time asymptotic behaviors are also presented.Wave breaking phenomenon to a nonlinear equation including the Fornberg-Whitham modelhttps://zbmath.org/1517.351772023-09-22T14:21:46.120933Z"Hong, Jin"https://zbmath.org/authors/?q=ai:hong.jin|hong.jin.1"Lai, Shaoyong"https://zbmath.org/authors/?q=ai:lai.shaoyongSummary: A nonlinear equation, which contains nonlocal dispersion term and high order advection term, is investigated. The \(L^2 ( \mathbb{R} )\) uniform bound of its solution is deduced when its initial value lies in \(L^2 ( \mathbb{R} )\). Utilizing this uniform bound leads to sufficient and necessary conditions to ensure the occurrence of wave breaking for the solutions.Existence of steady solutions for a general model for micropolar electrorheological fluid flowshttps://zbmath.org/1517.351782023-09-22T14:21:46.120933Z"Kaltenbach, Alex"https://zbmath.org/authors/?q=ai:kaltenbach.alex"Růžička, Michael"https://zbmath.org/authors/?q=ai:ruzicka.michaelSummary: In this paper, we study the existence of solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted Sobolev spaces. By deploying the Lipschitz truncation technique, we establish the existence of solutions without additional assumptions on the electric field.Regularity criteria and Liouville theorem for 3D inhomogeneous Navier-Stokes flows with vacuumhttps://zbmath.org/1517.351792023-09-22T14:21:46.120933Z"Kim, Jae-Myoung"https://zbmath.org/authors/?q=ai:kim.jaemyoungSummary: In this paper, we investigate the 3D inhomogeneous Navier-Stokes flows with vacuum, and obtain regularity criteria and Liouville type theorems in the Lorentz space if a smooth solution \((\rho, u)\) satisfies suitable conditions.A semi-discrete first-order low regularity exponential integrator for the ``good'' Boussinesq equation without loss of regularityhttps://zbmath.org/1517.351802023-09-22T14:21:46.120933Z"Li, Hang"https://zbmath.org/authors/?q=ai:li.hang"Su, Chunmei"https://zbmath.org/authors/?q=ai:su.chunmeiSummary: In this paper, we propose a semi-discrete first-order low regularity exponential-type integrator (LREI) for the ``good'' Boussinesq equation. It is shown that the method is convergent linearly in the space \(H^r\) for solutions belonging to \(H^{r+p(r)}\) where \(0\le p(r)\le 1\) is non-increasing with respect to \(r\), which means less additional derivatives might be needed when the numerical solution is measured in a more regular space. Particularly, the LREI presents the first-order accuracy in \(H^r\) with no assumptions of additional derivatives when \(r>5/2\). This is the first time to propose a low regularity method which achieves the optimal first-order accuracy without loss of regularity for the GB equation. The convergence is confirmed by extensive numerical experiments.Remarks on regularity criteria for the 3d Navier-Stokes equationshttps://zbmath.org/1517.351812023-09-22T14:21:46.120933Z"Liu, Qiao"https://zbmath.org/authors/?q=ai:liu.qiao"Pan, Meiling"https://zbmath.org/authors/?q=ai:pan.meilingSummary: We study regularity criteria of weak solutions to the three dimensional (3d) incompressible Navier-Stokes equations, and provide several Prodi-Serrin type regularity criteria involving only partial components of the velocity or its gradient on framework of the anisotropic Lebesgue spaces.Global smooth solutions to the 3D compressible viscous non-isentropic magnetohydrodynamic flows without magnetic diffusionhttps://zbmath.org/1517.351822023-09-22T14:21:46.120933Z"Li, Yongsheng"https://zbmath.org/authors/?q=ai:li.yongsheng"Xu, Huan"https://zbmath.org/authors/?q=ai:xu.huan"Zhai, Xiaoping"https://zbmath.org/authors/?q=ai:zhai.xiaopingSummary: How to construct the global smooth solutions to the compressible viscous, non-isentropic, non-resistive magnetohydrodynamic equations in \({\mathbb{T}}^3\) appears to be unknown. In this paper, we give a positive answer to this problem. More precisely, we prove a global stability result on perturbations near a strong background magnetic field to the 3D compressible viscous, non-isentropic, non-resistive magnetohydrodynamic equations. This stability result provides a significant example of the stabilizing effect of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit decay rate for the solutions to this nonlinear system.Stochastic primitive equations with horizontal viscosity and diffusivityhttps://zbmath.org/1517.351832023-09-22T14:21:46.120933Z"Saal, Martin"https://zbmath.org/authors/?q=ai:saal.martin"Slavík, Jakub"https://zbmath.org/authors/?q=ai:slavik.jakubSummary: We establish the existence and uniqueness of pathwise strong solutions to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise on a cylindrical domain \(M=(-h,0)\times G\), \(G\subset{\mathbb{R}^2}\) bounded and smooth, with the physical Dirichlet boundary conditions on the lateral part of the boundary. Compared to the deterministic case where the uniqueness of \(z\)-weak solutions holds in \({L^2} \), more regular initial data are necessary to establish uniqueness in the anisotropic space \(H_z^1 L_{xy}^2\) so that the existence of local pathwise solutions can be deduced from the Gyöngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.Darcy's law for porous media with multiple microstructureshttps://zbmath.org/1517.351842023-09-22T14:21:46.120933Z"Shen, Zhongwei"https://zbmath.org/authors/?q=ai:shen.zhongwei.1|shen.zhongweiSummary: In this paper we study the homogenization of the Dirichlet problem for the Stokes equations in a perforated domain with multiple microstructures. First, under the assumption that the interface between subdomains is a union of Lipschitz surfaces, we show that the effective velocity and pressure are governed by a Darcy law, where the permeability matrix is piecewise constant. The key step is to prove that the effective pressure is continuous across the interface, using Tartar's method of test functions. Secondly, we establish the sharp error estimates for the convergence of the velocity and pressure, assuming the interface satisfies certain smoothness and geometric conditions. This is achieved by constructing two correctors. One of them is used to correct the discontinuity of the two-scale approximation on the interface, while the other is used to correct the discrepancy between boundary values of the solution and its approximation.Unconditionally energy-stable finite element scheme for the chemotaxis-fluid systemhttps://zbmath.org/1517.351852023-09-22T14:21:46.120933Z"Tang, Yangyang"https://zbmath.org/authors/?q=ai:tang.yangyang"Zou, Guang-an"https://zbmath.org/authors/?q=ai:zou.guangan"Li, Jian"https://zbmath.org/authors/?q=ai:li.jian.11|li.jian.6|li.jian.8|li.jian.21|li.jian.3|li.jian.4|li.jian.7|li.jian.1|li.jian.5Summary: In this paper, we first deduce an improved chemotaxis-fluid system by introducing a chemotactic stress force, which can be used to describe the chemotactic movement of bacteria in a viscous fluid. Compared with the classical chemotaxis-Navier-Stokes system, the newly modified system obeys the law of energy dissipation. To solve such a chemotaxis-fluid system, we develop a linear, decoupled fully-discrete finite element scheme by combining the scalar auxiliary variable (SAV) approach, implicit-explicit (IMEX) scheme and pressure-projection method. The unconditional energy stability of the developed scheme is proved rigorously, and we further prove the optimal error estimates for the fully discrete scheme, especially for the pressure bound. Finally, some numerical examples are presented to verify the accuracy, energy stability and performance of the proposed numerical scheme.Local well-posedness and decay for some generalized shallow water equationshttps://zbmath.org/1517.351862023-09-22T14:21:46.120933Z"Xu, Runzhang"https://zbmath.org/authors/?q=ai:xu.runzhang"Yang, Yanbing"https://zbmath.org/authors/?q=ai:yang.yanbingSummary: Consideration herein is a big class of nonlocal partial differential equations including some classical shallow water models such as b-family equation, Novikov equation, rotation-Camassa-Holm equation, generalized Camassa-Holm equation and Camassa-Holm-Novikov equation over a whole line. Based on the local structure of the dynamics along the characteristics, a local well-posedness result is established for some initial data class containing certain non-smooth solitary wave solutions. Subsequently in this set certain decay persistence properties are also constructed. These partially extend some obtained results and bring the effect of nonlinearities to more extent.A novel dynamic model and the oblique interaction for ocean internal solitary waveshttps://zbmath.org/1517.351872023-09-22T14:21:46.120933Z"Yu, Di"https://zbmath.org/authors/?q=ai:yu.di"Zhang, Zongguo"https://zbmath.org/authors/?q=ai:zhang.zongguo"Dong, Huanhe"https://zbmath.org/authors/?q=ai:dong.huanhe"Yang, Hongwei"https://zbmath.org/authors/?q=ai:yang.hongwei(no abstract)Fully discrete discontinuous Galerkin numerical scheme with second-order temporal accuracy for the hydrodynamically coupled lipid vesicle modelhttps://zbmath.org/1517.351882023-09-22T14:21:46.120933Z"Zou, Guang-an"https://zbmath.org/authors/?q=ai:zou.guangan"Li, Zhaohua"https://zbmath.org/authors/?q=ai:li.zhaohua"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: In this paper, for the highly nonlinear hydrodynamically coupled elastic bending energy model of vesicle membranes, based on the discontinuous Galerkin (DG) method for spatial discretization, a linear, decoupled, and second-order time-accurate numerical scheme is constructed. The scheme combines several efficient approaches, including the scalar auxiliary variable (SAV) method for the linearization of the nonlinear energy potential, the implicit-explicit (IMEX) discretization method for dealing with the nonlinear coupling terms, and the projection method for the Navier-Stokes equations. We also rigorously establish the energy stability and optimal error estimates, and also carry out several numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed fully discrete DG scheme, numerically.Van der Waals interactions between two hydrogen atoms: the next ordershttps://zbmath.org/1517.351892023-09-22T14:21:46.120933Z"Cancès, Eric"https://zbmath.org/authors/?q=ai:cances.eric"Coyaud, Rafaël"https://zbmath.org/authors/?q=ai:coyaud.rafael"Scott, L. Ridgway"https://zbmath.org/authors/?q=ai:scott.larkin-ridgwaySummary: We extend a method [\textit{E. Cancès} and \textit{L. R. Scott}, SIAM J. Math. Anal. 50, No. 1, 381--410 (2018; Zbl 1386.35363)] to compute more terms in the asymptotic expansion of the van der Waals attraction between two hydrogen atoms. These terms are obtained by solving a set of modified Slater-Kirkwood partial differential equations. The accuracy of the method is demonstrated by numerical simulations and comparison with other methods from the literature. It is also shown that the scattering states of the hydrogen atom, that are the states associated with the continuous spectrum of the Hamiltonian, have a major contribution to the C\(_6\) coefficient of the van der Waals expansion.Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equationshttps://zbmath.org/1517.351902023-09-22T14:21:46.120933Z"Mohammed, Wael W."https://zbmath.org/authors/?q=ai:mohammed.wael-w"Al-Askar, Farah M."https://zbmath.org/authors/?q=ai:al-askar.farah-m"El-Morshedy, Mahmoud"https://zbmath.org/authors/?q=ai:el-morshedy.mahmoudSummary: In this article, the stochastic fractional Davey-Stewartson equations (SFDSEs) that result from multiplicative Brownian motion in the Stratonovich sense are discussed. We use two different approaches, namely the Riccati-Bernoulli sub-ordinary differential equations and sine-cosine methods, to obtain novel elliptic, hyperbolic, trigonometric, and rational stochastic solutions. Due to the significance of the Davey-Stewartson equations in the theory of turbulence for plasma waves, the discovered solutions are useful in explaining a number of fascinating physical phenomena. Moreover, we illustrate how the fractional derivative and Brownian motion affect the exact solutions of the SFDSEs using MATLAB tools to plot our solutions and display a number of three-dimensional graphs. We demonstrate how the multiplicative Brownian motion stabilizes the SFDSE solutions at around zero.Low regularity local well-posedness for the zero energy Novikov-Veselov equationhttps://zbmath.org/1517.351912023-09-22T14:21:46.120933Z"Adams, Joseph"https://zbmath.org/authors/?q=ai:adams.joseph-brian"Grünrock, Axel"https://zbmath.org/authors/?q=ai:grunrock.axelThe authors study the initial value problem \(u(x,y,0)=u_0(x,y)\) for the zero energy Novikov-Veselov equation
\[
\partial_{t}u+(\partial^{3}+\overline{{\partial}}^{3})u+3(\partial(u\overline{{\partial}}^{-1}\partial u)+\overline{{\partial}}(u\partial^{-1}\overline{{\partial}}u))=0,
\]
where either
\begin{itemize}
\item the data \(u_0\) and the solution \(u(t)\) at time \(t\) belong to some classical Sobolev space \(H^{s}(\mathbb{R}^{2})\) of functions defined on the whole plane (Cauchy problem, non-periodic case), or
\item \(u_0\) and \(u(t)\) are elements of \(H_0^s (\mathbb{T}^2)\), the Sobolev space of (in both directions) periodic functions on \(\mathbb{R}^{2}\) of mean zero, i.e., assume
\[
\int_{\mathbb{T}^2}u_0(x,y)dxdy=0,
\]
which is preserved under the evolution of (NV). Local well-posedness is shown in the non-periodic case with \(s>-\frac{3}{4}\) and in the periodic case with \(s>-\frac{1}{5}\) by the Fourier restriction norm method.
\end{itemize}
Reviewer: Ti-Jun Xiao (Fudan)Blowup and ill-posedness for the complex, periodic KdV equationhttps://zbmath.org/1517.351922023-09-22T14:21:46.120933Z"Bona, J. L."https://zbmath.org/authors/?q=ai:bona.jerry-l"Weissler, F. B."https://zbmath.org/authors/?q=ai:weissler.fred-bSummary: This paper is concerned with complex-valued solutions of the Korteweg-de Vries equation. Interest will be focused upon the initial-value problem with initial data that is periodic in space. Derived here are results of local and global well-posedness, singularity formation in finite time and, perhaps surprisingly, results of non-existence. The overall picture is notably different from the situation that obtains for real-valued solutions.Parametric analysis of dust ion acoustic waves in superthermal plasmas through non-autonomous KdV frameworkhttps://zbmath.org/1517.351932023-09-22T14:21:46.120933Z"Chadha, Naresh M."https://zbmath.org/authors/?q=ai:chadha.naresh-m"Tomar, Shruti"https://zbmath.org/authors/?q=ai:tomar.shruti"Raut, Santanu"https://zbmath.org/authors/?q=ai:raut.santanuSummary: In the presence of superthermal plasma, propagation of non-linear dust ion acoustic waves has been studied in the framework of the Damped Forced Korteweg-de Vries (DFKdV) Equation. A feasible range is obtained for the existence of the solitary wave solutions in terms of the spectral index, and the unaffected dust-to-ion density ratio (denoted by \(\kappa\) and \(\mu\), respectively). It is observed that the transition of solitary wave structures from compressive to rarefactive can be completely determined by mainly these two parameters. The effects of all other parameters involved in the model, namely, the strength \((f_0)\) and frequency \((\omega)\) of the external periodic force, and damping coefficient \((\nu_{\mathrm{id}0})\) are studied using the bifurcation analysis; \(f_0\) and \(\omega\) play a crucial role in the periodic and chaotic motions in the system. We also obtained certain critical values of the key parameters involved in the model for which the model exhibits periodic, quasi-periodic and chaotic behaviour.Global conservative solution for a dissipative Camassa-Holm type equation with cubic and quartic nonlinearitieshttps://zbmath.org/1517.351942023-09-22T14:21:46.120933Z"Deng, Wenjie"https://zbmath.org/authors/?q=ai:deng.wenjie"Yin, Zhaoyang"https://zbmath.org/authors/?q=ai:yin.zhaoyangSummary: This paper is devoted to the global conservative solutions of a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solutions of the semilinear system. Returning to the original variables, we get the global conservative solution of the equation. Finally, we show that the peakon solutions of the equation still conserve in \(H^1\).Norm inflation and ill-posedness for the Fornberg-Whitham equationhttps://zbmath.org/1517.351952023-09-22T14:21:46.120933Z"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu.1"Wu, Xing"https://zbmath.org/authors/?q=ai:wu.xing"Yu, Yanghai"https://zbmath.org/authors/?q=ai:yu.yanghai"Zhu, Weipeng"https://zbmath.org/authors/?q=ai:zhu.weipengSummary: In this paper, we prove that the Cauchy problem for the Fornberg-Whitham equation is not locally well-posed in \(B_{p , r}^s(\mathbb{R})\) with \((s, p, r) \in(1, 1 + \frac{ 1}{ p}) \times [2, \infty) \times [1, \infty]\) or \((s, p, r) \in \{1 \} \times [2, \infty) \times [1, 2]\) by showing norm inflation phenomena of the solution for some special initial data.On modified scattering for 1D quadratic Klein-Gordon equations with non-generic potentialshttps://zbmath.org/1517.351962023-09-22T14:21:46.120933Z"Lindblad, Hans"https://zbmath.org/authors/?q=ai:lindblad.hans"Lührmann, Jonas"https://zbmath.org/authors/?q=ai:luhrmann.jonas"Schlag, Wilhelm"https://zbmath.org/authors/?q=ai:schlag.wilhelm"Soffer, Avy"https://zbmath.org/authors/?q=ai:soffer.avrahamAuthors' abstract: We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein- Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to continue the investigation of the occurrence of a novel modified scattering behavior of the solutions that involves a logarithmic slow-down of the decay rate along certain rays. This phenomenon is ultimately caused by the threshold resonance of the linear Klein-Gordon operator. It was previously uncovered for the special case of the zero potential in [\textit{H. Lindblad} et al., Arch. Ration. Mech. Anal. 241, No. 3, 1459--1527 (2021; Zbl 1475.35305)]. The Klein-Gordon model considered in this paper is motivated by the asymptotic stability problem for kink solutions arising in classical scalar field theories on the real line.
Reviewer: Ti-Jun Xiao (Fudan)Local well-posedness for the gKdV equation on the background of a bounded functionhttps://zbmath.org/1517.351972023-09-22T14:21:46.120933Z"Palacios, José Manuel"https://zbmath.org/authors/?q=ai:palacios.jose-manuelThe author thoroughly discusses well-posedness, unconditional uniqueness and global-in-time existence of solutions for the generalized Korteweg-de Vries equation
\[
v_t+v_{xxx}+f(v)_x=0
\]
on the real line, for general nonlinearities \(f\) with suitable supplementary assumptions for global existence. This approach permits him to study perturbations of kink and periodic traveling wave solutions of various modifications of the classical Korteweg-de Vries equation. The results are generally obtained for the Sobolev spaces \(H^s(\mathbb R)\) with \(s>1/2\).
Reviewer: Piotr Biler (Wrocław)Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensionshttps://zbmath.org/1517.351982023-09-22T14:21:46.120933Z"Abeya, Asela"https://zbmath.org/authors/?q=ai:abeya.asela"Biondini, Gino"https://zbmath.org/authors/?q=ai:biondini.gino"Hoefer, Mark A."https://zbmath.org/authors/?q=ai:hoefer.mark-aSummary: The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation in two, three and higher spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is briefly outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonics and matter waves.Efficient manipulation of Bose-Einstein condensates in a double-well potentialhttps://zbmath.org/1517.351992023-09-22T14:21:46.120933Z"Adriazola, Jimmie"https://zbmath.org/authors/?q=ai:adriazola.jimmie"Goodman, Roy"https://zbmath.org/authors/?q=ai:goodman.roy-h"Kevrekidis, Panayotis"https://zbmath.org/authors/?q=ai:kevrekidis.panayotis-gSummary: We pose the problem of transferring a Bose-Einstein Condensate (BEC) from one side of a double-well potential to the other as an optimal control problem for determining the time-dependent form of the potential. We derive a reduced dynamical system using a Galerkin truncation onto a finite set of eigenfunctions and find that including \textit{three} modes suffices to effectively control the full dynamics, described by the Gross-Pitaevskii model of BEC. The functional form of the control is reduced to finite dimensions by using another Galerkin-type method called the chopped random basis (CRAB) method, which is then optimized by a genetic algorithm called differential evolution (DE). Finally, we discuss the extent to which the reduction-based optimal control strategy can be refined by means of including more modes in the Galerkin reduction.A focusing-defocusing intermediate nonlinear Schrödinger systemhttps://zbmath.org/1517.352002023-09-22T14:21:46.120933Z"Berntson, Bjorn K."https://zbmath.org/authors/?q=ai:berntson.bjorn-k"Fagerlund, Alexander"https://zbmath.org/authors/?q=ai:fagerlund.alexanderSummary: We introduce and study a system of coupled nonlocal nonlinear Schrödinger equations that interpolates between the mixed, focusing-defocusing Manakov system on one hand and a limiting case of the intermediate nonlinear Schrödinger equation on the other. We show that this new system, which we call the intermediate mixed Manakov (IMM) system, admits multi-soliton solutions governed by a complexification of the hyperbolic Calogero-Moser (CM) system. Furthermore, we introduce a spatially periodic version of the IMM system, for which our main result is a class of exact solutions governed by a complexified elliptic CM system.On well-posedness and concentration of blow-up solutions for the intercritical inhomogeneous NLS equationhttps://zbmath.org/1517.352012023-09-22T14:21:46.120933Z"Cardoso, Mykael"https://zbmath.org/authors/?q=ai:cardoso.mykael"Farah, Luiz Gustavo"https://zbmath.org/authors/?q=ai:farah.luiz-gustavo"Guzmán, Carlos M."https://zbmath.org/authors/?q=ai:guzman.carlos-mSummary: We consider the focusing inhomogeneous nonlinear Schrödinger (INLS) equation in
\({\mathbb{R}}^N\)
\[
i \partial_t u +\Delta u + |x|^{-b} |u|^{2\sigma }u = 0,
\]
where \(N\ge 2\) and \(\sigma , b>0\). We first obtain a small data global result in \(H^1\), which, in the two spatial dimensional case, improves the third author result [Nonlinear Anal., Real World Appl. 37, 249--286 (2017; Zbl 1375.35486)] on the range of \(b\). For \(N\ge 3\) and \(\frac{2-b}{N}<\sigma <\frac{2-b}{N-2} \), we also study the local well posedness in \(\dot{H}^{s_c}\cap \dot{H}^1 \), where \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma } \). Sufficient conditions for global existence of solutions in \(\dot{H}^{s_c}\cap \dot{H}^1\) are also established, using a Gagliardo-Nirenberg type estimate. Finally, we study the \(L^{\sigma_c} \)-norm concentration phenomenon, where \(\sigma_c=\frac{2N\sigma }{2-b} \), for finite time blow-up solutions in \(\dot{H}^{s_c}\cap \dot{H}^1\) with bounded \(\dot{H}^{s_c} \)-norm. Our approach is based on the compact embedding of \(\dot{H}^{s_c}\cap \dot{H}^1\) into a weighted \(L^{2\sigma +2}\) space.Wellposed spaces for hyperbolic equationshttps://zbmath.org/1517.352022023-09-22T14:21:46.120933Z"Furuya, Kiyoko"https://zbmath.org/authors/?q=ai:furuya.kiyokoSummary: In the previous work we posed the problem of wellposed function spaces for nonparabolic equations and discussed the problem for Schrödinger equations. In this paper we intend to discuss ``wellposed function spaces'' for hyperbolic equations of the simplest type. We get minimum (or maximum) wellposed space containing (or contained in) any given function space. Some applications to nonlinear problems will be found to oblique boundary conditions.Nondispersive solutions to the mass critical half-wave equation in two dimensionshttps://zbmath.org/1517.352032023-09-22T14:21:46.120933Z"Georgiev, Vladimir"https://zbmath.org/authors/?q=ai:georgiev.vladimir-s"Li, Yuan"https://zbmath.org/authors/?q=ai:li.yuan.8The purpose of this paper is to study two nondispersive phenomena connected with the focusing half-wave equation in two dimensions
\[
\left\{
\begin{array}
[c]{c}
i\partial_{t}u =Du-\left\vert u\right\vert u,\\
u\left( t_{0},x\right) =u_{0}\left( x\right) ,\text{ }u:I\times
\mathbb{R}^{2}\rightarrow\mathbb{C}\text{,}
\end{array}
\right.
\]
where \(t_{0}\in I\subset\mathbb{R}\) is an interval and \(D\) denotes the first-order nonlocal fractional derivative. First, in Theorem 1.1 the authors show the existence of traveling solitary waves of the form
\[
u\left( t,x\right) =e^{it\mu}Q_{\nu}\left( x-\nu t\right)
\]
with a profile \(Q_{\nu}\in H^{\frac{1}{2}}\left(\mathbb{R}\right),\) some \(\mu\in\mathbb{R}\) and traveling velocity \(0<\nu<1.\) Moreover, the authors prove that the mass \(\left\Vert Q_{\nu}\right\Vert _{L^{2}}\ \)tends to the mass of the ground state solution \(Q\) when \(\left\vert \nu\right\vert \rightarrow0\) and it vanishes when \(\left\vert \nu\right\vert \rightarrow1.\) The second result of the paper, given in Theorem 1.2, shows that there is a solution with ground state mass that blows up in finite time. The detailed asymptotics of this solution near the blow-up time is also presented. The delicate proof is based on complicated modulation and energy/virial estimates,
and a bootstrap argument.
Reviewer: Ivan Naumkin (Nice)Multi-soliton solutions for a higher-order coupled nonlinear Schrödinger system in an optical fiber via Riemann-Hilbert approachhttps://zbmath.org/1517.352042023-09-22T14:21:46.120933Z"Guo, Han-Dong"https://zbmath.org/authors/?q=ai:guo.handong"Xia, Tie-Cheng"https://zbmath.org/authors/?q=ai:xia.tie-cheng(no abstract)Rogue wave and multi-pole solutions for the focusing Kundu-Eckhaus equation with nonzero background via Riemann-Hilbert problem methodhttps://zbmath.org/1517.352052023-09-22T14:21:46.120933Z"Guo, Ning"https://zbmath.org/authors/?q=ai:guo.ning|guo.ning.1"Xu, Jian"https://zbmath.org/authors/?q=ai:xu.jian.1"Wen, Lili"https://zbmath.org/authors/?q=ai:wen.lili"Fan, Engui"https://zbmath.org/authors/?q=ai:fan.engui(no abstract)Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equationhttps://zbmath.org/1517.352062023-09-22T14:21:46.120933Z"Hendy, Ahmed S."https://zbmath.org/authors/?q=ai:hendy.ahmed-s"Zaky, Mahmoud A."https://zbmath.org/authors/?q=ai:zaky.mahmoud-a(no abstract)Continuum limit for the Ablowitz-Ladik systemhttps://zbmath.org/1517.352072023-09-22T14:21:46.120933Z"Killip, Rowan"https://zbmath.org/authors/?q=ai:killip.rowan"Ouyang, Zhimeng"https://zbmath.org/authors/?q=ai:ouyang.zhimeng"Visan, Monica"https://zbmath.org/authors/?q=ai:visan.monica"Wu, Lei"https://zbmath.org/authors/?q=ai:wu.lei.1|wu.lei.4|wu.lei.2|wu.lei.3|wu.leiSummary: We show that solutions to the Ablowitz-Ladik system converge to solutions of the cubic nonlinear Schrödinger equation for merely \(L^2\) initial data. Furthermore, we consider initial data for this lattice model that excites Fourier modes near both critical points of the discrete dispersion relation and demonstrate convergence to a \textit{decoupled} system of nonlinear Schrödinger equations.Hydrodynamic limits of Manton's Schrödinger systemhttps://zbmath.org/1517.352082023-09-22T14:21:46.120933Z"Kim, Jeongho"https://zbmath.org/authors/?q=ai:kim.jeongho"Moon, Bora"https://zbmath.org/authors/?q=ai:moon.boraSummary: We present the hydrodynamic limit for the Schrödinger equation coupled with the gauge fields introduced by \textit{N. S. Manton} [Ann. Phys. 256, No. 1, 114--131 (1997; Zbl 0932.58014)]. We consider two types of self-interacting potential functions inspired by the Gross-Pitaevskii and Ginzburg-Landau potentials. Each of these potentials is used to derive the hydrodynamic limit when the Planck constant converges to zero. The limiting hydrodynamic system become a compressible/incompressible Euler-type equation with rotating force depending on the choice of potential. Our analyses are based on the estimate of the modulated energy, which measures a deviation of macroscopic quantities of the Schrödinger-type system from the Euler-type equations.Well-posedness of the Cauchy problem for the kinetic DNLS on \(\mathbb{T}\)https://zbmath.org/1517.352092023-09-22T14:21:46.120933Z"Kishimoto, Nobu"https://zbmath.org/authors/?q=ai:kishimoto.nobu"Tsutsumi, Yoshio"https://zbmath.org/authors/?q=ai:tsutsumi.yoshioSummary: We consider the Cauchy problem for the \textit{kinetic derivative nonlinear Schrödinger equation} on the torus \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}: \partial_tu-i \partial_x^2u=\alpha \partial_x(|u |^2u)+\beta \partial_x[H(|u |^2)u]\) for \((t,x)\in[0,T]\times\mathbb{T}\), where the constants \(\alpha\), \(beta\) are such that \(\alpha\in \mathbb{R}\) and \(\beta<0\), and \(H\) denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces \(H^s\) for \(s>3/2\). However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when \(\beta=0\), cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem.
In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in \(H^s(\mathbb{T})\), \(s>1/2\). To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term \(\beta \partial_x[H(|u |^2)u]\), in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.The radial bi-harmonic generalized Hartree equation revisitedhttps://zbmath.org/1517.352102023-09-22T14:21:46.120933Z"Saanouni, Tarek"https://zbmath.org/authors/?q=ai:saanouni.tarek"Feng, Binhua"https://zbmath.org/authors/?q=ai:feng.binhuaSummary: This note studies the fourth-order generalized Hartree equation
\[
i\dot u+\Delta^2u\pm|u|^{p-2}(J_\gamma*|u|^p)u=0,\quad p\ge 2.
\]
Indeed, for both attractive and repulsive sign, the scattering is obtained in the inter-critical regime, which is given by \(0<s_c<2\), where the critical Sobolev exponent is given by the equality
\[
\kappa^\frac{4+\gamma}{2(p-1)}\|u(\kappa^4\cdot,\kappa t)\|_{\dot H^{s_c}}=\|u(\kappa t)\|_{\dot H^{s_c}}.
\]
In the focusing sign, the scattering follows the method due to \textit{B. Dodson} and \textit{J. Murphy} [Proc. Am. Math. Soc. 145, No. 11, 4859--4867 (2017; Zbl 1373.35287)]. This approach is based on a scattering criteria and a Morawetz estimate. This avoids the concentration-compactness method which requires a heavy machinery in order to obtain the desired space-time bounds. The Kenig-Merle road-map was used by the first author, in a previous paper, in order to obtain the scattering. One assumes here that the data is spherically symmetric. This condition will be removed in a paper in progress. In the defocusing regime, the scattering is based on the decay of solutions in some Lebesgue norms coupled with a Morawetz estimate. In order to prove the Morawetz estimates, one assume that the space dimension is \(N\ge 5\). Moreover, one assumes that \(p\ge 2\) in order to avoid a singularity of the source term. The energy scattering implies that the energy global solutions to the considered equation are asymptotic to \(e^{i\cdot\Delta^2}u_\pm\), when \(t\to\pm\infty\). This means that the source term has no effect for large time.Loss of physical reversibility in reversible systemshttps://zbmath.org/1517.352112023-09-22T14:21:46.120933Z"Sagiv, Amir"https://zbmath.org/authors/?q=ai:sagiv.amir"Ditkowski, Adi"https://zbmath.org/authors/?q=ai:ditkowski.adi"Goodman, Roy H."https://zbmath.org/authors/?q=ai:goodman.roy-h"Fibich, Gadi"https://zbmath.org/authors/?q=ai:fibich.gadiSummary: A dynamical system is said to be reversible if, given an output, the input can always be recovered in a well-posed manner. Nevertheless, we argue that reversible systems that have a time-reversal symmetry, such as the Nonlinear Schrödinger equation and the \(\phi^4\) equation can become ``physically irreversible''. By this, we mean that realistically-small experimental errors in measuring the output can lead to dramatic differences between the recovered input and the original one. The loss of reversibility reveals a natural ``arrow of time'', reminiscent of the thermodynamic one, which is the direction in which the radiation is emitted outward. Our results are relevant to imaging and reversal applications in nonlinear optics.Ground states in spatially discrete non-linear Schrödinger modelshttps://zbmath.org/1517.352122023-09-22T14:21:46.120933Z"Stefanov, Atanas G."https://zbmath.org/authors/?q=ai:stefanov.atanas-g"Ross, Ryan M."https://zbmath.org/authors/?q=ai:ross.ryan-m"Kevrekidis, Panayotis G."https://zbmath.org/authors/?q=ai:kevrekidis.panayotis-gSummary: In the seminal work [Nonlinearity 12, No. 3, 673--691 (1999; Zbl 0984.35147)], \textit{M. I. Weinstein} considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on \(\mathbb{Z}^d\). More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed power \(P\) (i.e. \(l^2\) mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein's work, as well as the innovative variational methods introduced for this problem in
[\textit{E. W. Laedke} et al., Phys. Rev. Lett. 73, No. 8, 1055--1059 (1994; \url{doi:10.1103/PhysRevLett.73.1055}); Phys. Rev. E (3) 54, No. 4, 4299--4311 (1996; \url{doi:10.1103/PhysRevE.54.4299})]
in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g. \(l^2\) supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis-Shatah-Strauss/Vakhitov-Kolokolov (GSS/VK) quantity \(\partial_\omega\Vert\varphi_\omega\Vert^2_{l^2}\). In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameter \(\omega\) varies, in contrast with the corresponding continuous NLS model.Random data final-state problem of fourth-order inhomogeneous NLShttps://zbmath.org/1517.352132023-09-22T14:21:46.120933Z"Tao, Liying"https://zbmath.org/authors/?q=ai:tao.liying"Zhao, Tengfei"https://zbmath.org/authors/?q=ai:zhao.tengfeiSummary: We consider the fourth-order inhomogeneous nonlinear Schrödinger equation with mass-subcritical nonlinearity \(i u_t + \Delta^2 u + \lambda | x |^{- \alpha} | u |^\beta u = 0\). For \(u_+ \in L^2( \mathbb{R}^d)\), we perform a physical-space randomization and get a random final state \(u_+^\omega \in L^2( \mathbb{R}^d)\). By establishing the smoothing estimate, time-space-weighted inhomogeneous Strichartz estimate and random Strichartz estimate, we prove that for almost every \(\omega \), there exists a unique, global solution to the fourth-order inhomogeneous nonlinear Schrödinger equation that scatters to \(u_+^\omega \in L^2\).Riemann-Hilbert method and multi-soliton solutions of the Kundu-nonlinear Schrödinger equationhttps://zbmath.org/1517.352142023-09-22T14:21:46.120933Z"Yan, Xue-Wei"https://zbmath.org/authors/?q=ai:yan.xuewei(no abstract)The dynamics of the smooth positon and b-positon solutions for the NLS-MB equationshttps://zbmath.org/1517.352152023-09-22T14:21:46.120933Z"Yuan, Feng"https://zbmath.org/authors/?q=ai:yuan.feng(no abstract)Stable and oscillating solitons of \(\mathcal{PT}\)-symmetric couplers with gain and loss in fractional dimensionhttps://zbmath.org/1517.352162023-09-22T14:21:46.120933Z"Zeng, Liangwei"https://zbmath.org/authors/?q=ai:zeng.liangwei"Shi, Jincheng"https://zbmath.org/authors/?q=ai:shi.jincheng"Lu, Xiaowei"https://zbmath.org/authors/?q=ai:lu.xiaowei"Cai, Yi"https://zbmath.org/authors/?q=ai:cai.yi"Zhu, Qifan"https://zbmath.org/authors/?q=ai:zhu.qifan"Chen, Hongyi"https://zbmath.org/authors/?q=ai:chen.hongyi"Long, Hu"https://zbmath.org/authors/?q=ai:long.hu"Li, Jingzhen"https://zbmath.org/authors/?q=ai:li.jingzhen(no abstract)Increasing stability in the inverse scattering problem for a nonlinear Schrödinger equation with multiple frequencieshttps://zbmath.org/1517.352172023-09-22T14:21:46.120933Z"Zhao, Yue"https://zbmath.org/authors/?q=ai:zhao.yueSummary: This paper is concerned with the inverse scattering problem of determining the unknown coefficients for a nonlinear two-dimensional Schrödinger equation. We establish for the first time the increasing stability of the inverse scattering problem from the multi-frequency far-field pattern for nonlinear equations. To achieve this goal, we prove the existence of a holomorphic region and an upper bound for the solution with respect to the complex wavenumber, which also leads to the well-posedness of the direct scattering problem. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the unknown coefficients, where the latter decreases as the upper bound of the frequency increases.Averaging principle for stochastic complex Ginzburg-Landau equationshttps://zbmath.org/1517.352182023-09-22T14:21:46.120933Z"Cheng, Mengyu"https://zbmath.org/authors/?q=ai:cheng.mengyu"Liu, Zhenxin"https://zbmath.org/authors/?q=ai:liu.zhenxin"Röckner, Michael"https://zbmath.org/authors/?q=ai:rockner.michaelSummary: Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. In this paper, we study three types of averaging principle for stochastic complex Ginzburg-Landau equations. Firstly, we prove that the solution of the original equation converges to that of the averaged equation on finite intervals as the time scale \(\varepsilon\) goes to zero when the initial data are the same. Secondly, we show that there exists a unique recurrent solution (in particular, periodic, almost periodic, almost automorphic, etc.) to the original equation in a neighborhood of the stationary solution of the averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as \(\varepsilon\) goes to zero.A Liouville type result and quantization effects on the system \(-\Delta u = u J'(1-|u|^2)\) for a potential convex near zerohttps://zbmath.org/1517.352192023-09-22T14:21:46.120933Z"De Maio, Umberto"https://zbmath.org/authors/?q=ai:de-maio.umberto"Hadiji, Rejeb"https://zbmath.org/authors/?q=ai:hadiji.rejeb"Lefter, Catalin"https://zbmath.org/authors/?q=ai:lefter.catalin-george"Perugia, Carmen"https://zbmath.org/authors/?q=ai:perugia.carmenSummary: We consider a Ginzburg-Landau type equation in \(\mathbb{R}^2\) of the form \(-\Delta u = u J'(1-|u|^2)\) with a potential function \(J\) satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context the results concerning quantization of finite potential solutions of \textit{H. Brézis} et al. [Arch. Ration. Mech. Anal. 126, No. 1, 35--58 (1994; Zbl 0809.35019)] who treat the case when \(J\) behaves polinomially near \(0,\) as well as a result of Th. Cazenave, found in the same reference, and concerning the form of finite energy solutions.Resonant double Hopf bifurcation in a diffusive Ginzburg-Landau model with delayed feedbackhttps://zbmath.org/1517.352202023-09-22T14:21:46.120933Z"Huang, Yuxuan"https://zbmath.org/authors/?q=ai:huang.yuxuan"Zhang, Hua"https://zbmath.org/authors/?q=ai:zhang.hua.5"Niu, Ben"https://zbmath.org/authors/?q=ai:niu.ben(no abstract)Dynamic behavior and stability analysis of nonlinear modes in the fourth-order generalized Ginzburg-Landau model with near \(\mathcal{PT}\)-symmetric potentialshttps://zbmath.org/1517.352212023-09-22T14:21:46.120933Z"Zhang, Jia-Rui"https://zbmath.org/authors/?q=ai:zhang.jiarui"Zhang, Jia-Qi"https://zbmath.org/authors/?q=ai:zhang.jiaqi"Zheng, Zhao-Lin"https://zbmath.org/authors/?q=ai:zheng.zhao-lin"Lin, Da"https://zbmath.org/authors/?q=ai:lin.da"Shen, Yu-Jia"https://zbmath.org/authors/?q=ai:shen.yujia(no abstract)Analysis of a quasilinear coupled magneto-quasistatic model: solvability and regularity of solutionshttps://zbmath.org/1517.352222023-09-22T14:21:46.120933Z"Chill, Ralph"https://zbmath.org/authors/?q=ai:chill.ralph"Reis, Timo"https://zbmath.org/authors/?q=ai:reis.timo"Stykel, Tatjana"https://zbmath.org/authors/?q=ai:stykel.tatjanaThis article studies a quasilinear MQS approximation of Maxwell's equations, which is coupled to an integral equation. By leveraging magnetic energy, the system can be rearticulated as an abstract differential-algebraic equation that encompasses subgradients. The authors of this study have obtained novel well-posedness and regularity results for such equations, which are applied to the coupled MQS system.
Reviewer: Hongyu Liu (Hong Kong)The heterogeneous Helmholtz problem with spherical symmetry: Green's operator and stability estimateshttps://zbmath.org/1517.352232023-09-22T14:21:46.120933Z"Sauter, Stefan"https://zbmath.org/authors/?q=ai:sauter.stefan-a"Torres, Céline"https://zbmath.org/authors/?q=ai:torres.celineThis article examines the modeling of wave propagation phenomena in the frequency domain using the Helmholtz equation in heterogeneous media, with a specific focus on media with discontinuous and highly oscillating wave speed. The authors specifically analyze problems with spherical symmetry and provide explicit representations of the Green's operator, as well as stability estimates in terms of both frequency and wave speed.
Reviewer: Hongyu Liu (Hong Kong)The homogenized quasi-static model of a thermoelastic composite stitched with reinforcing threadshttps://zbmath.org/1517.352242023-09-22T14:21:46.120933Z"Fankina, Irina V."https://zbmath.org/authors/?q=ai:fankina.irina-vladimirovna"Furtsev, Alexey I."https://zbmath.org/authors/?q=ai:furtsev.alexey-igorevich"Rudoy, Evgeny M."https://zbmath.org/authors/?q=ai:rudoy.evgeny-mikhailovich"Sazhenkov, Sergey A."https://zbmath.org/authors/?q=ai:sazhenkov.sergey-alexandrovichSummary: The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating many thin inclusions, each of which geometrically is a straight line segment with the endpoints on the body edge. The inclusions (i.e. threads, filaments) are parallel to each other and the problem contains a small positive parameter \(\epsilon \), which describes the distance between two neighboring inclusions. Relying on the variational formulation of the problem, we investigate the behavior of solutions as \(\epsilon\) tends to zero. As the result, we derive a well-posed homogenized model, which describes effective behavior on the macroscopic scale, i.e., on the scale where there is no need to take into account each individual inclusion. The limiting passage as \(\epsilon \to 0\) is based on the use of the two-scale convergence theory.Mathematical modeling of heat transfer in biological fluids during laser thermotherapy of cystshttps://zbmath.org/1517.352252023-09-22T14:21:46.120933Z"Tereshko, D. A."https://zbmath.org/authors/?q=ai:tereshko.dmitry-a"Kudryashov, A. P."https://zbmath.org/authors/?q=ai:kudryashov.a-pSummary: The paper proposes a mathematical model of heat transfer by a jet of hot liquid that occurs during laser thermotherapy of cysts. This model is semiempirical, since the hot jet is set using sources obtained from the analysis of experimental data. Based on this model, a series of computational experiments were carried out for different laser powers.The time asymptotic expansion of the bipolar hydrodynamic model for semiconductorshttps://zbmath.org/1517.352262023-09-22T14:21:46.120933Z"Wu, Xiao-chun"https://zbmath.org/authors/?q=ai:wu.xiaochunSummary: In [J. Differ. Equations 192, No. 2, 326--359 (2003; Zbl 1045.35087)], \textit{I. Gasser} et al. showed that the solution to the bipolar hydrodynamic model for semiconductors (HD model) without doping function time-asymptotically converges to the diffusion wave of the porous media equation (PME) for the switch-off case. Motivated by the work of \textit{F. Huang} and \textit{X. Wu} [``The time asymptotic expansion for the compressible Euler equations with damping'', Preprint, \url{arXiv:2210.13157}], we will confirm that the time-asymptotic expansion proposed by \textit{S. Geng} et al. [``The time asymptotic expansion for the compressible Euler equations with time-dependent damping'', Preprint, \url{arXiv:2202.13385}] around the diffusion wave is a better asymptotic profile for the HD model in this paper, where we mainly adopt the approximate Green function method and the energy method.Constrained consensus-based optimizationhttps://zbmath.org/1517.352272023-09-22T14:21:46.120933Z"Borghi, Giacomo"https://zbmath.org/authors/?q=ai:borghi.giacomo"Herty, Michael"https://zbmath.org/authors/?q=ai:herty.michael-matthias"Pareschi, Lorenzo"https://zbmath.org/authors/?q=ai:pareschi.lorenzoIn this work, the authors investigate numerical methods for high-dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. Some constrained optimization problems of the type \(\min_{x\in\mathbb{R}^d}j(x)\) subject to \(x\in\mathcal{M}\) with the continuous cost functional \(j\in\mathcal{C}\) are considered. The feasible set \(\mathcal{M}\subset\mathbb{R}^d\) is assumed to have a boundary of zero Lebesgue-measure, that is, \(dx(\partial \mathcal{M})=0\). Here it is not required \(\mathcal{M}\) to be convex, contrary to the work of [\textit{H.-O. Bae} et al., Appl. Math. Comput. 416, Article ID 126726, 10 p. (2022; Zbl 1510.90208)]. The set \(\mathcal{M}\) is typically defined as the set of points \(x\in\mathbb{R}^d\) satisfying some constraints \(g_i(x) \leq 0\), \(h_l (x) = 0\) (\(i = 1,\ldots , m\), \(l = 1,\ldots , n\)). In this case, a simple not necessary, condition to ensure \(dx(\partial \mathcal{M})=0\) consists of taking \(g_i,h_l\in\mathcal{C}^1(\mathbb{R}^d)\). As a consequence, \(g_i^{-1}(0),h_l^{-1}(0)\) are submanifolds of \(\mathbb{R}^d\) with null Lebesgue-measure. Then the condition \(dx(\partial \mathcal{M})=0\) follows from \(\partial \mathcal{M}\subset (\bigcup_ig_i^{-1}(0))\cup (\bigcup_lh_l^{-1}(0))\). The affine constraints and implicitly defined manifolds, like a sphere or a torus, all satisfy such a condition too. Next it is introduced a consensus-based optimization (CBO) approach combined with suitable penalization techniques. The method applied by authors here relies on a reformulation of the constrained minimization problem in an unconstrained problem for a penalty function. Moreover it is extended to the constrained settings of the class of CBO methods. Exact penalization is employed. Since the optimal penalty parameter is unknown, an iterative strategy is proposed that successively updates the parameter based on the constrained violation. The convergence of the proposed method is shown to the minimum for general nonlinear constrained problems. The authors establish a new algorithm with some properties of interest. Several numerical examples in high dimensions illustrate the theory.
Reviewer: Dimitar A. Kolev (Sofia)Kinetic modeling of a leader-follower system in crowd evacuation with collective learninghttps://zbmath.org/1517.352282023-09-22T14:21:46.120933Z"Liao, Jie"https://zbmath.org/authors/?q=ai:liao.jie"Ren, Yi'ang"https://zbmath.org/authors/?q=ai:ren.yiang"Yan, Wenbin"https://zbmath.org/authors/?q=ai:yan.wenbinSummary: A kinetic modeling of crowd evacuation with leaders and followers is considered in this paper, in which the followers may not know the full information about the walking environment and the evacuation strategy, but they follow the leaders and learn the walking strategy to get out of the walking venue. Based on the kinetic theory of active particles, the learning dynamics are considered by introducing an activity variable \(u\), which represents the learning level of the followers and measures how much knowledge a follower has learned about the walking strategy, the walking environment, or the geometry of the walking venue. Several fundamental factors are considered in this leader-follower learning system of crowd evacuation, including: (1) the \textit{rational motion} of all pedestrians, i.e. the trend to the exit or to a preferred direction, the ability to avoid collisions with walls or obstacles, and the tendency to search for less crowded direction with minimal density gradient, (2) the \textit{irrational motion} of followers to follow other pedestrians induced by panic, (3) the \textit{learning dynamics} of the followers who learn the walking strategy during interaction with others, and, (4) the \textit{transition} from a follower to a leader when one's activity reaches the highest level of learning. A numerical comparison of a metro platform evacuation with and without learning shows a reasonably good predictive ability of the model that the learning effect plays a significant role in the evacuation dynamics.Generalized solution and eventual smoothness in a logarithmic Keller-Segel system for criminal activitieshttps://zbmath.org/1517.352292023-09-22T14:21:46.120933Z"Li, Bin"https://zbmath.org/authors/?q=ai:li.bin.7"Xie, Li"https://zbmath.org/authors/?q=ai:xie.liSummary: This paper focuses on a simplified variant of the Short \textit{et al.} model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction-diffusion-like equations -- one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller-Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller-Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller-Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well.Convergence of a particle method for a regularized spatially homogeneous Landau equationhttps://zbmath.org/1517.352302023-09-22T14:21:46.120933Z"Carrillo, José A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonio"Delgadino, Matias G."https://zbmath.org/authors/?q=ai:delgadino.matias-gonzalo"Wu, Jeremy S. H."https://zbmath.org/authors/?q=ai:wu.jeremy-s-hSummary: We study a regularized version of the Landau equation, which was recently introduced in [\textit{J. A. Carrillo} et al., J. Comput. Phys. 7, Article ID 100066, 24 p. (2020; \url{doi:10.1016/j.jcpx.2020.100066})] to numerically approximate the Landau equation with good accuracy at reasonable computational cost. We develop the existence and uniqueness theory for weak solutions, and we reinforce the numerical findings in the above-mentioned paper by rigorously proving the validity of particle approximations to the regularized Landau equation.Measure optimal controls for models inspired by biologyhttps://zbmath.org/1517.352312023-09-22T14:21:46.120933Z"Coclite, Giuseppe M."https://zbmath.org/authors/?q=ai:coclite.giuseppe-maria"Garavello, Mauro"https://zbmath.org/authors/?q=ai:garavello.mauroSummary: The paper is concerned with optimal control problems for a parabolic system, coupled with zero Neumann boundary conditions and with nonlinear source terms. Inspired by applications in biology and medicine, the system aims to describe two species in competition in the same spatial region and is supplemented with measure valued distributed controls, acting as source terms. Introducing general cost functionals, one can study optimal control problems. We prove the existence of solutions for the parabolic equations with measure valued controls, together with suitable stability estimates. Moreover, the existence of optimal solutions in a distributional sense is also established.Desensitizing control for the heat equation with respect to domain variationshttps://zbmath.org/1517.352322023-09-22T14:21:46.120933Z"Ervedoza, Sylvain"https://zbmath.org/authors/?q=ai:ervedoza.sylvain"Lissy, Pierre"https://zbmath.org/authors/?q=ai:lissy.pierre"Privat, Yannick"https://zbmath.org/authors/?q=ai:privat.yannickSummary: This article is dedicated to desensitizing issues for a quadratic functional involving the solution of the linear heat equation with respect to domain variations. This work can be seen as a continuation of [\textit{P. Lissy} et al., ESAIM, Control Optim. Calc. Var. 25, Paper No. 50, 21 p. (2019; Zbl 1442.93009)], insofar as we generalize several of the results it contains and investigate new related properties. In our framework, we consider variations of the spatial domain on which the solution of the PDE is defined at each time, and investigate three main issues: (i) approximate desensitizing, (ii) approximate desensitizing combined with an exact desensitizing for a finite-dimensional subspace, and (iii) exact desensitizing. We provide positive answers to questions (i) and (ii) and partial results to question (iii).Stationary solutions for the fractional Navier-Stokes-Coriolis system in Fourier-Besov spaceshttps://zbmath.org/1517.352352023-09-22T14:21:46.120933Z"Aurazo-Alvarez, Leithold L."https://zbmath.org/authors/?q=ai:aurazo-alvarez.leithold-lSummary: In this work we prove the existence of stationary solutions for the tridimensional fractional Navier-Stokes-Coriolis in critical Fourier-Besov spaces. We first deal with the non-stationary fractional Navier-Stokes-Coriolis and in this framework we get the existence of stationary solutions. Also we state a kind of stability of these non-stationary solutions which applied to the stationary case permits to conclude that, under suitable conditions, non-stationary solutions converge to the stationary ones when the time goes to infinity. Finally we establish a relation between the external force and the Coriolis parameter in order to get a unique solution for the stationary system.Structures of exact solutions for the modified nonlinear Schrödinger equation in the sense of conformable fractional derivativehttps://zbmath.org/1517.352462023-09-22T14:21:46.120933Z"Sağlam Özkan, Yeşim"https://zbmath.org/authors/?q=ai:saglam-ozkan.yesim"Ünal Yılmaz, Esra"https://zbmath.org/authors/?q=ai:yilmaz.esra-unalSummary: This paper is devoted to discuss analytically the conformable time-fractional modified nonlinear Schrödinger equation with the aid of efficient methods. The suggested model is a model used in ocean engineering to explain the propagation of water waves. At this stage, while using the proposed methods, the first step is to reduce the model defined by the conformable fractional derivative to the ordinary differential equation system with an appropriate transformation. We have obtained a variety of new families of exact traveling wave solutions including trigonometric, hyperbolic and exponential types. In related subject, the Adomian decomposition method is implemented to approximate the one of the solution of the underlying equation. For dynamic properties of the obtained solutions, we have depicted them graphically using computer programming to explain more efficiently the behavior of different shapes of solutions for the different values of free parameters with constraint conditions. Finally, a comparison is given for the solutions obtained in this study.Mathematical analysis of the Hadamard-type fractional Fokker-Planck equationhttps://zbmath.org/1517.352482023-09-22T14:21:46.120933Z"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen.10|wang.zhen.12|wang.zhen.8|wang.zhen.5|wang.zhen.7|wang.zhen|wang.zhen.9|wang.zhen.3|wang.zhen.2|wang.zhen.1"Sun, Luhan"https://zbmath.org/authors/?q=ai:sun.luhanSummary: This paper deals with a time-fractional Fokker-Planck equation, where the time-fractional derivative (denoted by \(_H\mathrm{D}_{a, t}^{1-\alpha}u\)) is in the Hadamard sense with order \(\alpha\in(0, 1)\). With the help of the modified Laplace transform and its inverse transform, the mild solutions of the considered equation are constructed. The existence and uniqueness of the mild solutions are proved by the contraction mapping principle, and some regularity estimates are satisfied. For \(\alpha\in(1/2, 1)\), the mild solution is shown to be the classical solution. The decay estimates of the solution \(u\) and \(_H\mathrm{D}_{a, t}^{1-\alpha}u\) are also investigated.A high-efficiency second-order numerical scheme for time-fractional phase field models by using extended SAV methodhttps://zbmath.org/1517.352502023-09-22T14:21:46.120933Z"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.12|zhang.hui.25"Jiang, Xiaoyun"https://zbmath.org/authors/?q=ai:jiang.xiaoyun(no abstract)Carleman estimates for a magnetohydrodynamics system and application to inverse source problemshttps://zbmath.org/1517.352602023-09-22T14:21:46.120933Z"Huang, Xinchi"https://zbmath.org/authors/?q=ai:huang.xinchi"Yamamoto, Masahiro"https://zbmath.org/authors/?q=ai:yamamoto.masahiroSummary: In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We first prove two kinds of Carleman estimates. This is done by combining the Carleman estimates for the parabolic and the elliptic equations. Then we apply the Carleman estimates to prove Hölder type stability results for some inverse source problems.Applications of Nijenhuis geometry. IV: Multicomponent KdV and Camassa-Holm equationshttps://zbmath.org/1517.370672023-09-22T14:21:46.120933Z"Bolsinov, Alexey V."https://zbmath.org/authors/?q=ai:bolsinov.alexey-v"Konyaev, Andrey Yu."https://zbmath.org/authors/?q=ai:konyaev.andrei-yu"Matveev, Vladimir S."https://zbmath.org/authors/?q=ai:matveev.vladimir-sSummary: We construct a new series of multi-component integrable PDE systems that contains as particular examples (with appropriately chosen parameters) and generalises many famous integrable systems including KdV, coupled KdV [\textit{M. Antonowicz} and \textit{A. P. Fordy}, Physica D 28, 345--357 (1987; Zbl 0638.35079)], Harry Dym, coupled Harry Dym [\textit{M. Antonowicz} and \textit{A. P. Fordy}, J. Phys. A, Math. Gen. 21, No. 5, L269--L275 (1988; Zbl 0673.35088)], Camassa-Holm, multicomponent Camassa-Holm [\textit{D. D. Holm} and \textit{R. I. Ivanov}, J. Phys. A, Math. Theor. 43, No. 49, Article ID 492001, 20 p. (2010; Zbl 1213.37097)], Dullin-Gottwald-Holm and Kaup-Boussinesq systems. The series also contains integrable systems with no low-component analogues.Explicit solutions and Darboux transformations of a generalized D-Kaup-Newell hierarchyhttps://zbmath.org/1517.370682023-09-22T14:21:46.120933Z"McAnally, Morgan"https://zbmath.org/authors/?q=ai:mcanally.morgan"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiu(no abstract)A new approach to investigate the nonlinear dynamics in a \((3 + 1)\)-dimensional nonlinear evolution equation via Wronskian condition with a free functionhttps://zbmath.org/1517.370692023-09-22T14:21:46.120933Z"Wu, Jianping"https://zbmath.org/authors/?q=ai:wu.jianping(no abstract)A novel Riemann-Hilbert approach via \(t\)-part spectral analysis for a physically significant nonlocal integrable nonlinear Schrödinger equationhttps://zbmath.org/1517.370702023-09-22T14:21:46.120933Z"Wu, Jianping"https://zbmath.org/authors/?q=ai:wu.jianpingThe paper is devoted to an integrable system obtained from Manakov's system of equations, which corresponds to a two-component vector nonlinear Schrödinger equation. This is done by a reduction procedure, which essentially corresponds to imposing a nonlocal constraint that connects the two components in a consistent manner, and which results in a single equation. As a consequence, the imposed constraint should be taken into account and dealt with carefully in the subsequent analysis, which is the main difficulty to analyze the integrable properties of the system. The main interest to investigate this system, unlike the usual known integrable systems, lies in a wider range of applicability in various physical phenomena.
To investigate the integrable properties of the system the author uses the Riemann-Hilbert method, which unlike the usual cases, is considered with respect to the time variable. Surprisingly, in this case, the resulting equations turn out to be simpler in comparison to the standard case. The analytical properties of the matrices involved in the formulation of the Riemann-Hilbert problem are carefully constructed. Under the assumption that there are only simple zeros with respect to the spectral parameter, the solution to the Riemann-Hilbert problem is given, and some specific cases are constructed and discussed. In addition, the author provides some numerical analysis which confirms the analytical results.
The paper is written in a very clear and easy to follow manner, and the main proofs are given.
Reviewer: Arsen Melikyan (Brasília)Darboux transformations for the \(\hat{A}_{2n}^{(2)}\)-KdV hierarchyhttps://zbmath.org/1517.370732023-09-22T14:21:46.120933Z"Terng, Chuu-Lian"https://zbmath.org/authors/?q=ai:terng.chuu-lian"Wu, Zhiwei"https://zbmath.org/authors/?q=ai:wu.zhiweiAuthors' abstract: The \(\hat{A}^{(2)}_{2n}\)-hierarchy can be constructed from a splitting of the Kac-Moody algebra of type \(\hat{A}^{(1)}_{2n}\) by an involution. By choosing certain cross section of the gauge action, we obtain the \(\hat{A}^{(2)}_{2n}\)-KdV hierarchy. They are the equations for geometric invariants of isotropic curve flows of type A, which gives a geometric interpretation of the soliton hierarchy. In this paper, we construct Darboux and Bäcklund transformations for the \(\hat{A}^{(2)}_{2n}\)-hierarchy, and use it to construct Darboux transformations for the \(\hat{A}^{(2)}_{2n}\)-KdV hierarchy and isotropic curve flows of type A. Moreover, explicit soliton solutions for these hierarchies are given.
Reviewer: Ti-Jun Xiao (Fudan)Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized \((2 + 1)\)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physicshttps://zbmath.org/1517.370742023-09-22T14:21:46.120933Z"Zhao, Xin"https://zbmath.org/authors/?q=ai:zhao.xin.1"Tian, Bo"https://zbmath.org/authors/?q=ai:tian.bo"Tian, He-Yuan"https://zbmath.org/authors/?q=ai:tian.he-yuan"Yang, Dan-Yu"https://zbmath.org/authors/?q=ai:yang.danyu(no abstract)Construction of abundant solutions for two kinds of \((3+1)\)-dimensional equations with time-dependent coefficientshttps://zbmath.org/1517.370752023-09-22T14:21:46.120933Z"Han, Peng-Fei"https://zbmath.org/authors/?q=ai:han.pengfei"Bao, Taogetusang"https://zbmath.org/authors/?q=ai:bao.taogetusang(no abstract)KAM tori for the two-dimensional completely resonant Schrödinger equation with the general nonlinearityhttps://zbmath.org/1517.370762023-09-22T14:21:46.120933Z"Zhang, Min"https://zbmath.org/authors/?q=ai:zhang.min.6"Si, Jianguo"https://zbmath.org/authors/?q=ai:si.jianguoThe authors deal with the quasi-periodic solutions \(u\) of the two-dimensional completely resonant Schrödinger equation with the general nonlinear term \(|u|^{2p}u\) \((p\in\mathbb{Z}^+)\) under periodic boundary conditions. They show that, for an appropriate choice of tangential sites, the considered two-dimensional Schrödinger equation has small amplitude analytic quasi-periodic solutions of specific form. To prove their result, the authors first rewrite the Schrödinger equation as a Hamiltonian system (in infinitely many coordinates) and then (using a symplectic change of coordinates) convert its Hamiltonian to a partial Birkhoff normal form, which is suitable to be treated with the help of an infinite-dimensional KAM (Kolmogorov-Arnold-Moser) theorem. The needed KAM theorem is stated and proved by a KAM iterative scheme.
Reviewer: Catalin Popa (Iaşi)Stability of pullback random attractors for stochastic 3D Navier-Stokes-Voight equations with delayshttps://zbmath.org/1517.370822023-09-22T14:21:46.120933Z"Zhang, Qiangheng"https://zbmath.org/authors/?q=ai:zhang.qianghengSummary: This paper is concerned with the limiting dynamics of stochastic retarded 3D non-autonomous Navier-Stokes-Voight (NSV) equations driven by Laplace-multiplier noise. We first prove the existence, uniqueness, forward compactness and forward longtime stability of pullback random attractors (PRAs). We then establish the upper semicontinuity of PRAs from non-autonomy to autonomy. Finally, we study the upper semicontinuity of PRAs under an analogue of Hausdorff semi-distance as the memory time tends to zero. Because of the solution has no higher regularity, the forward pullback asymptotic compactness of solutions in the state space is proved by the spectrum decomposition technique.Stability of the inverses of interpolated operators with application to the Stokes systemhttps://zbmath.org/1517.460182023-09-22T14:21:46.120933Z"Asekritova, I."https://zbmath.org/authors/?q=ai:asekritova.irina-u"Kruglyak, N."https://zbmath.org/authors/?q=ai:kruglyak.natan-ya"Mastyło, M."https://zbmath.org/authors/?q=ai:mastylo.mieczyslawSummary: We study the stability of isomorphisms between interpolation scales of Banach spaces, including scales generated by well-known interpolation methods. We develop a general framework for compatibility theorems, and our methods apply to general cases. As a by-product we prove that the interpolated isomorphisms satisfy uniqueness-of-inverses. We use the obtained results to prove the stability of lattice isomorphisms on interpolation scales of Banach function lattices and demonstrate their application to the Calderón product spaces as well as to the real method scales. We also apply our results to prove solvability of the Neumann problem for the Stokes system of linear hydrostatics on an arbitrary bounded Lipschitz domain with a connected boundary in \(\mathbb{R}^n\), \(n\geq 3\), with data in some Lorentz spaces \(L^{p,q}(\partial \Omega, \mathbb{R}^n)\) over the set \(\partial \Omega\) equipped with a boundary surface measure.Chernoff iterations as an averaging method for random affine transformationshttps://zbmath.org/1517.470702023-09-22T14:21:46.120933Z"Kalmetev, R. Sh."https://zbmath.org/authors/?q=ai:kalmetev.r-sh"Orlov, Yu. N."https://zbmath.org/authors/?q=ai:orlov.yurii-n"Sakbaev, V. Zh."https://zbmath.org/authors/?q=ai:sakbaev.vsevolod-zhSummary: For functions defined on a finite-dimensional vector space, we study compositions of their independent random affine transformations that represent a noncommutative analogue of random walks. Conditions on iterations of independent random affine transformations are established that are sufficient for convergence to a group solving the Cauchy problem for an evolution equation of shift along the averaged vector field and sufficient for convergence to a semigroup solving the Cauchy problem for the Fokker-Planck equation. Numerical estimates for the deviation of random iterations from solutions of the limit problem are presented. Initial-boundary value problems for differential equations describing the evolution of functionals of limit random processes are formulated and studied.Solution remapping method with lower bound preservation for Navier-Stokes equations in aerodynamic shape optimizationhttps://zbmath.org/1517.490252023-09-22T14:21:46.120933Z"Zhang, Bin"https://zbmath.org/authors/?q=ai:zhang.bin.2|zhang.bin.6|zhang.bin.4|zhang.bin.1"Yuan, Weixiong"https://zbmath.org/authors/?q=ai:yuan.weixiong"Wang, Kun"https://zbmath.org/authors/?q=ai:wang.kun.2"Wang, Jufang"https://zbmath.org/authors/?q=ai:wang.jufang"Liu, Tiegang"https://zbmath.org/authors/?q=ai:liu.tiegangSummary: It is found that the solution remapping technique proposed in [\textit{J. Wang} et al., Numer. Math., Theory Methods Appl. 13, No. 4, 863--880 (2020; Zbl 1474.49091)] and [\textit{J. Wang} and \textit{T. Liu}, J. Sci. Comput. 87, No. 3, Paper No. 79, 26 p. (2021; Zbl 1469.65151)] does not work out for the Navier-Stokes equations with a high Reynolds number. The shape deformations usually reach several boundary layer mesh sizes for viscous flow, which far exceed one-layer mesh that the original method can tolerate. The direct application to Navier-Stokes equations can result in the unphysical pressures in remapped solutions, even though the conservative variables are within the reasonable range. In this work, a new solution remapping technique with lower bound preservation is proposed to construct initial values for the new shapes, and the global minimum density and pressure of the current shape which serve as lower bounds of the corresponding variables are used to constrain the remapped solutions. The solution distribution provided by the present method is proven to be acceptable as an initial value for the new shape. Several numerical experiments show that the present technique can substantially accelerate the flow convergence for large deformation problems with 70\%--80\% CPU time reduction in the viscous airfoil drag minimization.Correction to: ``A geometric view on the generalized Proudman-Johnson and \(r\)-Hunter-Saxton equations''https://zbmath.org/1517.580032023-09-22T14:21:46.120933Z"Bauer, Martin"https://zbmath.org/authors/?q=ai:bauer.martin"Lu, Yuxiu"https://zbmath.org/authors/?q=ai:lu.yuxiu"Maor, Cy"https://zbmath.org/authors/?q=ai:maor.cyFrom the text: Our article [ibid. 32, No. 1, Paper No. 17, 18 p. (2022; Zbl 1482.58004)] concerns the study of two families of equations: the generalized inviscid Proudman-Johnson equation, and the \(r\)-Hunter-Saxton equation. There, we investigate these equations both on the real line and the circle and claim that they are equivalent for both cases. This statement was wrong, as the equations are only equivalent when considered on the real line. As a consequence the results on the circle (Section 3) are only valid for the \(r\)-Hunter-Saxton equations and do not hold for the generalized inviscid Proudman-Johnson equations. The results on the real line (Section 2) are valid as written.Lack of robustness and accuracy of many numerical schemes for phase-field simulationshttps://zbmath.org/1517.650642023-09-22T14:21:46.120933Z"Xu, Jinchao"https://zbmath.org/authors/?q=ai:xu.jinchao"Xu, Xiaofeng"https://zbmath.org/authors/?q=ai:xu.xiaofengSummary: In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that \textit{the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small.} More specifically, by rigorous analysis in most cases, we have the following conclusions:
\begin{itemize}
\item[(i)] The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution.
\item[(ii)] The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions.
\item[(iii)] For the second-order fully implicit and convex splitting schemes, for any time step size \(\delta t>0\), there exists an initial condition \(u_0\), with \(|u_0| >1\), such that the numerical solution converges to the wrong steady state solution.
\item[(iv)] For \(|u_0| \leq 1\), all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small.
\item[(v)] An unconditionally energy-stable scheme (such as the modified Crank-Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank-Nicolson scheme).
\end{itemize}
Most, if not all, of the above conclusions are expected to be true for more general Allen-Cahn and other phase-field models.Efficient third-order BDF finite difference scheme for the generalized viscous Burgers' equationhttps://zbmath.org/1517.650702023-09-22T14:21:46.120933Z"Guo, Tao"https://zbmath.org/authors/?q=ai:guo.tao"Xu, Da"https://zbmath.org/authors/?q=ai:xu.da"Qiu, Wenlin"https://zbmath.org/authors/?q=ai:qiu.wenlinSummary: This article presents a third-order backward differentiation formula (BDF3) finite difference scheme for the generalized viscous Burgers' equation. The discretization of time and space directions is accomplished by the BDF3 method and standard second-order difference formula, respectively, thereby constructing a fully-discrete scheme. For the proposed scheme, we yield the convergence of \(h^2 + \tau^3\) by means of the energy argument and the cut-off function method. Besides, a comparison of the time convergence rate and numerical accuracy with those of recent existing work shows the effectiveness and competitiveness of our approach. A numerical experiment is carried out to verify the theoretical predictions.Unconditional stability and convergence analysis of fully discrete stabilized finite volume method for the time-dependent incompressible MHD flowhttps://zbmath.org/1517.650792023-09-22T14:21:46.120933Z"Zhang, Tong"https://zbmath.org/authors/?q=ai:zhang.tong.1"Chu, Xiaochen"https://zbmath.org/authors/?q=ai:chu.xiaochen"Chen, Chuanjun"https://zbmath.org/authors/?q=ai:chen.chuanjunThe authors consider the fully discrete finite volume method for the incompressible MHD (magnetohydrodynamics) equations. The lowest equal-order mixed finite element pair is used to approximate the velocity, pressure, and magnetic fields. The local pressure projection method is introduced to overcome the restriction of discrete inf-sup condition. The time discretization is performed using the backward Euler semi-implicit scheme. The \(H^2\)-stability is proved and optimal error estimates are established as well. Some numerical examples are presented to confirm the reliability and accuracy of the considered numerical schemes.
Reviewer: Abdallah Bradji (Annaba)Tokamak free-boundary plasma equilibrium computations in presence of non-linear materialshttps://zbmath.org/1517.650832023-09-22T14:21:46.120933Z"Boulbe, Cédric"https://zbmath.org/authors/?q=ai:boulbe.cedric"Faugeras, Blaise"https://zbmath.org/authors/?q=ai:faugeras.blaise"Gros, Guillaume"https://zbmath.org/authors/?q=ai:gros.guillaume"Rapetti, Francesca"https://zbmath.org/authors/?q=ai:rapetti.francescaSummary: We consider the axisymmetric formulation of the equilibrium problem for a hot plasma in a tokamak. We adopt a non-overlapping mortar element approach, that couples \(\mathcal{C}^0\) piece-wise linear Lagrange finite elements in a region that does not contain the plasma and \(\mathcal{C}^1\) piece-wise cubic reduced Hsieh-Clough-Tocher finite elements elsewhere, to approximate the magnetic flux field on a triangular mesh of the poloidal tokamak section. The inclusion of ferromagnetic parts is simplified by assuming that they fit within the axisymmetric modeling and a new formulation of the Newton algorithm for the problem solution is stated, both in the static and quasi-static evolution cases.Optimal pointwise convergence of the LDG method for singularly perturbed convection-diffusion problemhttps://zbmath.org/1517.650842023-09-22T14:21:46.120933Z"Cheng, Yao"https://zbmath.org/authors/?q=ai:cheng.yao"Wang, Xuesong"https://zbmath.org/authors/?q=ai:wang.xuesongSummary: A local discontinuous Galerkin (LDG) method is considered for a one-dimensional singularly perturbed convection-diffusion problem with an exponential boundary layer. Based on the technique of discrete Green's function, we establish optimal pointwise convergence (up to a logarithmic factor) of the LDG method on three typical families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. Numerical experiments are also given.Analysis of a fully-discrete, non-conforming approximation of evolution equations and applicationshttps://zbmath.org/1517.650862023-09-22T14:21:46.120933Z"Kaltenbach, A."https://zbmath.org/authors/?q=ai:kaltenbach.alex"Růžička, M."https://zbmath.org/authors/?q=ai:ruzicka.michaelSummary: In this paper, we consider a fully-discrete approximation of an abstract evolution equation deploying a non-conforming spatial approximation and finite differences in time (Rothe-Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Therefore, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We formulate the problem in the very general and abstract setting of so-called non-conforming Bochner pseudo-monotone operators, which allows for a unified treatment of several evolution problems. Our abstract results for non-conforming Bochner pseudo-monotone operators allow to establish (weak) convergence just by verifying a few natural assumptions on the operators time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be performed easily. We exemplify the applicability of our approach on several DG schemes for the unsteady \(p\)-Navier-Stokes problem. The results of some numerical experiments are reported in the final section.Theoretical and practical aspects of space-time DG-SEM implementationshttps://zbmath.org/1517.650922023-09-22T14:21:46.120933Z"Versbach, Lea Miko"https://zbmath.org/authors/?q=ai:versbach.lea-miko"Linders, Viktor"https://zbmath.org/authors/?q=ai:linders.viktor"Klöfkorn, Robert"https://zbmath.org/authors/?q=ai:klofkorn.robert"Birken, Philipp"https://zbmath.org/authors/?q=ai:birken.philippSummary: We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge-Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.Superconvergence analysis of nonconforming finite element method for two-dimensional time-fractional Allen-Cahn equationhttps://zbmath.org/1517.650932023-09-22T14:21:46.120933Z"Wei, Yabing"https://zbmath.org/authors/?q=ai:wei.yabing"Zhao, Yanmin"https://zbmath.org/authors/?q=ai:zhao.yanmin"Wang, Fenling"https://zbmath.org/authors/?q=ai:wang.fenling"Tang, Yifa"https://zbmath.org/authors/?q=ai:tang.yifaSummary: In this paper, we investigate the numerical solution of the time-fractional Allen-Cahn equation with variable diffusion coefficients. A fully discrete approximation is presented by the \(L2\)-\(1_{\sigma}\) scheme on graded meshes and spatial high precision nonconforming finite element method (FEM). The nonlinear term is treated with the Newton linearization method. Based on the modified discrete fractional Grönwall inequality, we rigorously prove that the proposed scheme can achieve optimal convergence accuracy in both time and space directions. Furthermore, the \(H^1\)-norm global superconvergence result is obtained through the interpolation postprocessing technique. Finally, numerical experiments confirm the correctness of our theoretical analysis.Two SAV numerical methods for the nonlocal Cahn-Hilliard-Hele-Shaw systemhttps://zbmath.org/1517.650952023-09-22T14:21:46.120933Z"Huang, Langyang"https://zbmath.org/authors/?q=ai:huang.langyang"Wang, Yanan"https://zbmath.org/authors/?q=ai:wang.yanan"Mo, Yuchang"https://zbmath.org/authors/?q=ai:mo.yuchang"Tang, Bo"https://zbmath.org/authors/?q=ai:tang.boSummary: The capability of nonlocal models in materials field has attracted the scientific community a great attention to characterize the effect of various types of material heterogeneities and defects. In this paper, we are concerned with construction of a energy stability method for the nonlocal Cahn-Hilliard-Hele-Shaw system under periodic boundary conditions. By employing the Fourier spectral method in spatial and the scalar auxiliary variable (SAV) approach with first/second-order backward differentiation formula in temporal, two fast and effective schemes are established. The unconditional energy stability analyses is rigorously derived. Numerical experiments are presented to verify our theoretical results and to show the robustness and accuracy of the proposed method.A novel approach for multi dimensional fractional coupled Navier-Stokes equationhttps://zbmath.org/1517.650972023-09-22T14:21:46.120933Z"Kumbinarasaiah, S."https://zbmath.org/authors/?q=ai:kumbinarasaiah.sSummary: This study proposed a new scheme called the Hermite wavelet method (HWM) to find the numerical solutions to the multidimensional fractional coupled Navier-Stokes equation (NSE). This approach is based on the Hermite wavelets approximation with collocation points. Here, we reduce the fractional NSE into a set of nonlinear algebraic equations involving Hermite wavelet unknown coefficients. Convergence analysis is explained through the theorems. Three examples are given to validate the proposed technique's efficiency and discussed the comparison between the present method solutions with the exact solution. The obtained results are represented through graphs and tables for both integer and fractional order. These results disclose that the existing algorithm offers a better result.High-order positivity-preserving entropy stable schemes for the 3-D compressible Navier-Stokes equationshttps://zbmath.org/1517.650992023-09-22T14:21:46.120933Z"Yamaleev, Nail K."https://zbmath.org/authors/?q=ai:yamaleev.nail-k"Upperman, Johnathon"https://zbmath.org/authors/?q=ai:upperman.johnathonThe authors extend a new family of high-order positivity-preserving, entropy stable spectral collocation schemes developed for the one-dimensional compressible Navier-Stokes equations in [\textit{J. Upperman} et al., J. Comput. Phys. 466, Article ID 111355, 21 p. (2022; Zbl 07561057)] to three spatial dimensions. The proposed schemes are constructed by using a flux-limiting technique that combines a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable finite volume-type scheme discretized on the same Legendre-Gauss-Lobatto grid points used for constructing the high-order discrete operators. The positivity-preserving and excellent discontinuity-capturing properties are achieved by adding an artificial dissipation in the form of the low- and high-order Brenner-Navier-Stokes diffusion operators. In addition to this, the new schemes are entropy conservative for smooth inviscid flows and freestream preserving. One of the main features is that these schemes guarantee both the pointwise positivity of thermodynamic variables and \(L^2\) stability for the 3-D compressible Navier-Stokes equations. Some numerical examples are presented to show the accuracy and positivity-preserving properties of the family of the schemes proposed by the authors.
Reviewer: Abdallah Bradji (Annaba)Abundant traveling wave and numerical solutions for Novikov-Veselov system with their stability and accuracyhttps://zbmath.org/1517.651002023-09-22T14:21:46.120933Z"Almatrafi, M. B."https://zbmath.org/authors/?q=ai:almatrafi.mohammed-bakheetSummary: Solutions such as symmetric, bright soliton and periodic solutions play a prominent role in the field of differential equations, and they can be used to investigate several phenomena in nonlinear sciences. Some waves such as ion and magneto-sound waves in plasma are investigated by using some partial differential equations (PDEs) such as Novikov-Veselov (NV) equations. In this work, the improved \(\exp(-\Upsilon(\eta))\)-expansion approach is utilized to extract numerous soliton solutions for NV system. Hamiltonian system is invoked to analyze the stability of some solutions. The finite difference approach is successfully applied to achieve the numerical simulations of the proposed equations. We also introduce the stability and the accuracy of the numerical scheme. In order to validate the correctness of the accomplished results, we compare the exact solutions with the numerical solutions analytically and graphically. The presented techniques are very convenient and adequate and can be employed to other types of nonlinear evolution equations.A reduced basis method for Darcy flow systems that ensures local mass conservation by using exact discrete complexeshttps://zbmath.org/1517.651022023-09-22T14:21:46.120933Z"Boon, Wietse M."https://zbmath.org/authors/?q=ai:boon.wietse-m"Fumagalli, Alessio"https://zbmath.org/authors/?q=ai:fumagalli.alessioIn this paper, a three-step solution procedure is proposed for Darcy flow systems based on the exact de Rham complex. The mass conservation equation is first solved and the flux field is subsequently corrected by adding a solenoidal vector field. The computational cost in constructing the correction is reduced by applying reduced basis methods based on proper orthogonal decomposition. In the third step, the pressure field is constructed with discretization methods capable of conserving mass locally. The procedure was extended to the setting of Darcy flow in fractured porous media by employing mixed-dimensional differential operators.
Reviewer: Bülent Karasözen (Ankara)The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problemhttps://zbmath.org/1517.651032023-09-22T14:21:46.120933Z"Henning, Patrick"https://zbmath.org/authors/?q=ai:henning.patrickSummary: In this paper, we consider the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector (GPE) problem. For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross-Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [\textit{W. Bao} and \textit{Q. Du}, SIAM J. Sci. Comput. 25, No. 5, 1674--1697 (2004; Zbl 1061.82025)] or the damped inverse iteration suggested in our paper with \textit{D. Peterseim} [SIAM J. Numer. Anal. 58, No. 3, 1744--1772 (2020; Zbl 1512.35538)]. Our analysis also reveals why the inverse iteration for the GPE does not react favorably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.Accurate simulation of guided waves in optical fibers using finite element method combined with exact non-reflecting boundary conditionhttps://zbmath.org/1517.651072023-09-22T14:21:46.120933Z"Dautov, Rafail Z."https://zbmath.org/authors/?q=ai:dautov.rafail-z"Karchevskii, Evgenii M."https://zbmath.org/authors/?q=ai:karchevskii.evgeniiSummary: We present an analysis of numerical results illustrating the potentials of a new method for calculating guided waves in optical fibers and dispersion curves of corresponding eigenvalues. The earlier proposed finite element method is based on a special exact non-reflecting boundary condition and mathematically justified. For linear Lagrangian elements, the analysis demonstrates that the speed of convergence of the presented algorithm is quadratic, which corresponds to previously obtained theoretical estimates.
For the entire collection see [Zbl 1495.65002].Robust multigrid methods for nearly incompressible elasticity using macro elementshttps://zbmath.org/1517.651092023-09-22T14:21:46.120933Z"Farrell, Patrick E."https://zbmath.org/authors/?q=ai:farrell.patrick-emmet"Mitchell, Lawrence"https://zbmath.org/authors/?q=ai:mitchell.lawrence"Scott, L. Ridgway"https://zbmath.org/authors/?q=ai:scott.larkin-ridgway"Wechsung, Florian"https://zbmath.org/authors/?q=ai:wechsung.florianThe paper addresses the multigrid method for linear elasticity equations, focusing on nearly-incompressible cases where the first-order term dominates over the second-order term. The Scott-Vogelius finite-element discretization is employed to ensure an appropriate correspondence between the trial spaces. The mesh has a macro-element structure, and the inf-sup condition is assumed to hold for each macro element. This assumption allows for the construction of Fortin operators on each element, followed by the creation of localized Fortin operators. Utilizing these operators, the paper introduces robust relaxation and prolongation operators for lower polynomial degrees than previous constructions. Unlike previous Scott-Vogelius discretizations, the proposed method does not necessitate explicitly constructing a local basis in a \(C^1\) space. The resulting method is mesh-independent and parameter robust, and its convergence is rigorously established. Numerical experiments conducted on two and three-dimensional problems demonstrate that the standard algebraic and geometric multigrid methods are inadequate and that both the proposed smoothing and the proposed prolongation operators are essential for achieving an efficient solution.
Reviewer: Dana Černá (Liberec)An adaptive edge finite element DtN method for Maxwell's equations in biperiodic structureshttps://zbmath.org/1517.651132023-09-22T14:21:46.120933Z"Jiang, Xue"https://zbmath.org/authors/?q=ai:jiang.xue"Li, Peijun"https://zbmath.org/authors/?q=ai:li.peijun.1"Lv, Junliang"https://zbmath.org/authors/?q=ai:lv.junliang"Wang, Zhoufeng"https://zbmath.org/authors/?q=ai:wang.zhoufeng"Wu, Haijun"https://zbmath.org/authors/?q=ai:wu.haijun"Zheng, Weiying"https://zbmath.org/authors/?q=ai:zheng.weiyingSummary: We consider the diffraction of an electromagnetic plane wave by a biperiodic structure. This paper is concerned with a numerical solution of the diffraction grating problem for three-dimensional Maxwell's equations. Based on the Dirichlet-to-Neumann (DtN) operator, an equivalent boundary value problem is formulated in a bounded domain by using a transparent boundary condition. An \textit{a posteriori} error estimate-based adaptive edge finite element method is developed for the variational problem with the truncated DtN operator. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the former is used for local mesh refinements and the latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to demonstrate the competitive behaviour of the proposed method.On the mixtures of MGT viscoelastic solidshttps://zbmath.org/1517.740132023-09-22T14:21:46.120933Z"Bazarra, Noelia"https://zbmath.org/authors/?q=ai:bazarra.noelia"Fernández, José R."https://zbmath.org/authors/?q=ai:fernandez.jose-ramon"Quintanilla, Ramón"https://zbmath.org/authors/?q=ai:quintanilla.ramonSummary: In this paper, we study, from both analytical and numerical points of view, a problem involving a mixture of two viscoelastic solids. An existence and uniqueness result is proved using the theory of linear semigroups. Exponential decay is shown for the one-dimensional case. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained and the linear convergence is derived under suitable regularity conditions. Finally, one- and two-dimensional numerical simulations are presented to demonstrate the convergence, the discrete energy decay and the behavior of the solution.Arched beams of Bresse type: observability and application in thermoelasticityhttps://zbmath.org/1517.740242023-09-22T14:21:46.120933Z"Moraes, Gabriel E. Bittencourt"https://zbmath.org/authors/?q=ai:moraes.gabriel-e-bittencourt"Jorge da Silva, Marcio A."https://zbmath.org/authors/?q=ai:jorge-silva.marcio-antonio(no abstract)Modeling and analysis of unsteady Casson fluid flow due to an exponentially accelerating plate with thermal and solutal convective boundary conditionshttps://zbmath.org/1517.740292023-09-22T14:21:46.120933Z"Endalew, Mehari Fentahun"https://zbmath.org/authors/?q=ai:endalew.mehari-fentahun"Sarkar, Subharthi"https://zbmath.org/authors/?q=ai:sarkar.subharthiSummary: We intend to analyze the consequence of considering thermal radiation on time-dependent flow of the Casson fluid due to an exponentially accelerated inclined surface along with thermal as well as solutal convective boundary conditions. Fundamental equations governing an isotropic incompressible radiative Casson fluid flow are defined through a set of linear partial differential equations, and exact solutions are derived by using the Laplace transform approach. The numerical findings, obtained using MATLAB software, are presented in graphical and tabular representations based on the obtained analytical solutions of the fundamental equations. This investigation shows that the increment in thermal radiation results in the increment in fluid velocity and temperature distribution including thermal and momentum boundary layer thicknesses. Most interestingly, increasing the mass transfer coefficient results in an increment in the species concentration, velocity profiles, and mass transfer rate. However, the fluid velocity diminishes near the plate upon the increase in plate inclination. The scientific community will benefit greatly from this work since the findings can serve as benchmark solutions using numerical approaches to solve fully nonlinear flow governing problems.Study on the lateral dynamic impedance of pile groups in transversely isotropic soil using Novak's plane modelhttps://zbmath.org/1517.740652023-09-22T14:21:46.120933Z"Liu, Linchao"https://zbmath.org/authors/?q=ai:liu.linchao"Yan, Qifang"https://zbmath.org/authors/?q=ai:yan.qifangSummary: In order to consider the effect of anisotropy of soil around the pile on the lateral vibration of pile groups, the soil around the pile is regarded as a transversely isotropic medium, and a lateral dynamic interaction model of pile-pile in transversely isotropic soil is established. According to Novak's plane assumption and wave propagation theory, the lateral vibration of transversely isotropic soil layer is solved by introducing potential function and using mathematical and physical means, and the attenuation function of lateral displacement of free field is given. The Dobry and Dazetas simplified solution of attenuation function is different from that of solution of plane model. The pile-pile horizontal dynamic interaction factor in transversely isotropic soil is obtained by using the initial parameter method and Krylov function. The horizontal dynamic impedance of pile groups is obtained by using the pile-pile superposition principle. The change rule of the lateral displacement attenuation function of transversely isotropic soil with frequency is related to the direction and frequency. The ratio \(G_{hv}\) of the shear modulus in the lateral plane to the shear modulus in the vertical plane and the pile spacing \(S/d\) have a great impact on the lateral vibration of pile groups, and when the pile spacing is large, the curves of attenuation function varying with frequency fluctuate greatly. The ratio of elastic modulus of pile to vertical plane shear modulus of soil \(E_p/G_v\) has an effect on the lateral stiffness of pile groups, which is related to frequency, while the effect on dynamic damping is not affected by frequency. The difference of mechanical properties on different surfaces of soil around the pile has a great influence on the lateral vibration of pile groups in transversely isotropic soil, and the influence of the anisotropy on the attenuation function of the lateral displacement and the dynamic impedance cannot be ignored.The tapering length of needles in martensite/martensite macrotwinshttps://zbmath.org/1517.740752023-09-22T14:21:46.120933Z"Conti, Sergio"https://zbmath.org/authors/?q=ai:conti.sergio"Zwicknagl, Barbara"https://zbmath.org/authors/?q=ai:zwicknagl.barbara-mariaSummary: We study needle formation at martensite/martensite macro interfaces in shape-memory alloys. We characterize the scaling of the energy in terms of the needle tapering length and the transformation strain, both in geometrically linear and in finite elasticity. We find that linearized elasticity is unable to predict the value of the tapering length, as the energy tends to zero with needle length tending to infinity. Finite elasticity shows that the optimal tapering length is inversely proportional to the order parameter, in agreement with previous numerical simulations. The upper bound in the scaling law is obtained by explicit constructions. The lower bound is obtained using rigidity arguments, and as an important intermediate step we show that the Friesecke-James-Müller geometric rigidity estimate holds with a uniform constant for uniformly Lipschitz domains.Non-isothermal non-Newtonian fluids: the stationary casehttps://zbmath.org/1517.760062023-09-22T14:21:46.120933Z"Grasselli, Maurizio"https://zbmath.org/authors/?q=ai:grasselli.maurizio"Parolini, Nicola"https://zbmath.org/authors/?q=ai:parolini.nicola"Poiatti, Andrea"https://zbmath.org/authors/?q=ai:poiatti.andrea"Verani, Marco"https://zbmath.org/authors/?q=ai:verani.marcoSummary: The stationary Navier-Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on \(p \in (1, 2)\) (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier-Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.A generalized mean-field game model for the dynamics of pedestrians with limited predictive abilitieshttps://zbmath.org/1517.760132023-09-22T14:21:46.120933Z"Cristiani, Emiliano"https://zbmath.org/authors/?q=ai:cristiani.emiliano"de Santo, Arianna"https://zbmath.org/authors/?q=ai:de-santo.arianna"Menci, Marta"https://zbmath.org/authors/?q=ai:menci.martaSummary: This paper investigates the model for pedestrian flow firstly proposed in [\textit{E. Cristiani} et al., SIAM J. Appl. Math. 75, No. 2, 605--629 (2015; Zbl 1316.35189)]. The model assumes that each individual in the crowd moves in a known domain, aiming at minimizing a given cost functional. Both the pedestrian dynamics and the cost functional itself depend on the position of the whole crowd. In addition, pedestrians are assumed to have predictive abilities, but limited in time, extending only up to \(\theta\) time units into the future, where \(\theta \in [0,\infty)\) is a model parameter. (1) For \(\theta=0\) (no predictive abilities), we recover the modeling assumptions of the Hughes's model, where people take decisions on the basis of the current position of the crowd only. (2) For \(\theta \to \infty\), instead, we recover the standard mean-field game (MFG) setting, where people are able to forecast the behavior of the others at any future time and take decisions on the basis of the current and future position of the whole crowd. (3) For very short values of \(\theta\) (typically coinciding with a single time step in a discrete-in-time setting), we recover instead the MFG setting joined to the instantaneous model predictive control technique. (4) For intermediate values of \(\theta\) we obtain something different: as in the Hughes's model, the numerical procedure to solve the problem requires to run an off-line procedure at any fixed time \(t\), which returns the current optimal velocity field at time \(t\) by solving an associated backward-in-time Hamilton-Jacobi-Bellman equation; but, differently from the Hughes's model, here the procedure involves a prediction of the crowd behavior in the sliding time window \([t, t+\theta)\), therefore the optimal velocity field is given by the solution to a forward-backward system which joins a Fokker-Planck equation with a Hamilton-Jacobi-Bellman equation as in the MFG approach. The fact that a different forward-backward system must be solved at any time \(t\) gives rise to new interesting theoretical questions. Numerical tests will give some clues about the well-posedness of the problem.On the Hamiltonian and geometric structure of Langmuir circulationhttps://zbmath.org/1517.760152023-09-22T14:21:46.120933Z"Yang, Cheng"https://zbmath.org/authors/?q=ai:yang.chengSummary: The Craik-Leibovich equation (CL) serves as the theoretical model for Langmuir circulation. We show that the CL equation can be reduced to the dual space of a certain Lie algebra central extension. On this space, the CL equation can be rewritten as a Hamiltonian equation corresponding to the kinetic energy. Additionally, we provide an explanation of the appearance of this central extension structure through an averaging theory for Langmuir circulation. Lastly, we prove a stability theorem for two-dimensional steady flows of the CL equation. The paper also contains two examples of stable steady CL flows.Long-time decay of Leray solution of 3D-NSE with exponential dampinghttps://zbmath.org/1517.760172023-09-22T14:21:46.120933Z"Blel, Mongi"https://zbmath.org/authors/?q=ai:blel.mongi"Benameur, Jamel"https://zbmath.org/authors/?q=ai:benameur.jamelThis paper deals with the uniqueness, the continuity in \(L^2\) and the temporal decay for the Leray solution to the 3D incompressible Navier-Stokes equations with nonlinear exponential damping. To prove uniqueness, the authors use the energy method and an approximate system. The proof of the asymptotic decay is based on Fourier splitting method.
Reviewer: Shangkun Weng (Pohang)Stationary solutions to the Navier-Stokes system in an exterior plane domain: 90 years of search, mysteries and insightshttps://zbmath.org/1517.760182023-09-22T14:21:46.120933Z"Korobkov, Mikhail"https://zbmath.org/authors/?q=ai:korobkov.mikhail-v"Ren, Xiao"https://zbmath.org/authors/?q=ai:ren.xiaoThe paper mainly consists in a review of results concerning boundary value problems for the stationary Navier-Stokes system posed in \(\mathbb{R}^{2}\): \( -\Delta w+(w\cdot \nabla )w+\nabla p=f\), \(\nabla \cdot w=0\), in \(\mathbb{R} ^{2}\), \(w\rightarrow w_{\infty }=\lambda e_{1}\) as \(\left\vert z\right\vert \rightarrow \infty \), or in a planar exterior domain: \(-\Delta w+(w\cdot \nabla )w+\nabla p=f\), \(\nabla \cdot w=0\), in \(\Omega \subset \mathbb{R}^{2}\) , \(w\mid _{\partial \Omega }=\lambda e_{1}\), \(w\rightarrow 0\) as \(\left\vert z\right\vert \rightarrow \infty \). For the second problem, the authors recall the definition of D-solution as a solution with finite Dirichlet integral \(\int_{\Omega }\left\vert \nabla w\right\vert ^{2}<\infty \), and the invading domains method proposed by Leray, which consists to introduce \( \Omega _{k}=B_{R_{k}}\cap \Omega \), \(k=1,2,3,\ldots \), with \( R_{k}\rightarrow +\infty \), and to consider the second problem posed in \( \Omega _{k}\) replacing the boundary conditions by \(w\mid _{\partial \Omega _{k}}=0\) and \(w_{k}=w_{\infty }\) for \(r=R_{k}\). The authors recall some existence results already obtained and still open problems. They also recall properties of D-solutions. For the first problem posed in \(\mathbb{R}^{2}\), they recall the blow-down method they proposed in their earlier paper [J. Math. Pures Appl. (9) 158, 71--89 (2022; Zbl 1513.76068)] and which proves an asymptotic estimate for a sequence of D-solutions to the Navier-Stokes equations in an annulus whose radii satisfy growth properties. They prove that if \(f\) is compactly supported in \(B_{R}\) for some \(R>0\) and \(\left\Vert f\right\Vert _{W^{-1,2}(B_{2R})}<+\infty \), there exists a universal constant \(\varepsilon _{1}>0\) such that, if \(\left\Vert f\right\Vert _{H^{-1}(B_{2R})}<\frac{\varepsilon _{1}}{ln^{\frac{1}{2}}(2+\frac{1}{ \lambda R})}\lambda \), and the Leray solution \(w_{L}\) satisfies \(w_{0}=w_{\infty }=\lambda e_{1}\). If, in addition, the total force \(\mathcal{F}=\int_{ \mathbb{R}^{2}}f=0\), then the factor \(ln^{\frac{1}{2}}(2+\frac{1}{\lambda R}) \) can be removed in the previous assumption. They also prove that if \(f\) is compactly supported in \(B_{R}\) for some \(R>0\), \(\left\Vert f\right\Vert _{W^{-1,2}(B_{2R})}<+\infty \), \(\mathcal{F}=\int_{\mathbb{R}^{2}}f=0\), and \( w_{\infty }=0\), the Leray solution \(w_{L}\) satisfies \(w_{0}=w_{\infty }=0\). In the last part of their paper, the authors describe the asymptotic form of D-solutions with nonzero limit velocity. They introduce the Oseen system. They build the Finn-Smith solution and prove that it is the unique solution to the obstacle problem under an hypothesis on the domain \(\Omega \). They also prove a uniqueness result for the stationary Navier-Stokes problem under further hypotheses on the force \(f\). They conclude with open questions and further remarks.
Reviewer: Alain Brillard (Riedisheim)Separable Hamiltonian PDEs and turning point principle for stability of gaseous starshttps://zbmath.org/1517.760322023-09-22T14:21:46.120933Z"Lin, Zhiwu"https://zbmath.org/authors/?q=ai:lin.zhiwu"Zeng, Chongchun"https://zbmath.org/authors/?q=ai:zeng.chongchunSummary: We consider stability of nonrotating gaseous stars modeled by the Euler-Poisson system. Under general assumptions on the equation of states, we proved a turning point principle (TPP) that the stability of the stars is entirely determined by the mass-radius curve parametrized by the center density. In particular, the stability can only change at extrema (i.e., local maximum or minimum points) of the total mass. For a very general equations of state, TPP implies that for increasing center density the stars are stable up to the first mass maximum and unstable beyond this point until the next mass extremum (a minimum). Moreover, we get a precise counting of unstable modes and exponential trichotomy estimates for the linearized Euler-Poisson system. To prove these results, we develop a general framework of separable Hamiltonian PDEs. The general approach is flexible and can be used for many other problems, including stability of rotating and magnetic stars, relativistic stars, and galaxies.A novel multivariate spectral local quasilinearization method (MV-SLQLM) for modelling flow, moisture, heat, and solute transport in soilhttps://zbmath.org/1517.760522023-09-22T14:21:46.120933Z"Mwakilama, Elias"https://zbmath.org/authors/?q=ai:mwakilama.elias"Magagula, Vusi"https://zbmath.org/authors/?q=ai:magagula.vusi-mpendulo"Gathungu, Duncan"https://zbmath.org/authors/?q=ai:gathungu.duncan-kioiSummary: Conventionally, the problem of studying the transport of water, heat, and solute in soil or groundwater systems has been numerically solved using finite difference (FD) or finite element (FE) methods. FE methods are attractive over FD methods because they are geometrically flexible. However, recent studies demonstrate that spectral collocation (SC) methods converge exponentially faster than FD or FE methods using a few grid points or on coarse grids. This work proposes and applies a multivariate spectral local quasilinearization method (MV-SLQLM) to model the transportation and interaction of soil moisture, heat, and solute concentration in a nonbare soil ridge. The MV-SLQLM uses a quasilinearization method (QLM) to linearize any nonlinear equations and then employs a local linearization method (LLM) to decouple the linearized system of PDEs to form a sequence of equations that are solved in a computationally efficient manner. The MV-SLQLM is thus an extension of the bivariate spectral local linearization method (BI-SLLM) that fails to deal with a 2D problem and is a modification of the MV-SQLM whose efficiency is compromised when operating on high dense solution matrices. We use the residual error norms of the difference between successive iterations to affirm convergence to the expected solution. To illustrate the application and check the solution accuracy, we conduct systematic analyses of the effect of model parameters on distribution profiles. Findings are in good agreement with theory and literature, thereby revealing suitability of the MV-SLQLM to solve coupled nonlinear PDEs with environmental fluid dynamics applications.Entropy dissipation estimates for the Boltzmann equation without cut-offhttps://zbmath.org/1517.760582023-09-22T14:21:46.120933Z"Chaker, Jamil"https://zbmath.org/authors/?q=ai:chaker.jamil"Silvestre, Luis"https://zbmath.org/authors/?q=ai:silvestre.luis-eSummary: We prove the the entropy production of the Boltzmann equation, in the non cutoff regime, is bounded from below by a weighted \(L^p\) norm of the solution. The estimate holds for a wide range of potentials including soft potentials as well as very soft potentials. We discuss applications of this estimate for weak solutions of the Boltzmann equation. In particular, we obtain that weak solutions must be belong to the space \(L^1([0,T], L^p_q (\mathbb{R}^d))\) for some precise exponents \(p\) and \(q\).Regularity of minimizers for a model of charged dropletshttps://zbmath.org/1517.760682023-09-22T14:21:46.120933Z"de Philippis, Guido"https://zbmath.org/authors/?q=ai:de-philippis.guido"Hirsch, Jonas"https://zbmath.org/authors/?q=ai:hirsch.jonas"Vescovo, Giulia"https://zbmath.org/authors/?q=ai:vescovo.giuliaSummary: We investigate properties of minimizers of a variational model describing the shape of charged liquid droplets. The model, proposed by \textit{C. B. Muratov} and \textit{M. Novaga} [Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 472, No. 2187, Article ID 20150808, 12 p. (2016; Zbl 1371.76163)], takes into account the regularizing effect due to the screening of free counterionions in the droplet. In particular we prove partial regularity of minimizers, a first step toward the understanding of further properties of minimizers.Global well-posedness to the 3D nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and large initial velocityhttps://zbmath.org/1517.760792023-09-22T14:21:46.120933Z"Zhou, Ling"https://zbmath.org/authors/?q=ai:zhou.ling"Tang, Chun-Lei"https://zbmath.org/authors/?q=ai:tang.chun-leiThe authors study the well-posedness for the three-dimensional (3D) nonhomogeneous viscous resistive magnetohydrodynamic equations with density-dependent viscosity and initial vacuum in a bounded domain, and prove the global existence and uniqueness of strong solutions, provided that the initial density in \(L^1\)-norm and initial magnetic field in \(L^2\)-norm are sufficiently small, where, in particular, the initial velocity can be arbitrarily large. Moreover, the authors also obtain exponential time-decay rates of the solution. The key ingredient in the proof of this paper is to get the time-independent bounds on the \(L^1(0,T;L^\infty)\)-norm of the velocity gradient and then obtain the \(L^\infty (0, T;L^q)\)-norm of the viscosity gradient, and such bounds are obtained by careful combining energy method with the structure of the system under consideration. As a direct application, the global strong solutions of the 3D nonhomogeneous Navier-Stokes equations with density-dependent viscosity are obtained as long as the initial mass is properly small. This work improves the result [\textit{Y. Liu}, Z. Angew. Math. Phys. 70, No. 4, Paper No. 107, 18 p. (2019; Zbl 1420.35249)], and extends the local existence result in [\textit{S. Song}, Z. Angew. Math. Phys. 69, No. 2, Paper No. 23, 27 p. (2018; Zbl 1392.35238)] to be a global one.
Reviewer: Song Jiang (Beijing)Green's function and pointwise behaviors of the one-dimensional modified Vlasov-Poisson-Boltzmann systemhttps://zbmath.org/1517.760802023-09-22T14:21:46.120933Z"Li, Yanchao"https://zbmath.org/authors/?q=ai:li.yanchao"Zhong, Mingying"https://zbmath.org/authors/?q=ai:zhong.mingyingSummary: The pointwise space-time behaviors of the Green's function and the global solution to the modified Vlasov-Poisson-Boltzmann (mVPB) system in one-dimensional space are studied in this paper. It is shown that, the Green's function admits the diffusion wave, the Huygens's type sound wave, the singular kinetic wave and the remainder term decaying exponentially in space-time. These behaviors are similar to the Boltzmann equation [\textit{T.-P. Liu} and \textit{S.-H. Yu}, Commun. Pure Appl. Math. 57, No. 12, 1543--1608 (2004; Zbl 1111.76047)]. Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the nonlinear mVPB system in terms of the Green's function.Supersymmetric solitonshttps://zbmath.org/1517.810082023-09-22T14:21:46.120933Z"Shifman, Misha"https://zbmath.org/authors/?q=ai:shifman.mikhail-a"Yung, A."https://zbmath.org/authors/?q=ai:yung.alexeiPublisher's description: In the last decade methods and techniques based on supersymmetry have provided deep insights in quantum chromodynamics and other non-supersymmetric gauge theories at strong coupling. This book summarizes major advances in critical solitons in supersymmetric theories, and their implications for understanding basic dynamical regularities of non-supersymmetric theories. After an extended introduction on the theory of critical solitons, including a historical introduction, the authors focus on three topics: non-Abelian strings and confined monopoles; reducing the level of supersymmetry; and domain walls as D-brane prototypes. They also provide a thorough review of issues at the cutting edge, such as non-Abelian flux tubes. The book presents an extensive summary of the current literature so researchers in this field can understand the background and related issues. First published in 2009, this title has been reissued as an Open Access publication on Cambridge Core.
See the review of the original edition in [Zbl 1182.81003].Universality near the gradient catastrophe point in the semiclassical sine-Gordon equationhttps://zbmath.org/1517.810552023-09-22T14:21:46.120933Z"Lu, Bing-Ying"https://zbmath.org/authors/?q=ai:lu.bing-ying"Miller, Peter"https://zbmath.org/authors/?q=ai:miller.peter-dSummary: We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of \textit{B. Dubrovin} et al. [J. Nonlinear Sci. 19, No. 1, 57--94 (2009; Zbl 1220.37048)], we found that in a \(\mathcal{O}(\epsilon^{4/5})\) neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painlevé I tritronquée solution. A linear map can be explicitly made from the tritronquée solution to this neighborhood. Under this map: away from the tritronquée poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquée solution; localized defects appear at locations mapped from the poles of the tritronquée solution; the defects are proved universally to be a two-parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann-Hilbert problems, substantially generalizing [5] to establish universality beyond the context of solutions of a single equation.Memory functions, projection operators, and the defect technique. Some tools of the trade for the condensed matter physicisthttps://zbmath.org/1517.820052023-09-22T14:21:46.120933Z"Kenkre, V. M. (Nitant)"https://zbmath.org/authors/?q=ai:kenkre.vasudev-mangeshThis book covers a wide range of topics related to phenomena exhibiting memory features, which can be derived from either dynamic to stochastic points of view. A mathematical treatment of such problems is supplied with an extensive set of physical examples. This makes the book very useful for physicists who need for building models of observable phenomena and understanding the respective mathematical apparatus.
The first chapter provides a quite clear general introduction to the utility of memory functions considered by examples of transitions between ordinary differential equations describing a range of solutions between the exponential decay and harmonic oscillations as well as partial differential equations for a variety of processes from diffusion to wave motion. In continuation, it is illustrated how memory functions emerge when one considers only a part of a general multistate system. This multistate concept is explained in more mathematical detail leading to generalized master equations (GME), memory functions, etc. via the projection operators. The picture is refined in the third chapter, where the coarse-grained procedure is introduced and the memory function appears in the Fourier transform-like manner. This route leading to the memory functions of irreversible processes is illustrated by a number of practical cases of molecular optical and acoustical phenomena.
The fourth chapter specifies the relation of memory functions within the theory of generalized master equations to the physical case of exciton transport considering various approaches to the problem. In addition to the variant considered in the previous chapters, those which were developed by \textit{H. Haken} and \textit{P. Reineker} [Z. Phys. 249, 253--268 (1972; \url{doi:10.1007/BF01400230})] and \textit{R. Silbey} [Ann. Rev. Phys. Chem. 27, 203--223 (1976; \url{doi:10.1146/annurev.pc.27.100176.001223})] are analysed and all together unified. Of special interest, is the discussion of the GME's equivalence to the Continuous Time Random Walks. Model issues are illustrated with physical examples; this is continued in the fifth chapter, where the transport coherence of Frenkel excitons is explored, and the sixth chapter addresses peculiar temperature-dependent mobility of photo-injected holes and electrons in naphthalene.
The next chapters are devoted to theoretical (although, observable in principle) situations rather than experimental data. The seventh chapter mainly considers the specificity of relaxation effects originating from the interaction with a thermal bath and the eighth chapter reviews various applications such as the quantum theory of electrical resistivity and dynamical localization, nuclear magnetic resonance.
Further, the consideration goes to spatial memory effects. Compactification of granular media is discussed in the ninth chapter, the discrete Schrödinger equation for chains and nonlinear reaction-diffusion are addressed in the tenth chapter, and molecular crystals from the point of view of the Montroll defect technique in the eleventh. After this, in the twelfth chapter, the analysis of the defect technique is applied to continual media of different spatial dimensions; the chapter is resumed with the discussion of interplay with the Smoluchowski theory.
The next two chapters are focused on the effective medium approximation, memory functions and relaxation corresponding to different distributions as well as spatial long-rand and finite-size effects: as the basics of the respective theory (Chapter 13) and in the application to molecular movement in cell membranes (Chapter 14), which are modelled as systems with barriers. The concluding Chapters 15 and 16 cover mathematical issues related to some non-physical problems among which the ones related to mathematical epidemiology and, also provide an overview of other either non-covered or open issues.
Reviewer: Eugene Postnikov (Kursk)Origin of the spontaneous oscillations in a simplified coagulation-fragmentation system driven by a sourcehttps://zbmath.org/1517.820272023-09-22T14:21:46.120933Z"Fortin, Jean-Yves"https://zbmath.org/authors/?q=ai:fortin.jean-yvesSummary: We consider a system of aggregated clusters of particles, subjected to coagulation and fragmentation processes with mass dependent rates. Each monomer particle can aggregate with larger clusters, and each cluster can fragment into individual monomers with a rate directly proportional to the aggregation rate. The dynamics of the cluster densities is governed by a set of Smoluchowski equations, and we consider the addition of a source of monomers at constant rate. The whole dynamics can be reduced to solving a unique non-linear differential equation which displays self-oscillations in a specific range of parameters, and for a number of distinct clusters in the system large enough. This collective phenomenon is due to the presence of a fluctuating damping coefficient and is closely related to the Liénard self-oscillation mechanism observed in a more general class of physical systems such as the van der Pol oscillator.Correction to: ``On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds''https://zbmath.org/1517.820292023-09-22T14:21:46.120933Z"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Ko, Dongnam"https://zbmath.org/authors/?q=ai:ko.dongnam"Ryoo, Seung-Yeon"https://zbmath.org/authors/?q=ai:ryoo.sang-wooCorrection to the authors' paper [ibid. 172, No. 5, 1427--1478 (2018; Zbl 1407.82029)].
``In this article the author's name Seung-Yeon Ryoo was incorrectly written as Sang Woo Ryoo.
The original article has been corrected.''Jet quenching in glasmahttps://zbmath.org/1517.820342023-09-22T14:21:46.120933Z"Carrington, Margaret E."https://zbmath.org/authors/?q=ai:carrington.margaret-e"Czajka, Alina"https://zbmath.org/authors/?q=ai:czajka.alina"Mrówczyński, Stanisław"https://zbmath.org/authors/?q=ai:mrowczynski.stanislawSummary: We discuss the transverse momentum broadening of hard probes traversing an evolving glasma, which is the earliest phase of the matter produced in relativistic heavy-ion collisions. The coefficient \(\hat{q}\) is calculated using the Fokker-Planck equation, and an expansion in the proper time \(\tau\) which is applied to describe the temporal evolution of the glasma. The correlators of the chromodynamic fields that determine the Fokker-Planck collision terms, which in turn provide \(\hat{q}\), are computed to fifth order in \(\tau\). The momentum broadening is shown to rapidly grow in time and reach a magnitude of several \(\mathrm{GeV}^2\)/fm. We show that the transient pre-equilibrium phase provides a contribution to the energy loss of hard probes which is comparable to that of the long lasting, hydrodynamically evolving, equilibrium phase.Linearized Boltzmann collision operator. I: Polyatomic molecules modeled by a discrete internal energy variable and multicomponent mixtureshttps://zbmath.org/1517.820352023-09-22T14:21:46.120933Z"Bernhoff, Niclas"https://zbmath.org/authors/?q=ai:bernhoff.niclasSummary: The linearized Boltzmann collision operator appears in many important applications of the Boltzmann equation. Therefore, knowing its main properties is of great interest. This work extends some classical results for the linearized Boltzmann collision operator for monatomic single species to the case of polyatomic single species, while also reviewing corresponding results for multicomponent mixtures of monatomic species. The polyatomicity is modeled by a discrete internal energy variable, that can take a finite number of (given) different values. Results concerning the linearized Boltzmann collision operator being a nonnegative symmetric operator with a finite-dimensional kernel are reviewed.
A compactness result, saying that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a splitting, such that the terms can be shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators and as such being compact operators. Moreover, bounds on -- including coercivity of -- the collision frequency are obtained for a hard sphere like model, from which Fredholmness of the linearized collision operator follows, as well as its domain.Weak convergence of directed polymers to deterministic KPZ at high temperaturehttps://zbmath.org/1517.820362023-09-22T14:21:46.120933Z"Chatterjee, Sourav"https://zbmath.org/authors/?q=ai:chatterjee.sourav.1|chatterjee.sourav.2Summary: It is shown that when \(d\ge 3\), the growing random surface generated by the \((d+1)\)-dimensional directed polymer model at sufficiently high temperature, after being smoothed by taking microscopic local averages, converges to a solution of the deterministic KPZ equation in a suitable scaling limit.Anomalous transport in nodal ring semimetal slabs induced by vertical confinementhttps://zbmath.org/1517.820392023-09-22T14:21:46.120933Z"Chen, Jia-Di"https://zbmath.org/authors/?q=ai:chen.jia-di"Yang, Mou"https://zbmath.org/authors/?q=ai:yang.mouSummary: The nodal ring semimetal (NRSM) is a family member of gapless topological materials with the gap being closed at a nodal ring in momentum space. We partition the Fermi surface, a pipe ring, into two halves, so that the inter-half relaxation time is much longer than the in-half one. By calculating the parameter dependence of the relaxation times, we find that, the in-half relaxation results in the usual Drude conductance, and the inter-half relaxation, combined with the special energy bands of NRSM discretized by the vertical confinement, leads to an anomalous conductance that is inversely proportional to the slab thickness. The anomalous transport effect could be observed in ultra-thin NRSM slabs at low energies and can be regarded as the electronic transport signal of NRSMs.Mixing in anharmonic potential wellhttps://zbmath.org/1517.820412023-09-22T14:21:46.120933Z"Moreno, M."https://zbmath.org/authors/?q=ai:moreno.manuel-crescenio|moreno.maria.1|moreno.matias|moreno.manuel-crescencio|moreno.m-g-m|moreno.marta-susana|moreno.michael-r|moreno.m-a|moreno.m-h|moreno.miguel-vera|moreno.mabel"Rioseco, P."https://zbmath.org/authors/?q=ai:rioseco.paola"Van Den Bosch, H."https://zbmath.org/authors/?q=ai:van-den-bosch.hanne|van-den-bosch.h-l-aSummary: We prove phase-space mixing for solutions to Liouville's equation for integrable systems. Under a natural non-harmonicity condition, we obtain weak convergence of the distribution function with rate \(\langle\mathrm{time}\rangle^{-1}\). In one dimension, we also study the case where this condition fails at a certain energy, showing that mixing still holds but with a slower rate. When the condition holds and functions have higher regularity, the rate can be faster.
{\copyright 2022 American Institute of Physics}Transverse domain walls in thin ferromagnetic stripshttps://zbmath.org/1517.820452023-09-22T14:21:46.120933Z"Morini, M."https://zbmath.org/authors/?q=ai:morini.massimiliano"Muratov, C. B."https://zbmath.org/authors/?q=ai:muratov.cyrill-b"Novaga, M."https://zbmath.org/authors/?q=ai:novaga.matteo"Slastikov, V. V."https://zbmath.org/authors/?q=ai:slastikov.valeriy-vSummary: We present a characterization of the domain wall solutions arising as minimizers of an energy functional obtained in a suitable asymptotic regime of the micromagnetics for infinitely long thin film ferromagnetic strips in which the magnetization is forced to lie in the film plane. For the considered energy, we provide the existence, uniqueness, monotonicity, and symmetry of the magnetization profiles in the form of \(180^\circ\) and \(360^\circ\) walls. We also demonstrate how this energy arises as a \(\Gamma \)-limit of the reduced two-dimensional thin film micromagnetic energy that captures the non-local effects associated with the stray field, and characterize its respective energy minimizers.KPZ-type fluctuation exponents for interacting diffusions in equilibriumhttps://zbmath.org/1517.820472023-09-22T14:21:46.120933Z"Landon, Benjamin"https://zbmath.org/authors/?q=ai:landon.benjamin-a"Noack, Christian"https://zbmath.org/authors/?q=ai:noack.christian"Sosoe, Philippe"https://zbmath.org/authors/?q=ai:sosoe.philippeSummary: We consider systems of \(N\) diffusions in equilibrium interacting through a potential \(V\). We study a ``height function,'' which, for the special choice \(V(x) = \mathrm{e}^{-x}\), coincides with the partition function of a stationary semidiscrete polymer, also known as the (stationary) O'Connell-Yor polymer. For a general class of smooth convex potentials (generalizing the O'Connell-Yor case), we obtain the order of fluctuations of the height function by proving matching upper and lower bounds for the variance of order \(N^{2/3}\), the expected scaling for models lying in the KPZ universality class. The models we study are not expected to be integrable, and our methods are analytic and nonperturbative, making no use of explicit formulas or any results for the O'Connell-Yor polymer.Theoretical and numerical investigations of the energy states and absorption coefficients of quantum dots and quantum anti-dots in the presence of a magnetic fieldhttps://zbmath.org/1517.820492023-09-22T14:21:46.120933Z"Rahimi, Fatemeh"https://zbmath.org/authors/?q=ai:rahimi.fatemeh"Ghaffary, Tooraj"https://zbmath.org/authors/?q=ai:ghaffary.toorajSummary: Present study focuses on analyzing the role of magnetic field on electronic spectra and absorption coefficients for the transitions \(1s\to 2p\) of \(GaAs/Ga_{1-x}Al_x As/GaAs\) spherical multilayer quantum dot (MLQD) and \(Ga_{1-x}Al_x As/GaAs/Ga_{1-x}Al_x As\) spherical multilayer quantum anti-dot (MLQAD) with hydrogenic impurity, via a comparative view. These nano structure systems have been studied both theoretically (Perturbation Theory), and numerically (Finite Difference Method). The wave functions and energy eigenvalues have been calculated using the finite difference method. In this paper it will be shown that a new degeneracy in presence of the uniform magnetic field in the MLQAD model will be appeared. Furthermore, the effects of the magnetic field, core radius size and potential confinement on \(1s\to 2p_0\) absorption coefficient and also on \(1s\) and \(2p\) energy states of these nano spherical structures have been discussed.Torsion-induced chiral magnetic current in equilibriumhttps://zbmath.org/1517.830692023-09-22T14:21:46.120933Z"Amitani, Tatsuya"https://zbmath.org/authors/?q=ai:amitani.tatsuya"Nishida, Yusuke"https://zbmath.org/authors/?q=ai:nishida.yusukeSummary: We study equilibrium transport properties of massless Dirac fermions at finite temperature and chemical potential in spacetime accompanied by torsion, which in four dimensions couples with Dirac fermions as an axial gauge field. In particular, we compute the current density at the linear order in the torsion as well as in an external magnetic field with the Pauli-Villars regulatization, finding that an equilibrium current akin to the chiral magnetic current is locally induced. Such torsion can be realized in condensed matter systems along a screw dislocation line, around which localized and extended current distributions are predicted so as to be relevant to Dirac and Weyl semimetals. Furthermore, we compute the current density at the linear order in the torsion as well as in a Weyl node separation, which turns out to vanish in spite of being allowed from the symmetry perspective. Contrasts of our findings with torsion-induced currents from previous work are also discussed.A Birman-Schwinger principle in galactic dynamics: ESI, Vienna, 07--11 February 2022https://zbmath.org/1517.830832023-09-22T14:21:46.120933Z"Kunze, Markus"https://zbmath.org/authors/?q=ai:kunze.markus|kunze.markus-christianSummary: These are the (somewhat extended) lecture notes for four lectures delivered at the spring school during the thematic programme `Mathematical Perspectives of Gravitation beyond the Vacuum Regime' at ESI Vienna in February 2022.Critique of the use of geodesics in astrophysics and cosmologyhttps://zbmath.org/1517.830852023-09-22T14:21:46.120933Z"Mannheim, Philip D."https://zbmath.org/authors/?q=ai:mannheim.philip-dSummary: Since particles obey wave equations, in general one is not free to postulate that particles move on the geodesics associated with test particles. Rather, for this to be the case one has to be able to derive such behavior starting from the equations of motion that the particles obey, and to do so one can employ the eikonal approximation. To see what kind of trajectories might occur we explore the domain of support of the propagators associated with the wave equations, and extend the results of some previous propagator studies that have appeared in the literature. For a minimally coupled massless scalar field the domain of support in curved space is not restricted to the light cone, while for a conformally coupled massless scalar field the curved space domain is only restricted to the light cone if the scalar field propagates in a conformal to flat background. Consequently, eikonalization does not in general lead to null geodesics for curved space massless rays even though it does lead to straight line trajectories in flat spacetime. Equal remarks apply to the conformal invariant Maxwell equations. However, for massive particles one does obtain standard geodesic behavior this way, since they do not propagate on the light cone to begin with. Thus depending on how big the curvature actually is, in principle, even if not necessarily in practice, the standard null-geodesic-based gravitational bending formula and the general behavior of propagating light rays are in need of modification in regions with high enough curvature. We show how to appropriately modify the geodesic equations in such situations. We show that relativistic eikonalization has an intrinsic light-front structure, and show that eikonalization in a theory with local conformal symmetry leads to trajectories that are only globally conformally symmetric. Propagation of massless particles off the light cone is a curved space reflection of the fact that when light travels through a refractive medium in flat spacetime its velocity is modified from its free flat spacetime value. In the presence of gravity spacetime itself acts as a medium, and this medium can then take light rays off the light cone. This is also manifest in a conformal invariant scalar field theory propagator in two spacetime dimensions. It takes support off the light cone, doing so in fact even if the geometry is conformal to flat. We show that it is possible to obtain eikonal trajectories that are exact without approximation, and show that normals to advancing wavefronts follow these exact eikonal trajectories, with these trajectories being the trajectories along which energy and momentum are transported. In general then, in going from flat space to curved space one does not generalize flat space geodesics to curved space geodesics. Rather, one generalizes flat space wavefront normals (normals that are geodesic in flat space) to curved space wavefront normals, and in curved space normals to wavefronts do not have to be geodesic.Global high-order numerical schemes for the time evolution of the general relativistic radiation magneto-hydrodynamics equationshttps://zbmath.org/1517.850072023-09-22T14:21:46.120933Z"Izquierdo, M. R."https://zbmath.org/authors/?q=ai:izquierdo.m-r"Pareschi, L."https://zbmath.org/authors/?q=ai:pareschi.lorenzo"Miñano, B."https://zbmath.org/authors/?q=ai:minano.borja"Massó, J."https://zbmath.org/authors/?q=ai:masso.joan"Palenzuela, C."https://zbmath.org/authors/?q=ai:palenzuela.carlosSummary: Modeling correctly the transport of neutrinos is crucial in some astrophysical scenarios such as core-collapse supernovae and binary neutron star mergers. In this paper, we focus on the truncated-moment formalism, considering only the first two moments (M1 scheme) within the \textit{grey} approximation, which reduces Boltzmann seven-dimensional equation to a system of \(3 + 1\) equations closely resembling the hydrodynamic ones. Solving the M1 scheme is still mathematically challenging, since it is necessary to model the radiation-matter interaction in regimes where the evolution equations become stiff and behave as an advection-diffusion problem. Here, we present different global, high-order time integration schemes based on Implicit-Explicit Runge-Kutta methods designed to overcome the time-step restriction caused by such behavior while allowing us to use the explicit Runge-Kutta commonly employed for the magneto-hydrodynamics and Einstein equations. Finally, we analyze their performance in several numerical tests.Consumption and investment demand when health evolves stochasticallyhttps://zbmath.org/1517.912102023-09-22T14:21:46.120933Z"Bolin, Kristian"https://zbmath.org/authors/?q=ai:bolin.kristian"Caputo, Michael R."https://zbmath.org/authors/?q=ai:caputo.michael-rSummary: The health capital model of \textit{M. Grossman} [``On the concept of health capital and the demand for health'', J. Polit. Econ. 80, No. 2, 223--255 (1972; \url{doi:10.1086/259880})] is extended to account for uncertainty in the rate at which a stock of health depreciates. Two general versions of the model are contemplated, one with a fully functioning financial market and the other in its absence. The comparative dynamics of the feedback form of the consumption and health-investment demand functions are studied in these general settings, where it is shown that the key to deriving refutable results is to determine how a parameter or state variable affects the expected lifetime marginal utilities of health and wealth. To add further reach to the results, a simplified stochastic control problem is explicitly solved, yielding estimable structural feedback demand functions.An alternative method for analytical solutions of two-dimensional Black-Scholes-Merton equationhttps://zbmath.org/1517.912432023-09-22T14:21:46.120933Z"Yu, Jun"https://zbmath.org/authors/?q=ai:yu.jun.1|yu.jun|yu.jun.3|yu.jun.2|yu.jun.4"Tomas, Michael J."https://zbmath.org/authors/?q=ai:tomas.michael-jSummary: We present a method of deriving analytical solutions for a two-dimensional Black-Scholes-Merton equation. The method consists of three changes of variables in order to reduce the original partial differential equation (PDE) to a normal form and then solve it. Analytical solutions for two cases of option pricing on the minimum and maximum of two assets are derived using our method and are shown to agree with previously published results. The advantage of our solution procedure is the ability of splitting the original problem into several components in order to demonstrate some solution properties. The solutions of the two cases have a total of five components; each is a particular solution of the PDE itself. Due to the linearity of the two-dimensional Black-Scholes-Merton equation, any linear combination of these components constitutes another solution. Some other possible solutions as well as the solution properties are discussed.Extensions of Dupire formula: stochastic interest rates and stochastic local volatilityhttps://zbmath.org/1517.912472023-09-22T14:21:46.120933Z"Ögetbil, Orcan"https://zbmath.org/authors/?q=ai:ogetbil.orcan"Hientzsch, Bernhard"https://zbmath.org/authors/?q=ai:hientzsch.bernhardSummary: We derive generalizations of the Dupire formula to the case of general stochastic drift and/or the case of general stochastic local volatility. First, we handle the case in which the drift is given as a difference of two stochastic short rates. Such a setting is natural in a foreign exchange context where the short rates correspond to the short rates of the two currencies, in an equity single-currency context with stochastic dividend yield, or in a commodity context with stochastic convenience yield. We present the formula in both a call surface formulation and a total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes, and we present limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion, including one or more stochastic local volatility terms. In the general setting, our derivation allows for the computation and cali ration of the leverage function for stochastic local volatility models. Despite being implicit, the generalized Dupire formula can be used numerically in a fixed-point iterative scheme.Exploiting ergodicity in forecasts of corporate profitabilityhttps://zbmath.org/1517.912742023-09-22T14:21:46.120933Z"Mundt, Philipp"https://zbmath.org/authors/?q=ai:mundt.philipp"Alfarano, Simone"https://zbmath.org/authors/?q=ai:alfarano.simone"Milaković, Mishael"https://zbmath.org/authors/?q=ai:milakovic.mishaelSummary: Theory suggests that competition tends to equalize profit rates through the process of capital reallocation, and numerous studies have confirmed that profit rates are indeed persistent and mean-reverting. Recent empirical evidence further shows that fluctuations in the profitability of surviving corporations are well approximated by a stationary Laplace distribution. Here we show that a parsimonious diffusion process of corporate profitability that accounts for all three features of the data achieves better out-of-sample forecasting performance across different time horizons than previously suggested time-series and cross-sectional models. As a consequence of replicating the empirical distribution of profit rates, the model prescribes a particular strength or speed for the mean-reversion of all returns, which leads to superior forecasts of individual time-series when we exploit information from the cross-sectional collection of firms. The new model should appeal to managers, analysts, investors, and other groups of corporate stakeholders who are interested in accurate forecasts of profitability. To the extent that mean-reversion in profitability is the source of predictable variation in earnings, our approach can also be used in forecasts of earnings and is thus useful for firm valuation.Null boundary controllability for some biological and chemical diffusion problemshttps://zbmath.org/1517.930112023-09-22T14:21:46.120933Z"Ismailov, Mansur I."https://zbmath.org/authors/?q=ai:ismailov.mansur-i"Oner, Isil"https://zbmath.org/authors/?q=ai:oner.isilSummary: We examine the null boundary controllability of diffusion problems (with non-local boundary conditions) arising from the morphogen concentration process and the electrochemistry. Two different types of semi-periodic conditions are addressed. Also, towards further applications to biological and chemical diffusion problems, we comment on anti-semi-periodic boundary conditions. Such boundary conditions make the auxiliary spectral problems non-self-adjoint and, therefore the classical eigenfunctions expansion method does not work. The systems of eigenfunctions form a Riesz basis in the \(L_2\) space by adding associated eigenfunctions. For the controllability, we determine the admissible classes of initial data in terms of their Fourier coefficients. Finally, we present the null boundary controllability of these problems by the reduction to the moment problem.Controllability of suspension bridge model proposed by Lazer and McKenna under the influence of impulses, delays, and non-local conditionshttps://zbmath.org/1517.930122023-09-22T14:21:46.120933Z"Zouhair, Walid"https://zbmath.org/authors/?q=ai:zouhair.walid"Leiva, Hugo"https://zbmath.org/authors/?q=ai:leiva.hugoSummary: The main purpose of this paper is to prove the controllability of the model proposed by Lazer and Mckenna under the influence of impulses, delay, and non-local conditions. First, we study approximate controllability by employing a technique that pulls back the control solution to a fixed curve in a short time interval. Subsequently, based on Banach Fixed Point Theorem we investigate the exact controllability.Finite-time non-fragile boundary feedback control for a class of nonlinear parabolic systemshttps://zbmath.org/1517.930332023-09-22T14:21:46.120933Z"Wei, Chengzhou"https://zbmath.org/authors/?q=ai:wei.chengzhou"Li, Junmin"https://zbmath.org/authors/?q=ai:li.junmin(no abstract)PDE evolutions for M-smoothers in one, two, and three dimensionshttps://zbmath.org/1517.940212023-09-22T14:21:46.120933Z"Welk, Martin"https://zbmath.org/authors/?q=ai:welk.martin"Weickert, Joachim"https://zbmath.org/authors/?q=ai:weickert.joachimSummary: Local M-smoothers are interesting and important signal and image processing techniques with many connections to other methods. In our paper, we derive a family of partial differential equations (PDEs) that result in one, two, and three dimensions as limiting processes from M-smoothers which are based on local order-\(p\) means within a ball the radius of which tends to zero. The order \(p\) may take any nonzero value \(>-1\), allowing also negative values. In contrast to results from the literature, we show in the space-continuous case that mode filtering does not arise for \(p \rightarrow 0\), but for \(p \rightarrow -1\). Extending our filter class to \(p\)-values smaller than \(-1\) allows to include, e.g. the classical image sharpening flow of Gabor. The PDEs we derive in 1D, 2D, and 3D show large structural similarities. Since our PDE class is highly anisotropic and may contain backward parabolic operators, designing adequate numerical methods is difficult. We present an \(L^\infty\)-stable explicit finite difference scheme that satisfies a discrete maximum-minimum principle, offers excellent rotation invariance, and employs a splitting into four fractional steps to allow larger time step sizes. Although it approximates parabolic PDEs, it consequently benefits from stabilisation concepts from the numerics of hyperbolic PDEs. Our 2D experiments show that the PDEs for \(p<1\) are of specific interest: Their backward parabolic term creates favourable sharpening properties, while they appear to maintain the strong shape simplification properties of mean curvature motion.