Recent zbMATH articles in MSC 37Lhttps://www.zbmath.org/atom/cc/37L2022-05-16T20:40:13.078697ZWerkzeugAsymptotic autonomy of bi-spatial attractors for stochastic retarded Navier-Stokes equationshttps://www.zbmath.org/1483.350442022-05-16T20:40:13.078697Z"Zhang, Qiangheng"https://www.zbmath.org/authors/?q=ai:zhang.qiangheng"Li, Yangrong"https://www.zbmath.org/authors/?q=ai:li.yangrongSummary: We establish semi-convergence of a non-autonomous bi-spatial random attractor towards to an autonomous attractor under the topology of the regular space when time-parameter goes to infinity, where the criteria are given by forward compactness of the attractor in the terminal space as well as forward convergence of the random dynamical system in the initial space. We then apply to both non-autonomous and autonomous stochastic 2D Navier-Stokes equations with general delays (including variable and distribution delays). The forward-pullback asymptotic compactness in the space of continuous Sobolev-valued functions is proved by the method of spectrum decomposition.A determining form for the 2D Rayleigh-Bénard problemhttps://www.zbmath.org/1483.351602022-05-16T20:40:13.078697Z"Cao, Yu"https://www.zbmath.org/authors/?q=ai:cao.yu"Jolly, Michael S."https://www.zbmath.org/authors/?q=ai:jolly.michael-s"Titi, Edriss S."https://www.zbmath.org/authors/?q=ai:titi.edriss-salehSummary: We construct a determining form for the 2D Rayleigh-Bénard (RB) system in a strip with solid horizontal boundaries, in the cases of no-slip and stress-free boundary conditions. The determining form is an ODE in a Banach space of trajectories whose steady states comprise the long-time dynamics of the RB system. In fact, solutions on the global attractor of the RB system can be further identified through the zeros of a scalar equation to which the ODE reduces for each initial trajectory. The twist in this work is that the trajectories are for the velocity field only, which in turn determines the corresponding trajectories of the temperature.Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in \(\mathbb{R}^2\)https://www.zbmath.org/1483.351922022-05-16T20:40:13.078697Z"Alouini, Brahim"https://www.zbmath.org/authors/?q=ai:alouini.brahimSummary: We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in \(\mathbb{R}^2\) that reads
\[
u_t +i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right) +ig(|u|^2)u+\gamma u = f\, ,\quad (t,(x,y))\in \mathbb{R}\times \mathbb{R}^2.
\]
We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.Scattering in the weighted \( L^2 \)-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearityhttps://www.zbmath.org/1483.351982022-05-16T20:40:13.078697Z"Bensouilah, Abdelwahab"https://www.zbmath.org/authors/?q=ai:bensouilah.abdelwahab"Dinh, Van Duong"https://www.zbmath.org/authors/?q=ai:dinh.van-duong"Majdoub, Mohamed"https://www.zbmath.org/authors/?q=ai:majdoub.mohamedSummary: We investigate the defocusing inhomogeneous nonlinear Schrödinger equation
\[ i \partial_tu + \Delta u = |x|^{-b} \left(\mathrm{e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0) = u_0, \quad x \in \mathbb{R}^2, \]
with \( 0<b<1 \) and \( \alpha = 2\pi(2-b) \). First we show the decay of global solutions by assuming that the initial data \( u_0 \) belongs to the weighted space \( \Sigma(\mathbb{R}^2) = \{u\in H^1(\mathbb{R}^2) : |x|u\in L^2(\mathbb{R}^2)\} \). Then we combine the local theory with the decay estimate to obtain scattering in \( \Sigma \) when the Hamiltonian is below the value \( \frac{2}{(1+b)(2-b)} \).Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLShttps://www.zbmath.org/1483.352102022-05-16T20:40:13.078697Z"Genovese, Giuseppe"https://www.zbmath.org/authors/?q=ai:genovese.giuseppe"Lucà, Renato"https://www.zbmath.org/authors/?q=ai:luca.renato"Tzvetkov, Nikolay"https://www.zbmath.org/authors/?q=ai:tzvetkov.nikolaySummary: The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on \(L^2(\mathbb{T})\) with covariance \([1+(-\Delta)^s]^{- 1}\) under these transformations for any \(s > \frac{1}{2}\). This extends previous achievements by \textit{A. R. Nahmod} et al. [Math. Res. Lett. 18, No. 5, 875--887 (2011; Zbl 1250.60018)] and the first author et al. [Math. Ann. 374, No. 3--4, 1075--1138 (2019; Zbl 1420.35354)], who proved the result for integer values of the regularity parameter \(s\).The essential spectrum of periodically stationary solutions of the complex Ginzburg-Landau equationhttps://www.zbmath.org/1483.352342022-05-16T20:40:13.078697Z"Zweck, John"https://www.zbmath.org/authors/?q=ai:zweck.john-w"Latushkin, Yuri"https://www.zbmath.org/authors/?q=ai:latushkin.yuri"Marzuola, Jeremy L."https://www.zbmath.org/authors/?q=ai:marzuola.jeremy-l"Jones, Christopher K. R. T."https://www.zbmath.org/authors/?q=ai:jones.christopher-k-r-tSummary: We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically stationary pulses in ultrafast fiber lasers.Optimal control of Clarke subdifferential type fractional differential inclusion with non-instantaneous impulses driven by Poisson jumps and its topological propertieshttps://www.zbmath.org/1483.370942022-05-16T20:40:13.078697Z"Durga, N."https://www.zbmath.org/authors/?q=ai:durga.nagarajan"Muthukumar, P."https://www.zbmath.org/authors/?q=ai:muthukumar.palanisamySummary: This article is devoted to studying the topological structure of a solution set for Clarke subdifferential type fractional non-instantaneous impulsive differential inclusion driven by Poisson jumps. Initially, for proving the solvability result, we use a nonlinear alternative of Leray-Schauder fixed point theorem, Gronwall inequality, stochastic analysis, a measure of noncompactness, and the multivalued analysis. Furthermore, the mild solution set for the proposed problem is demonstrated with nonemptyness, compactness, and, moreover, \(R_\delta\)-set. By employing Balder's theorem, the existence of optimal control is derived. At last, an application is provided to validate the developed theoretical results.Random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations on unbounded domainshttps://www.zbmath.org/1483.370952022-05-16T20:40:13.078697Z"Shu, Ji"https://www.zbmath.org/authors/?q=ai:shu.ji"Zhang, Jian"https://www.zbmath.org/authors/?q=ai:zhang.jianSummary: This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with \(\alpha\in(0,1)\). We prove the existence and uniqueness of tempered pullback random attractors for the equations in \(L^2(\mathbf{R}^3)\). In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in \(L^2(\mathbf{R}^3)\) by the tail-estimates of solutions.Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equations on unbounded domainshttps://www.zbmath.org/1483.370962022-05-16T20:40:13.078697Z"Sun, Yaqing"https://www.zbmath.org/authors/?q=ai:sun.yaqing"Gao, Hongjun"https://www.zbmath.org/authors/?q=ai:gao.hongjunSummary: In this paper, we investigate the Wong-Zakai approximations induced by a stationary process and the long term behavior of the fractional stochastic reaction-diffusion equation driven by a white noise. Precisely, one of the main ingredients in this paper is to establish the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of fractional stochastic reaction-diffusion equations. Thereafter the upper semi-continuity of attractors for the Wong-Zakai approximation of the equation as \(\delta\rightarrow0\) is proved.Forward-backward approximation of nonlinear semigroups in finite and infinite horizonhttps://www.zbmath.org/1483.470882022-05-16T20:40:13.078697Z"Contreras, Andrés"https://www.zbmath.org/authors/?q=ai:contreras.andres-a"Peypouquet, Juan"https://www.zbmath.org/authors/?q=ai:peypouquet.juanThe authors consider the problem
\[
\begin{aligned}
-&\dot{u}(t)\in\left( A+B\right) u(t) \text{ for a.e. }t>0,\\
&u(0)=u_{0}\in D(A),
\end{aligned}
\]
in a class of Banach spaces, where \(A\) is \(m\)-accretive and \(B\) is coercive. First, the approximation of solutions is investigated. Solutions are approximated by trajectories constructed by interpolation of sequences generated using forward-backward iteration and these are shown to converge uniformly on a finite time interval, proving existence and uniqueness of solutions. Second, asymptotic equivalence results are given that connect the behaviour of forward-backward iterations as the number of iterations goes to infinity with the behaviour of the solution as time goes to infinity, for step sizes that are sufficiently small. These results are based on a certain inequality which the authors trace back to \textit{E. Hille} [Fysiogr. Sällsk. Lund Förh. 21, No. 14, 130--142 (1951; Zbl 0044.32902)].
Reviewer: Daniel C. Biles (Nashville)