Asymptotic Compactness Research Articles

Existence and upper semicontinuity of random pullback attractors for 2D and 3D non-autonomous stochastic convective Brinkman-Forchheimer equations on whole space

In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counterpart with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u} + (\boldsymbol{u}\cdot\nabla)\boldsymbol{u} + \alpha\boldsymbol{u} + \beta|\boldsymbol{u}|^{r-1} \boldsymbol{u} + \nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0, $$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$ and $d=3$ with $r > 3$ (for any $\beta,\mu > 0$), and $d=r=3$ with $2\beta\mu\geq1$. For the same cases, we show the existence of a unique random pullback attractor for non-autonomous stochastic CBF equations with linear multiplicative white noise. In addition, a remark on the existence of a unique weak pullback mean random attractor for the stochastic CBF equations is also provided. Finally, we establish the upper semicontinuity of the random pullback attractors, that is, the random pullback attractors converge towards the global pullback attractor when the noise intensity approaches to zero. Since we do not have compact Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of the random dynamical system is proved by the method of energy equations given by Ball. For the case of Navier-Stokes equations defined on $\mathbb{R}^d$, such results are not available and the presence of Darcy term $\alpha\boldsymbol{u}$ helps us to establish the above mentioned results for CBF equations on whole space.

Existence of Random Attractors for a Stochastic Strongly Damped Plate Equations with Multiplicative Noise

In this article, we study the asymptotic dynamics of a stochastic strongly damped plate system with homogeneous Neumann boundary conditions and multiplicative noise. First, we investigate the existence and uniqueness of solutions in infinite-dimensional dynamical systems using the notion of mild solutions, and then we examine the presence of a bounded absorbing set. Finally, we investigate the asymptotic compactness by using the decomposition technique to prove the existence of a random attractor.

Forward dynamics of 3D double time-delayed MHD-Voight equations

This paper is concerned with the long-term behaviour of 3D double time-delayed magnetohydrodynamics (MHD)-Voight equations with non-autonomous forcing term. First, we establish the existence, forward compactness and forward stability of pullback attractors by the forward equi-continuity and forward pullback asymptotic compactness of solutions. We then study the upper semi-convergence of pullback attractors as the time parameter tends to positive infinity. Since the solution of this equation has no high regularity, we use the spectrum decomposition technique to prove the forward pointwise pullback asymptotic compactness of solutions.

Dynamical behavior of Benjamin-Bona-Mahony system with finite distributed delay in 3D

<abstract><p>We study the Benjamin-Bona-Mahony model with finite distributed delay in 3D, which depicts the dispersive impact of long waves. Based on the well-posedness of model, the family of pullback attractors for the evolutionary processes generated by a global weak solution has been obtained, which is unique and minimal, via verifying asymptotic compactness in functional space with delay $ C_V $ and topological space $ V\times C_V $, where the energy equation method and a retarded Gronwall inequality are utilized.</p></abstract>

Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the solutions to a class of non-autonomous <i>fractional</i> stochastic <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian equation driven by linear additive noise on the entire space <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. We firstly prove the existence of a continuous non-autonomous cocycle for the equation as well as the uniform estimates of solutions. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains. Finally, we establish the upper semi-continuity of the random attractors when noise intensity approaches zero.</p>

Dynamics of stochastic 3D Brinkman-Forchheimer equations on unbounded domains

<abstract><p>This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations.</p></abstract>

Dynamics for subcritical fractional nonclassical diffusion equations with nonlinear Wong-Zakai noise and delays

The multi-scale stability of random attractors in the space of continuous functions from the delay interval into the fractional Sobolev space for a class of nonclassical diffusion equations on $ \mathbb{R}^n $ with time-delay and nonlinear Wong-Zakai noise is studied. First, we prove the existence of a unique pullback random attractor, which is a family of compact sets depending on two parameters: the step size of noise and the current time. Secondly, we study the semicontinuity of the pullback random attractor as the current time goes to positive infinity. Thirdly, we consider the semicontinuity of the pullback random attractor as the delay time approach zero. Finally, we prove the semicontinuity of the pullback random attractor as the step size of noise tends to positive infinity. In order to overcome the lack of compact Sobolev embeddings on $ \mathbb{R}^n $ and the weak dissipation of equations, we employ the spectral method and idea of uniform tail-estimates to prove that the solution operator is pullback asymptotic compactness over a time-uniformly tempered attracting universe.

Asymptotic behavior of non-autonomous fractional stochastic <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian equations with colored noise and delay on <inline-formula><tex-math id="M2">$ \mathbb{R}^n $</tex-math></inline-formula>

This paper deals with the asymptotic behavior of the solutions to a class of non-autonomous fractional stochastic $ p $-Laplacian equations with delay driven by nonlinear colored noise on the entire space $ \mathbb{R}^n $. We firstly considered the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the time delay. We then showed pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains.

Asymptotic Behavior of Random Reaction-diffusion Delay Equations Driven by Colored Noise

In this paper, we consider the asymptotic behavior of random reaction-diffusion delay equations driven by colored noise defined on unbounded domains. We firstly establish the existence and uniqueness of pullback random attractors for the continuous cocycle associated with the equation, and then consider the convergence of random attractors when colored noise approximates white noise. The methods of uniform tail-estimate and operator decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embeddings on unbounded domains.

Pullback D$$ \mathcal{D} $$‐attractors for doubly nonlinear parabolic equations

In this paper, we study a class of doubly nonlinear parabolic equations (1.1). The nonlinearity brings great difficulties to the study of the problem. First, we show that the problem has a unique solution. Then, we prove that the process corresponding to the problem is norm‐to‐weak continuous. After that, by using Legendre transform, we obtain uniform estimates and asymptotic compactness properties that allow us to ensure the existence of pullback ‐attractors for the associated process to the problem.

Asymptotically autonomous dynamics for fractional subcritical nonclassical diffusion equations driven by nonlinear colored noise

The asymptotically autonomous dynamics in $$H^s(\mathbb R^n)$$ for any $$s\in (0,1)$$ and $$n\in \mathbb {N}$$ are discussed for a class of highly nonlinear nonclassical diffusion equations perturbed by colored noise. The main feature of this model is that the time-dependent drift and diffusion terms have subcritical and superlinear polynomial growth of orders p and q, respectively, satisfying $$\begin{aligned} 2\le 2q<p<\infty \text { if }n=1\text { and }s\in [\frac{1}{2},1);\text { otherwise, }\, 2\le 2q<p<\frac{2n}{n-2s}. \end{aligned}$$ The number $$\frac{2n}{n-2s}$$ is called the fractional critical Sobolev embedding exponent. Under this setting, we prove the existence, time-dependent uniform compactness and asymptotically autonomous convergence of pathwise random attractors of the equations in $$H^s(\mathbb R^n)$$ when the time-dependent nonlinearities satisfy some new conditions. The time-dependent uniform pullback asymptotical compactness of the solution operators in $$H^s(\mathbb R^n)$$ is proved by virtue of a cut-off technique [38], a spectral decomposition approach and uniform estimates in a time-uniformly tempered attracting universe in order to overcome several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains, the weak dissipativeness of the systems and the unknown measurability of attractors.

Global attractor for a degenerate Klein–Gordon–Schrödinger-type system

In this paper, we study the long-time behaviour of solutions of a degenerate Klein–Gordon–Schrödinger-type system which is defined in a bounded domain. First, we proved the existence, uniqueness and continuity of the solutions on the initial data, then the asymptotic compactness of the solutions and finally the existence of a global compact attractor.

Invariant measures for stochastic 3D Lagrangian-averaged Navier–Stokes equations with infinite delay

In this paper we investigate stochastic dynamics and invariant measures for stochastic 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay and additive noise. We first use Galerkin approximations, a priori estimates and the standard Gronwall lemma to show the well-posedness for the corresponding random equation, whose solution operators generate a random dynamical system. Next, the asymptotic compactness for the random dynamical system is established via the Ascoli–Arzelà theorem. Besides, we derive the existence of a global random attractor for the random dynamical system. Moreover, we prove that the random dynamical system is bounded and continuous with respect to the initial values. Eventually, we construct a family of invariant Borel probability measures, which is supported by the global random attractor.

Dynamical stability of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise

In this paper, two problems related to FitzHugh–Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation time of Wong–Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delayed lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli–Arzelà theorem and uniform tail-estimates. We then show the upper semicontinuity of attractors as the correlation time tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model and study the convergence of attractors when the delay approaches zero. That is, the upper semicontinuity of attractors for the delayed system to the non-delayed one is proved.

Limiting dynamics of stochastic heat equations with memory on thin domains

This paper is concerned with the limiting behavior of a stochastic integro-differential equation driven by additive noise defined on thin domains. We prove the existence and uniqueness of random attractors for the equation in an -dimensional narrow domain. We also establish the upper-semicontinuity of these attractors when a family of -dimensional thin domains collapses onto an n-dimensional domain. The main difficulty of this paper is the non-compactness of the generated RDS based on the fact that the memory term includes the whole past history of the phenomenon. To solve this, a splitting method is employed to prove the asymptotic compactness.

The Long-Time Behavior of 2D Nonautonomous g -Navier-Stokes Equations with Weak Dampness and Time Delay

In this paper, we discuss the long-time behavior of g -Navier-Stokes equations with weak dampnesss and time delay. The uniformly attracting sets of processes are obtained. On the basis of the method with asymptotic compactness, the existence of the uniform attractor for the equation is proved with the restriction of the forcing term belonging to translational compacted function space.

Backward regularity of attractors for lattice FitzHugh-Nagumo system with double random coefficients

In this paper, we investigate the backward compact dynamical behavior for the stochastic lattice FitzHugh-Nagumo system with double random coefficients and multiplicative noise. It is proved, under some suitable conditions on the nonlinear term and the body force, that the system has a unique backward compact bi-spatial random (ℓσ2×ℓσ2,ℓσp×ℓσ2)-attractor for any p>2. By such a bi-spatial attractor we mean an invariant backward compact set in ℓσp×ℓσ2 that pullback attracts all nonempty subsets of ℓσ2×ℓσ2 under another topology of ℓσp×ℓσ2. The time-dependent uniform pullback asymptotical compactness of the solution operators in ℓσp×ℓσ2 is proved by virtue of a cut-off technique and considering two different attracting universes to overcome several difficulties caused by the lack of compact Sobolev embeddings in the case of a lattice system and the unknown measurability of random attractor.

Well-posedness and long term behavior of supercritical wave equations driven by nonlinear colored noise on [formula omitted

This paper is concerned with the well-posedness and long term behavior of the non-autonomous random wave equations driven by nonlinear colored noise on Rn with n≤5. The drift nonlinearity has a supercritical growth exponent (n+2)/(n−2). We first prove the existence and uniqueness of solutions in the energy space by showing the non-concentration of energy via the Morawetz identity and the uniform Strichartz estimates. We then prove the existence and uniqueness of tempered pullback random attractors of the non-autonomous random dynamical system associated with the equation. The asymptotic compactness of solutions is obtained by the idea of energy equation due to Ball and the uniform tail-ends estimates in order to circumvent the difficulty caused by the lack of compactness of Sobolev embeddings on unbounded domains.

Ergodicity for the Randomly Forced Navier–Stokes System in a Two-Dimensional Unbounded Domain

The ergodic properties of the randomly forced Navier–Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier–Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers Kuksin et al. (Geom Funct Anal 30(1):126–187, 2020) and Shirikyan (J Eur Math Soc, 2020) and using the asymptotic compactness of the dynamics.

The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

Abstract In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher-order Kirchhoff equation with variable coefficients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of D k {{\mathcal{D}}}_{k} -absorbing sets of the stochastic dynamical system Φ k {\Phi }_{k} is obtained, and the asymptotic compactness of Φ k {\Phi }_{k} is proved by the decomposition method. Finally, the D k {{\mathcal{D}}}_{k} -stochastic attractor family of the stochastic dynamical system Φ k {\Phi }_{k} in V m + k ( Ω ) × V k ( Ω ) {V}_{m+k}\left(\Omega )\times {V}_{k}\left(\Omega ) is obtained.