Non-autonomous Random Dynamical System Research Articles

Stability of regular attractors for non-autonomous random dynamical systems and applications to stochastic Newton–Boussinesq equations with delays

In this paper, we establish theoretical results on the stability of random regular attractors. First, we introduce a backward regular attractor, which is a new type of attractor defined by a minimal backward pullback attracting set. We then establish an existence theorem for such an attractor, and prove it is long time stable. Eventually, we prove the long time stability of regular pullback random attractors. As an application, we consider stochastic non-autonomous Newton–Boussinesq equations with variable and distributed delays. Since solutions of the equations have no higher regularity, we prove their regular asymptotic compactness via the spectrum decomposition technique.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

We prove some conditions for bounding the fractal dimension of random invariant sets of non-autonomous random dynamical systems on separable Banach spaces. Then we apply these conditions to prove the finiteness of fractal dimension of random attractor for stochastic 2D hydrodynamical type equations with linear additive white noise in bounded domains or unbounded domains satisfying the Poincaré inequality.

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

AbstractThis paper is mainly concerned with the long‐term random dynamics for the nonautonomous 3D globally modified Navier–Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville‐type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero.

Asymptotic behavior of plate equations driven by colored noise on unbounded domains

This paper investigates mainly the asymptotic behavior of the nonautonomous random dynamical systems generated by the plate equations driven by colored noise defined on mathbb{R}^{n}. First, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.

Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise

<p style='text-indent:20px;'>We mainly consider the existence of random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise. We first give an existence criterion for a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) defined on the space of infinite sequences with complex-valued components. Then we transfer the considered stochastic Schrödinger lattice system into a random system with random parameters and quasi-periodic forces without white noise through Ornstein-Uhlenbeck process, whose solutions generate a jointly continuous NRDS. Thirdly, we prove the existence of a uniform absorbing random set for this NRDS. Fourthly, we construct a bounded closed random set and verify the Lipschitz continuity of the NRDS on this random set, and next we decompose the difference between two solutions into a sum of two parts including some random variables with bounded expectation. Finally, we obtain the existence of a random uniform exponential attractor for the considered system.</p>

Random attractors and invariant measures for 3D stochastic globally modified Navier-Stokes equations with time-dependent delay and coefficient

In this article, we investigate the dynamics of three-dimensional, non-autonomous, and globally modified Navier-Stokes equations with time-dependent delay and additive noise. Based on the well-posedness of solutions, we demonstrate that the solution operators of such equations generate a non-autonomous random dynamical system. We further prove the existence and uniqueness of random attractors as well as the periodicity and the upper semicontinuity for this random dynamical system. We also prove the existence of invariant measures supported on the random attractors. Moreover, we rigorously find that every limit of a sequence of invariant measures must be an invariant measure of the corresponding limiting equations as the noise intensity $ \varepsilon\rightarrow0 $ or the modification parameter $ N\rightarrow N_0\in (0, +\infty) $.

Random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice systems in weighted space

<abstract><p>We mainly study the existence of random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice system with multiplicative white noise and quasi-periodic forces in weighted spaces. Firstly, the stochastic Schrödinger system is transformed into a random system without white noise by the Ornstein-Uhlenbeck process, whose solution generates a jointly continuous non-autonomous random dynamical system $ \Phi $. Secondly, we prove the existence of a uniform absorbing random set for $ \Phi $ in weighted spaces. Finally, we obtain the existence of a random uniform exponential attractor for the considered system $ \Phi $ in weighted space.</p></abstract>

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.

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.

Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equations

<p style='text-indent:20px;'>A non-autonomous random dynamical system is called to be controllable if there is a pullback random attractor (PRA) such that each fibre of the PRA converges upper semi-continuously to a nonempty compact set (called a controller) as the time-parameter goes to minus infinity, while the PRA is called to be asymptotically autonomous if there is a random attractor for another (autonomous) random dynamical system as a controller. We establish the criteria for ensuring the existence of the minimal controller and the asymptotic autonomy of a PRA respectively. The abstract results are illustrated in possibly non-autonomous stochastic p-Laplace lattice equations with tempered convergent external forces.</p>

Multi-valued random dynamics of stochastic wave equations with infinite delays

<p style='text-indent:20px;'>This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.</p>

Asymptotic behavior of plate equations with memory driven by colored noise on unbounded domains

<abstract><p>The paper investigates mainly the asymptotic behavior of the non-autonomous random dynamical systems generated by the plate equations with memory driven by colored noise defined on $ \mathbb{R}^n $. Firstly, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.</p></abstract>

Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in L^2 is established by the squeezing approach and that in H_0^1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

Random attractor and exponentially stability for the second order nonautonomous retarded lattice systems with multiplicative white noise

In this paper, we first present some conditions for the existence of a random attractor of continuous non-autonomous random dynamical systems (cocycle) and locally coupled stochastic retarded lattice dynamical systems in a space of continuous functions from finite closed interval into a product space of finite weighted spaces of infinite sequences. Then we consider the existence, singleton and exponentially stability of the random attractor for the second order non-autonomous retarded lattice systems with multiplicative white noise.

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

Morse decompositions of uniform random attractors

In this paper, we study the Morse theory on uniform random attractors for non-autonomous random dynamical systems. To handle the non-invariance of uniform random attractors, we construct the Morse sets of uniform random attractor as the projections of the Morse sets on pullback attractor of the skew product semiflow. First, we obtain the Morse decomposition of uniform random attractors in probability. Second, we describe a decaying energy level on such attractor by Lyapunov function with probability one. Furthermore, we study the stability of Morse decompositions of deterministic uniform attractor under a small random disturbance. Finally, we discuss and collect the results on the Morse decomposition under random setting.

Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise

<p style='text-indent:20px;'>This paper is concerned with the pullback random attractors of nonautonomous nonlocal fractional stochastic <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian equation with delay driven by multiplicative white noise defined on bounded domain. We first prove the existence of a continuous nonautonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise. We then show pullback asymptotical compactness of solutions and the existence of tempered random attractors by utilizing the Arzela-Ascoli theorem and appropriate uniform estimates of the solutions. Finally, we establish the upper semicontinuity of the random attractors when time delay approaches zero.</p>

Random pullback and uniform exponential attractors for stochastic non-autonomous Navier-Stokes equations

We first prove the existence of a random pullback exponential attractor for the 2D Navier-Stokes equation with additive noise and non-autonomous forcing. Then, we introduce the definition and existence conditions of a random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system, and prove the existence of a random uniform exponential attractor for the 2D Navier-Stokes equation with additive noise and quasi-periodic external forces.

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.

Random Exponential Attractor for the 3D Non-autonomous Stochastic Damped Navier–Stokes Equation

In this paper, we prove the existence of a random exponential attractor (a positively invariant, compact, random set with finite fractal dimension that attracts any trajectory exponentially) for the 3D non-autonomous damped Navier–Stokes equation with additive noise, which implies that the asymptotic behavior of solutions for the equation can be described by finite independent parameters. The key and difficult point of this proof lies in proving the Lipschitz continuity and the random squeezing property in mean sense for the non-autonomous random dynamical system generated by solutions of the equation.