Existence Of Random Attractor Research Articles

Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping

Abstract The incompressible 2D stochastic Navier-Stokes equations with linear damping are considered in this paper. Based on some new calculation estimates, we obtain the existence of random attractor and the upper semicontinuity of the random attractors as ε → 0 + \varepsilon \to {0}^{+} on the two-dimensional space.

Existence of attractors for stochastic diffusion equations with fractional damping and time-varying delay

This paper deals with the well-posedness and existence of attractors of a class of stochastic diffusion equations with fractional damping and time-varying delay on unbounded domains. We first prove the well-posedness and the existence of a continuous non-autonomous cocycle for the equations and the uniform estimates of solutions and the derivative of the solution operators with respect to the time-varying delay. We then show pullback asymptotic compactness of solutions and the existence of random attractors by utilizing the Arzelà–Ascoli theorem and the uniform estimates for the derivative of the solution operator in the fractional Sobolev space Hα(Rn), with 0 < α < 1.

Attractors for 2D quasi-geostrophic equations with and without colored noise in W2α−,p(ℝ2)

The asymptotic behavior of stochastic modified quasi-geostrophic equations with damping driven by colored noise is analyzed. In fact, the existence of random attractors is established in [Formula: see text] In particular, we prove also the existence of a global compact attractor for autonomous quasi-geostrophic equations with damping in [Formula: see text] Here, we do not add any modifying factor on the nonlinear term.

Random Attractors for 2D Stochastic Hydrodynamical-Type Systems

We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive noise that covers numerous 2D hydrodynamical models, such as the 2D Navier–Stokes equations, 2D Boussinesq equations, 2D MHD equations, etc., and also some 3D models, like the 3D Leray 𝛼-model. We prove the existence of random attractors for the associated continuous random dynamical systems. We also establish the upper semicontinuity of the random attractors as the parameter tends to zero.

A Random Attractor Family of the High Order Beam Equations with White Noise

In this paper, we studied a class of damped high order Beam equation stochas-tic dynamical systems with white noise. First, the Ornstein-Uhlenbeck process is used to transform the equation into a noiseless random equation with random variables as parameters. Secondly, by estimating the solution of the equation, we can obtain the bounded random absorption set. Finally, the isomorphism mapping method and compact embedding theorem are used to obtain the system. It is progressively compact, then we can prove the existence of ran-dom attractors.

Random attractors for the stochastic coupled suspension bridge equations of Kirchhoff type

This paper is devoted to the dynamical behavior of stochastic coupled suspension bridge equations of Kirchhoff type. For the deterministic cases, there are many classical results such as existence and uniqueness of a solution and long-term behavior of solutions. To the best of our knowledge, the existence of random attractors for the stochastic coupled suspension bridge equations of Kirchhoff type is not yet considered. We intend to investigate these problems. We first obtain the dissipativeness of a solution in higher-energy spaces H^{3}(U)times H_{0}^{1}(U)times (H^{2}(U)cap H_{0}^{1}(U))times H_{0}^{1}(U). This implies that the random dynamical system generated by the equation has a random attractor in (H^{2}(U)cap H_{0}^{1}(U))times L^{2}(U) times H_{0}^{1}(U)times L^{2}(U), which is a tempered random set in the space in H^{3}(U)times H_{0}^{1}(U)times (H^{2}(U)cap H_{0} ^{1}(U))times H_{0}^{1}(U).

Upper Semicontinuity of Random Attractors for Nonautonomous Stochastic Reversible Selkov System with Multiplicative Noise

In this paper, the existence of random attractors for nonautonomous stochastic reversible Selkov system with multiplicative noise has been proved through Ornstein-Uhlenbeck transformation. Furthermore, the upper semicontinuity of random attractors is discussed when the intensity of noise approaches zero. The main difficulty is to prove the asymptotic compactness for establishing the existence of tempered pullback random attractor.

Random Attractor Family for a Class of Stochastic Higher-Order Kirchhoff Equations

A class of stochastic dynamical systems with strong damped stochastic higher order Kirchhoff equation solutions with white noise is studied. Firstly, the equation is transformed into a stochastic equation with random variables as parameters and without noise by using Ornstein-Uhlenbeck process. Secondly, the bounded stochastic absorption set is obtained by estimating the solution of the equation. Finally, the stochastic dynamical system is obtained by using the isomorphic mapping method and the compact embedding theorem. It is progressively compact, thus proving the existence of random attractors.

Random Attractors of Stochastic Non-Autonomous Nonclassical Diffusion Equations with Linear Memory on a Bounded Domain

In this article, we discuss the long-time dynamical behavior of the stochastic non-autonomous nonclassical diffusion equations with linear memory and additive white noise in the weak topological space . By decomposition method of the solution, we give the necessary condition of asymptotic compactness of the solutions, and then prove the existence of random attractor, while the time-dependent forcing term only satisfies an integral condition.

Random attractors for stochastic parabolic equations with additive noise in weighted spaces

In this paper, we establish the existence of random attractors for stochastic parabolic equations driven by additive noise as well as deterministic non-autonomous forcing terms in weighted Lebesgue spaces \begin{document}$ L_{\delta}^r(\mathcal{O})$\end{document} , where \begin{document}$ 1 is the distance from \begin{document}$ x$\end{document} to the boundary. The nonlinearity \begin{document}$ f(x,u)$\end{document} of equation depending on the spatial variable does not have the bound on the derivative in \begin{document}$ u$\end{document} , and then causes critical exponent. In both subcritical and critical cases, we get the well-posedness and dissipativeness of the problem under consideration and, by smoothing property of heat semigroup in weighted space, the asymptotical compactness of random dynamical system corresponding to the original system.

Dynamics of the stochastic low concentration trimolecular oscillatory chemical system with jumps

This paper is devoted to discern long time dynamics through the stochastic low concentration trimolecular oscillatory chemical system with jumps. By Lyapunov technique, this system is proved to have a unique global positive solution, and the asymptotic stability in mean square of such model is further established. Moreover, the existence of random attractor and Lyapunov exponents are obtained for the stochastic homeomorphism flow generated by the corresponding global positive solution. And some numerical simulations are given to illustrate the presented results.

Well-posedness and dynamics of a fractional stochastic integro-differential equation

In this paper we investigate the well-posedness and dynamics of a fractional stochastic integro-differential equation describing a reaction process depending on the temperature itself. Existence and uniqueness of solutions of the integro-differential equation is proved by the Lumer–Phillips theorem. Besides, under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity, the existence of random attractor is achieved by obtaining first some a priori estimates. Moreover, the random attractor is shown to have finite Hausdorff dimension.

Random attractors for the three dimensional stochastical planetary geostrophic equations of large-scale ocean circulation

The objection of this paper is to study the asymptotical behavior of solutions for the three dimensional planetary geostrophic equations of large-scale ocean circulation with small additive noise. In this paper, we prove the existence of random attractors for the three dimensional planetary geostrophic equations of large-scale ocean circulation with small additive noise. Furthermore, we prove the random attractor of the three dimensional planetary geostrophic equations of large-scale ocean circulation with small additive noise will be close to the global attractor of the three dimensional planetary geostrophic equations of large-scale ocean circulation for any when the parameter of the perturbation goes to zero.

Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises

In this paper, we study the long-term dynamical behavior of stochastic Boisso nade systems with time-dependent deterministic forces, additive white noise and multiplicative white noise. We first prove the existence of random attrac tor for the considered systems. And then we establish the upper semi-continui ty of random attractors for the systems as the coefficient of quadratic term tends to zero and intensities of the noises approach zero, respectively. At last, we obtain an upper bound of fractal dimension of the random attractors for both systems without quadratic term.

Existence of Random Attractor for Stochastic Fractional Long-Short Wave Equations with Periodic Boundary Condition

We consider the asymptotic behaviors of stochastic fractional long-short equations driven by a random force. Under a priori estimates in the sense of expectation, using Galerkin approximation by the stopping time and the Borel-Cantelli lemma, we prove the existence and uniqueness of solutions. Then a global random attractor and the existence of a stationary measure are obtained via the Birkhoff ergodic theorem and the Chebyshev inequality.

Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients

In this paper, we consider the existence of random attractors in a weighted space $ l_\rho ^2 $ for first-order non-autonomous stochastic lattice system with random coupled coefficients and multiplicative/additive white noise, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Random Attractors for Non-autonomous Stochastic Lattice FitzHugh-Nagumo Systems with Random Coupled Coefficients

In this paper, we study the non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and multiplicative white noise. We consider the existence of random attractors in a weighted space $l_\rho^2 \times l_\rho^2$ for this system, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Existence and Upper Semicontinuity of Attractors for Nonautonomous Stochastic Sine-Gordon Lattice Systems with Random Coupled Coefficients

We study nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise. We first consider the existence of random attractors in a weighted space for this system and then establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Hausdorff Dimension of a Random Attractor for Stochastic Boussinesq Equations with Double Multiplicative White Noises

This paper investigates the existence of random attractor for stochastic Boussinesq equations driven by multiplicative white noises in both the velocity and temperature equations and estimates the Hausdorff dimension of the random attractor.

Random attractor of stochastic Brusselator system with multiplicative noise

Asymptotic dynamics of stochastic Brusselator system with multiplicative noise is investigated in this work. The existence of random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimations than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the attracting regularity to be an $(L^2\times L^2,H^1\times H^1)$ random attractor.