In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system driven by colored noise with a nonlinear diffusion term. We demonstrate that the colored noise is much easier to deal with than the white noise for studying the pathwise dynamics of stochastic systems. In addition, we show the attractors of the random FitzHugh-Nagumo system driven by a linear multiplicative colored noise converge to that of the corresponding stochastic system driven by a linear multiplicative white noise.
This paper is devoted to the study of long term behavior of the twodimensional random Navier-Stokes equations driven by colored noise defined in bounded and unbounded domains. We prove the existence and uniqueness of pullback random attractors for the equations with Lipschitz diffusion terms. In the case of additive noise, we show the upper semi-continuity of these attractors when the correlation time of the colored noise approaches zero. When the equations are defined on unbounded domains, we establish the pullback asymptotic compactness of the solutions by Ball's idea of energy equations in order to overcome the difficulty introduced by the noncompactness of Sobolev embeddings.Given τ ∈ R and ω ∈ Ω, consider the following non-autonomous two-dimensional random Navier-Stokes equations driven by colored noise:along with homogeneous Dirichlet boundary condition, where ν is a positive constant, f is a given function defined on R × O, G is a nonlinear function satisfying certain conditions, and ζ δ (θ t ω) is the colored noise with correlation time δ > 0.The Navier-Stokes equations are used to model the flow of an incompressible viscous fluid of constant density enclosed in a region O with rigid boundary ∂O, where u(t, x) ∈ R 2 and p(t, x) ∈ R are, respectively, the velocity and the pressure of the fluid at point x ∈ O and time t > τ , ν > 0 is the kinematic viscosity of the fluid and f = f (t, x) ∈ R 2 is the time dependent external force. The existence of global attractors for the Navier-Stokes equations has been extensively studied in the literature, see, e.g., [5,7,12,13,19,34,46,48,51] for the deterministic case, and [11,22,24,56] for the stochastic case. More work on random attractors can be found in [8,9,14,15,16,17,20,21,22,24,25,26,27,29,37,42,47,53,54] for the autonomous stochastic equations; and in [18,23,30,31,55,57,58] for the nonautonomous stochastic systems. In this paper, we will investigate pullback random attractors of the non-autonomous random system (2) driven by colored noise.The random system (2) can be considered as an approximation of the following Stratonovich stochastic equations: ∂u ∂t − ν∆u + (u · ∇)u = f (t, x) − ∇p + G(t, x, u) • dW dt , x ∈ Q, t > τ, div u = 0, x ∈ Q, t > τ,
185 186 ANHUI GU, KENING LU AND BIXIANG WANGWe are interested in the existence of random attractors for equation (1) when R is a nonlinear Lipschitz continuous function, for which we need to define a random dynamical system via the solution operators because the attractors theory of stochastic equations is formulated in terms of random dynamical systems. However, for a general Lipschitz nonlinearity R, the existence of such a random dynamical system for (1) is unknown (see, e.g., [22]), and hence the random attractors theory does not apply in this case directly. As far as the authors are aware, the existence of random attractors for the autonomous version of (1) was proved only for a special form of R and is still open for a general Lipschitz diffusion term (see, e.g., [8,18,21]). In order to solve this problem, in the present paper, we propose to study the pathwise dynamics of the stochastic equation (1) by the Wong-Zakai approximations defined as a stationary process given by the Wiener shift (see Section 2 for details). Indeed, as we will see later, the corresponding random equations driven by the Wong-Zakai approximations generate a random dynamical system for a general Lipschitz diffusion term R, and have a unique tempered random attractor (see Theorem 2.3). This indicates that random equations with pathwise approximations are more amenable to analysis than stochastic equations driven by white noise, as far as pathwise dynamics is concerned. Furthermore, we will prove the solutions of the approximate equations converge to that of the corresponding stochastic equations driven by linear multiplicative noise or additive noise when the step size of the approximations tends to zero. The continuity of random attractors for the approximate equations will also be established in these cases.The Wong-Zakai approximations were first proposed in [61,62] where the authors developed the idea of using pathwise deterministic equations to approximate stochastic ones driven by one-dimensional Brownian motions. Currently, the Wong-Zakai approximations have been extended to higher-dimensional Brownian motions as well as martingales and semimartingales, see, e.g.], and the references therein. In the present paper, we will use the idea of Wong-Zakai approximations to study the existence and uniqueness of tempered random attractors of (1). Such attractors have been extensively studied in the literature, see [4,5,9,11,12,13,17,18,21,24,25,26,27,33,37,49,51,57] for autonomous stochastic equations; and [14, 20, 58, 59, 60] for non-autonomous stochastic equations. In this paper, we will deal with the non-autonomous stochastic equations (1).In the next section, we prove the existence and uniqueness of random attractors for the Navier-Stokes equations driven by the Wong-Zakai approximations. In the last two sections, we prove the convergence of solutions and attractors of the approximate equations when the step size of approximations approaches zero, for linear multiplicative noise and additive noise, respectively. 2.Random attractors ...
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