Long term behavior of random Navier-Stokes equations driven by colored noise

Abstract

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,

Random dynamics of fractional nonclassical diffusion equations driven by colored noise

“…We remark that for stochastic PDEs with standard Laplacian, random attractors have been investigated in [8,12,13,14,15,18,20,21,23,31,35,36,47,48,53,57,58] for the autonomous case, and in [1,9,10,24,25,19,30,33,34,37,49,62,63] for the non-autonomous case. The reader is referred to [26,28,38,55] for attractors of random systems with standard Laplacian driven by colored noise or approximations of white noise.…”

“…where z δ (θ t ω) = (I + (−∆) s ) −1 hy δ (θ t ω) with y δ (θ t ω) = e −t t −∞ e s ζ δ (θ s ω)ds being the stationary solution of the random equation: dy + ydt = ζ δ (θ t ω). By [26], we have…”

Convergence and center manifolds for differential equations driven by colored noise

“…For any T > 0, in what follows, we shall show that the solutions of equation ( 16) converge in mean square to the solutions of equation (12) uniformly on [0, T ] as δ → 0 + . From now on, we use K to denote a generic positive constant whose value may change from line to line, but does not depend on δ.…”

“…The colored noise has been widely used in physics and biology to study the dynamical behavior of solutions of random systems, see, e.g., [10,20,23,43,36,45] and the references therein. Recently, the attractors and invariant manifolds of random differential equations driven by additive or linear multiplicative colored noise have been studied in [11,12] and [18,19], respectively.…”

“…du = f (u)dt + σ(u) • dW, u(0) = x, x ∈ R n (12) and its corresponding equation driven by colored noise of the forṁ…”