Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations

Anhui Gu, Kening Lu, Bixiang Wang
2019 Discrete and Continuous Dynamical Systems. Series A  
In this paper, we investigate the asymptotic behavior of the solutions of the two-dimensional stochastic Navier-Stokes equations via the stationary Wong-Zakai approximations given by the Wiener shift. We prove the existence and uniqueness of tempered pullback attractors for the random equations of the Wong-Zakai approximations with a Lipschitz continuous diffusion term. Under certain conditions, we also prove the convergence of solutions and random attractors of the approximate equations when
more » ... te equations when the step size of approximations approaches zero. 185 ANHUI GU, KENING LU AND BIXIANG WANG We 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 for Wong-Zakai approximations. 2.1. Definition of continuous cocycles. In this section, we will define a continuous cocycle for the random Navier-Stokes equations when the white noise in (1) is replaced by a Wong-Zakai approximation. To describe such an approximation, we consider the canonical probability space (Ω, F, P), where Ω = C 0 (R, R) := {ω ∈ C(R, R) : ω(0) = 0} with the open compact topology, F is its Borel σ-algebra, and P is the Wiener measure. The Brownian motion has the form W (t, ω) = ω(t). Consider the Wiener
doi:10.3934/dcds.2019008 fatcat:tvatgzelljc3flz7k3oeexzfhq