The asymptotic behavior of the stochastic Ginzburg–Landau equation with additive noise

Abstract

We show that the stochastic Ginzburg–Landau equation with additive noise can be solved pathwise and the unique solution generates a random dynamic system. Then we prove that the system possesses a compact random attractor in L²(D) when the spatial dimension of D is one and two, respectively.

Introduction

The Ginzburg–Landau equation arises in various areas of physics and chemistry. It has been studied extensively as a deterministic model [1], [2], [3]. However, some perturbations may neglect in the derivation of this ideal model (such as molecular collisions in gases and liquids and electric fluctuations in resistors [8]). When considering the perturbations of each microscopic units to the models, which will lead to a very large complex system, people usually represent the micro effects by random perturbations in the dynamics of the macro observables. Thus, it is interesting to study the following stochastic Ginzburg–Landau equation with additive white noise

$$\mathrm{d}u=(\lambda +\mathrm{i}\mu )\mathrm{\Delta }u\mathrm{d}t-(\kappa +\mathrm{i}\beta )|u{|}^{2}u\mathrm{d}t+\gamma u\mathrm{d}t+\sum _{j=1}^m{\phi }_j\mathrm{d}{w}_j(t)\text{,}$$

with boundary-initial conditions

$$u{|}_{\mathrm{\partial }D}=0\text{,}$$

$$u(t_0)=u_0\text{,}$$

where λ, μ, κ, β, γ are real coefficients, $D\subset {\mathbb{R}}^n(n=1\text{,}2)$ is an open bounded set with boundary ∂D sufficiently regular, the functions φ_j, j = 1, … , m are time independent, and w_j(t), j = 1, … , m are independent two-sided real-valued standard wiener processes on a complete probability space $(\Omega \text{,}\mathcal{F}\text{,}P)$.

Our main aim is to study the long time behavior of the stochastic system (1.1), (1.2), (1.3). It is well known that attractors are quite well investigated to describe the long time behavior for deterministic systems (see [1]). But all analysis breaks down as soon as one wants to take random influences on the system under investigation into account. In particular, when subjecting the system to additive white noise, there is no chance that bounded subsets of the state space remain invariant. White noise pushes the system out of every bounded set with probability one. Recently, the notion of random attractors for a stochastic dynamical system has been introduced in [4], [5]. Random attractors are in fact compact invariant sets – however, they are not fixed, but they depend on chance, and they move with time (in a coherent, “stationary” manner). It seems to be a good generalization of the classical concept of global attractors for deterministic dynamical systems, and it has been successfully applied to many infinite dimensional stochastic dynamical systems (see [4], [5], [6], [9]).

In [6], the author studied the Ginzburg–Landau equation perturbed by multiplicative white noise

$$\mathrm{d}u=(\lambda +\mathrm{i}\mu )\mathrm{\Delta }u\mathrm{d}t-(\kappa +\mathrm{i}\beta )|u{|}^{2}u\mathrm{d}t+\gamma u\mathrm{d}t+\sigma u\mathrm{d}w(t)\text{.}$$

Under the conditions that w(t) is a two-sided standard wiener process and σ is a positive constant, he obtained the existence of the random attractor in spacial dimension one and two.

Although the main conceptual line in proving the existence of global attractors in the additive noise case (in this paper) and the multiplicative case (see [6]) is similar, due to the difference of the translating variants in the two cases, we have to deal with spacial dimension one and two respectively at technical level in the present paper, and we also obtain the existence of the random attractors.

The paper is arranged as follows. In Section 2, we present some preliminaries of random dynamical system (RDS) and the solution of (1.1), (1.2), (1.3), and get the corresponding RDS. In Section 3, we prove the existence of the random attractors in dimension one and two, respectively.