WONG-ZAKAI APPROXIMATIONS AND ASYMPTOTIC BEHAVIOR OF STOCHASTIC GINZBURG-LANDAU EQUATIONS

. In this paper we discuss the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for stochastic Ginzburg-Landau equations driven by a white noise. We ﬁrst apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions. Consequently, we show that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. At last, when the stochastic Ginzburg-Landau equation is driven by a linear multiplicative noise, we estab- lish the convergence of solutions of Wong-Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.


1.
Introduction. The generalized complex Ginzburg-Landau equation is one of the most important equations in mathematical physics, which can describe turbulent dynamics and has a long history in physics as a generic amplitude equation near the onset of instabilities in fluid mechanical systems, as well as in the theory of phase transitions and superconductivity [3,9]. The global existence and long time behavior of Ginzburg-Landau equation were studied in [10,17,30]. The random attractors of stochastic Ginzburg-Landau equations were got in [29,28,31,44].
In this paper we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for the stochastic Ginzburg-Landau equations driven by a white noise. Specifically, we discuss the dynamics of the following stochastic Ginzburg-Landau equations defined in a bounded domain O : where u = u(x, t) is a unknown complex-valued function, i is the imaginary unit, λ ∈ R, ρ > 0, the nonlinear term f (u) is a complex-valued function, g(t, x) ∈ L 2 loc (R, L 2 (O)), h is a nonlinear function with certain properties, W is a two-sided real-valued wiener process on a probability space. The symbol • indicates that the equation is understood in the sense of Stratonovich's integration.
We are interested in studying the dynamics of Eq.(1) for almost all sample paths, including the existence and uniqueness of random attractors. For this purpose, one needs to define a random dynamical system (or cocycle) based on the solution operator of the equation. However, the existence of such a random dynamical system is unknown in general for a nonlinear function h in (1)(see, e.g., [11,15]), which results in a serious challenge for studying the pathwise dynamics of the equation. At present, the existence of random attractors for (1) has been established only when h is u or independent of u (see, e.g., [29,28,31,44]). In the nonlinear case, some progress has been made for a class of stochastic PDEs driven by a fractional Brownian motion by using rough path analysis, see Gao et al. [14]. Due to the difficulty directly dealing with the stochastic equation, in this paper, we investigate the Wong-Zakai approximations of (1) by using a stationary process via the Wiener shift. Consequently, the approximate equation generates a random dynamical system. Thus one can discuss its sample-wise (or pathwise) dynamics.
The standard probability space (Ω, F, P) will be used in this paper, where 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 shift θ t defined on the probability space (Ω, F, P) by θ t ω(·) = ω(t + ·) − ω(t).
By the properties of Brownian motions, it implies that G δ (θ t ω) is a stationary stochastic process with a normal distribution and is unbounded in t for almost all ω. G δ (θ t ω) may be viewed as an approximation of white noise. Indeed, we will give in Lemma 2.1 that converges to W (t, ω) almost surely uniformly on any finite time interval as δ → 0, which suggests that the dynamics of stochastic equation (1.1) could be approached by the following Wong-Zakai approximation of the equation as δ → 0: ∂u ∂t − (1 + iλ)∆u + ρu = f (u) + g(t, x) + h(t, x, u)G δ (θ t ω).
We will show, for a wide class of nonlinearity h, the random Eq.(3) generates a continuous random dynamical system in L 2 (O) and it has a unique tempered random attractor (see Theorem 2.5). This is in contrast with the stochastic equation (1) where the existence of random attractors is only known when h is linear in its third argument or independent of u. Furthermore, we will prove that when Eq. (1) is driven by a linear multiplicative noise (i.e., h(t, x, u) = u) , the solutions of (3) converge to that of (1) in L 2 (O) as δ → 0 (see Corollary 3). As for the long term dynamics, we will prove the random attractors of Eq.(3) approach that of (1) in terms of the Hausdorff semi-distance in L 2 (O) as δ → 0 (see Theorems 3.10). Using deterministic differential equations to approximate stochastic differential equations was introduced by Wong and Zakai in their pioneer work [60,59] in which they discussed both piecewise linear approximations and piecewise smooth approximations for one-dimensional Brownian motions. Their work was later generalized to stochastic differential equations of higher dimensions, for example, by McShane [37], Stroock and Varadhan [45], Sussmann [46,47], Ikeda et al. [22], Ikeda and Watanabe [23], and recently by Kelly and Melbourne [24], and Shen and Lu [43] in which the same approximations as this paper were studied. The results of the Wong-Zakai approximations have also been extended to stochastic differential equations driven by martingales and semimartingales, see for example, Nakao and Yamato [39], Konecny [25], Protter [41], Nakao [38], and Kurtz and Protter [26,27].
In the current paper, we use W δ (t, ω) = t 0 G δ (θ s ω)ds to approximate the Brownian motion W (t, ω). For such approximations, the corresponding approximate Eq.(3) generates a random dynamical system, which allows us to investigate the pathwise dynamics such as random attractors. Such approximation was also used in Lu and Wang [36,35], Wang et al. [58], and Shen et al. [43] where they studied the chaotic behavior of random differential equations driven by a multiplicative noise of G δ (θ t ω) and long term behavior of stochastic reaction-diffsion equations driven by multiplicative noise .
The concept of random attractor for autonomous stochastic equations was introduced in Crauel and Flandoli [7], Flandoli and Schmalfuss [12], and Schmalfuss [42]. After then, there is an extensive literature on this subject for autonomous SPDEs. In the non-autonomous case, random attractors have been studied, for example, in Caraballo et al. [6], Li [32,33] and Wang [57,56,55]. In this paper we will follow the framework of Wang [57], Lu and Wang [35] to deal with the non-autonomous random attractors of (1) and (3).
When we prove the existence and uniqueness of random attractors in L 2 (O), the nonlinearity f (u) is special form, i.e., f (u) = −(1 + iµ)|u| 2 u, µ ∈ R, which is consistent with the general physical background for the Ginzburg-Landau equation [3,9]. This paper is organized as follows. In Section 2, we show the existence of random attractors for the Wong-Zakai approximate Eq.(3) in L 2 (O) with general nonlinear function h. We first apply Galerkin method to prove the existence and uniqueness of solutions for non-autonomous stochastic Ginzburg-Landau equation. And then we derive uniform estimates for solutions and the existence of a pullback random attractor is proved. In Section 3, we consider the case h(t, x, u) = u for t ∈ R, x ∈ O and u ∈ R, and show that the solutions of (3) converge to that of the stochastic equation (1) in L 2 (O) as δ → 0. We also obtain the upper semicontinuity of random attractors of (3) in this case.
In this section, to define a continuous cocycle for the Wong-Zakai approximate Eq.(3) which is a non-autonomous stochastic Ginzburg-Landau equation driven by a stationary process, we first apply Galerkin method to prove the existence and uniqueness of solutions for the equation, and then prove the existence of pullback random attractors in L 2 (O) for a wide class of nonlinear functions h.
Recall that for δ = 0, the random variable G δ is defined by From [1], It follows that there exists a θ t -invariant set Ω ⊆ Ω of full P measure such that for each ω ∈ Ω, Hereafter, for simplicity, we will write Ω as Ω. From (9) we find (11) By (11) and the continuity of ω, we obtain for all t ∈ R, In fact, the convergence (12) holds true uniformly with respect to t on any finite interval [τ, τ + T ] with τ ∈ R and T > 0 which is stated below.
By (11) we may rewrite Eq.(4) as To define a continuous cocycle for the Wong-Zakai approximate Eq.(3), we need to prove the existence and uniqueness of solutions for the equation. In fact, we will prove that under conditions (7)-(8), for every τ ∈ R,ω ∈ Ω and u τ ∈ L 2 (O), the solution u of (16) supplement with (5)-(6) is well posed in L 2 (O).
In addition, this solution is continuous with respect to initial data in L 2 (O) and is (F, B(L 2 (O)))-measurable in ω ∈ Ω.
In this following, we will prove this theorem by the Galerkin method and compactness argument, see [9,28].
Proof. The proof proceeds in four distinct steps. We first construct a sequence of approximate solutions from finite-dimensional systems, then derive uniform estimates, and further take the limit of those approximate solutions, finally prove the uniqueness.
Step (4) : Uniqueness of solutions. Let u 1 and u 2 be solutions of (4)-(6) and u = u 1 − u 2 . Then we have Applying the interpolation inequality the first term on the right hand side of (36) can be bounded by Noting that for some δ ∈ (0, 1) so the second term on the right hand side of (36) can be bounded by where which along with the Gronwall Lemma, the continuity of ω(t) in t and (17) imply the uniqueness and continuity of solutions in the initial data in L 2 (O). Finally, we prove the measurability of solutions in ω ∈ Ω. By (34) and the uniqueness of solutions, we find that for every ω ∈ Ω, the whole sequence u n (t 0 , τ, ω, u τ ) → u(t 0 , τ, ω, u τ ) weakly in L 2 (O) for any fixed t 0 ∈ [τ, T ] and ω ∈ Ω. Then the measurability of u(t, τ, ω, u τ ) follows from that of u n (t, τ, ω, u τ ). This completes the proof.
We will study tempered random attractors of Φ in L 2 (O). Let D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} be a family of bounded nonempty subsets of L 2 (O). Such a family D is called tempered if for every c > 0,τ ∈ R and ω ∈ Ω, where we use the notation D = sup u∈D u L 2 (O) for a subset D of L 2 (O). Hereafter, we will use D to denote the collection of all tempered families of bounded nonempty subsets of L 2 (O), i.e., When deriving uniform estimates, we assume that, for every τ ∈ R, For deriving the existence of tempered random attractors, we further assume for every c > 0, We find that both conditions (46) and (47) do not require that g be bounded with respect to their first argument in L 2 (O) at ±∞.
As an immediate consequence of Lemma 2.4, we obtain the existence of Dpullback absorbing sets for Eq. (16). τ ∈ R, ω ∈ Ω} ∈ D in L 2 (O) where for every τ ∈ R and ω ∈ Ω, and with M being as in (48).
Note that Lemma 2.3 implies the D-pullback asymptonic compactness of Φ in L 2 (O) more precisely, we have the following result.
We are now ready to prove the existence of D-pullback attractors of Φ. where K is the D-pullback absorbing set of Φ as given by (56).
If, in addition, there exists T > 0 such that h, g are all T-periodic in their first argument, then the attractor A is also T-periodic, that is, for every τ ∈ R and ω ∈ Ω.
Proof. Since Φ has a closed measurable D-pullback absorbing set K by Corollary 1 and is D-pullback asymptotically compact in L 2 (O) by Corollary 2, then the existence of D-pullback attractor A of Φ follows from Proposition 2.10 in [56] immediately. Moreover, this attractor is unique and its structure is given as above.

Stochastic ginzburg-landau equations driven by linear multiplicative noise.
In this section, we consider Eq. (1) with multiplicative noise: which is supplemented with boundary condition (5) and initial condition (6). To discuss random attractors in this case, we need to convert (64) into a pathwise deterministic equation that can be done via the standard transformation v(t, τ, ω) = e −ω(t) u(t, τ, ω). From (64) we find v satisfies with boundary condition v(t, x) = 0, x ∈ ∂O and t > τ, where v τ (x) = e −ω(τ ) u τ (x). Given ω ∈ Ω, τ ∈ R and v τ ∈ L 2 (O), problem (65)-(67) is a deterministic system, and hence, as in the previous section, one can prove, this system has a unique solution In addition, v(·, τ, ω, v τ ) is continuous in v τ with respect to the norm of L 2 (O) and is (F, B(L 2 (O)))-measurable in ω ∈ Ω. This enables us to define a cocycle Φ 0 : for the stochastic equation (64) via the solutions of (65). Given t ∈ R + , τ ∈ R, ω ∈ Ω and u τ ∈ L 2 (O), let where v τ = e ω(−τ ) u τ .Then we find that Φ 0 given by (68) is a continuous cocycle on L 2 (O) over (Ω, F, P, {θ t } t∈R ). We want to show the stochastic equation (64) has a D-pullback attractor in L 2 (O). To that end, we must derive uniform estimates of the solutions which are given below.
As a consequence of inequality (71) we have the following useful estimates.
Proof. The proof is similar to that of Corollary 2.6. First, by the argument of Lemma 3.1, we can show that there exist T = T (τ, ω, D) > 0 and c = c(τ, ω) > 0 such that for all t ≥ T and u 0 ∈ D(τ − t, θ −t ω), Therefore, there is N = N (τ, ω, D) > 0 such that for all n ≥ N , which together with (76) concludes the proof.
In this following, we present the existence of D-pullback attractors of Eq. (64). (64) has a unique D-pullback If, in addition, there exists T > 0 such that g is T -periodic in their first argument, then the attractor A 0 is also T-periodic.
Proof. Given τ ∈ R and ω ∈ Ω, define a subset K 0 (τ, ω) by Then by Lemma 3.1 we find that for every τ ∈ R, ω ∈ Ω and D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D, there exists T = T (τ, ω, D) > 0 such that for all t ≥ T , Moreover, by (47), one can check K 0 is tempered in L 2 (O) , i.e., K 0 ∈ D. Therefore, K 0 is a closed measurable D-pullback absorbing set, which along with Lemma 3.3 implies the existence and uniqueness of D-pullback attractor A 0 of Φ 0 . In the case g is T -periodic in the first argument, we find Φ 0 and K 0 are also T -periodic. Thus by Proposition 2.11 in [56], it follows that the attractor A 0 is T -periodic.
We now approximate the solutions of the stochastic Ginzburg-Landau equation (64) by the following pathwise Wong-Zakai approximated equation: together with the boundary condition (5) and initial condition (6). Note that the solution of Eq.(78) is written as u δ from now on to imply its dependence on δ. By the previous section, we see for every δ = 0, Eq.(78) defines a continuous cocycle Φ δ in L 2 (O) which has a unique D-pullback attractor A δ . In what follows, we will investigate the convergence of solutions of (78) as δ → 0; more precisely, we will show that the solutions of Eq. (78) approach that of the stochastic Ginzburg-Landau equation (64) in L 2 (O) as δ → 0. Furthermore, we will obtain the upper semicontinuity of random attractors A δ as δ → 0. These results partially justify the idea to approximate a stochastic Ginzburg-Landau equation by replacing the white noise by a process G δ (θ t ω) with small δ, and thus, under certain circumstances, we could use the methods of deterministic dynamical systems to discuss the dynamics of stochastic Ginzburg-Landau equations.
To better understand the relations between the solutions of (64) and (78), we need a similar transformation for (78) as we did for (64). Let Then we have from (78): with boundary and initial conditions Next we derive the following estimates on the solutions of system (80)-(81) on a finite time interval.
As a result of Lemma 3.6, we get a D-pullback absorbing set of Eq. (78) immediately.
Next, we consider the convergence of solutions of (78) as δ → 0.
We finally give the upper semicontinuity of random attractors as δ → 0.