STOCHASTIC DYNAMICS OF 2D FRACTIONAL GINZBURG-LANDAU EQUATION WITH MULTIPLICATIVE NOISE

. In this work, we analyze the stochastic fractional Ginzburg-Landau equation with multiplicative noise in two spatial dimensions with a particular interest in the asymptotic behavior of its solutions. To get started, we ﬁrst transfer the stochastic fractional Ginzburg-Landau equation into a random equation whose solutions generate a random dynamical system. The existence of a random attractor for the resulting random dynamical system is explored, and the Hausdorﬀ dimension of the random attractor is estimated.


1.
Introduction. Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator. The last few decades have seen an enormous growth in the development of related dynamical concepts and in the applicability of nonlinear fractional models in a rather diverse scientific fields, such as fluid mechanics [23], biology [8], kinetic theories of systems with chaotic dynamics [18], pseudochaotic dynamics [25], dynamics in a complex or porous media [12,14], and quantum theory [9]. As described by Wheatcraft and Meerschaert [23], a fractional conservation of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear. In complex media, the propagation of acoustical waves, e.g. biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon has been described by a causal wave equation which incorporates fractional time derivatives [8]. Quite many other types of nonlinear fractional models have also been proposed and studied in physics and engineering, which include the fractional Schrödinger equation [4,6], fractional Landau-Lifshitz equation [7], fractional Landau-Lifshitz-Maxwell equation [16], and fractional Ginzburg-Landau equation [21] etc. However, in some physical and biological phenomena, it is commonly believed that perturbations may be neglected in the derivation of the ideal model such as molecular collisions in gases, liquids and electric fluctuations in resistors [5]. When considering the perturbations of each microscopic unit to the model, which will lead to a large complex system, people usually represent the micro effects by random perturbations in the dynamics of the macro observable. Recently, Stochastic and fractional partial differential equations and its applications have attracted continuous attention in the fields of mathematical physics and mathematical biology. One of the most interesting problems of stochastic partial differential equations is the asymptotic behavior of random dynamical systems. Some fundamental properties have been established, for example, by Crauel, Debussche and Flandoli [1,2] who developed the theory for the existence of random attractors for stochastic systems that closely parallels the deterministic theory [22], and by Debussche [3] who proved that the Hausdorff dimension of the random attractor could be estimated by using the global Lyapunov exponents.
For fractional partial differential equations, much attention has been devoted to the well-posedness of solutions [7,16] and the existence of positive solutions [20]. However, for stochastic fractional partial differential equations, very little has been undertaken as far as we know. It appears that the study just started. This motivates us to start our study by considering the asymptotic behavior of solutions for an interesting stochastic fractional Ginzburg-Landau equation. As we know, the fractional Ginzburg-Landau equation was initially derived from the variational Euler-Lagrange equation for the fractal media, which was used to describe the dynamical processes in a medium with fractal dispersion [21]. In [17], Pu and Guo considered a one-dimensional fractional complex Ginzburg-Landau equation The well-posedness of solutions was obtained by applying the semigroup method under the condition The existence of global attractors in L 2 was also presented under the condition σ = 1. In [10], we once studied the well-posedness and asymptotic behaviors of solutions of the fractional complex GinzburgLandau equation with the initial and periodic boundary conditions in two spatial dimensions. Estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor were presented. In this paper, we consider the two-dimensional stochastic fractional Ginzburg-Landau equation with multiplicative noise under the initial and periodic boundary conditions where u(x, t) is a complex-valued function on R 2 × [0, +∞). In (1.1), i is the imaginary unit, ρ > 0, σ > 0, β > 0, ν and µ are real constants, and α ∈ (1/2, 1). The white noise described by a two-sided Wiener process W (t) on a complete probability space results from the fact that small irregularity has to be taken account into some circumstances. The study in this paper is a continuation of the previous work by Lu and Lü [11] in which asymptotic behaviors of solutions to equation (1.1) were discussed in the case of σ = 1 in one spatial dimension.
As we see, for dynamical properties of random attractors, discussions are usually carried out in L 2 and do not include the estimate of the Hausdorff dimension of random attractors. In this study, we investigate the existence of the random attractor in H 1 p for the stochastic fractional Ginzburg-Landau equation with multiplicative noise in two spatial dimensions, and furthermore estimate the Hausdorff dimension of the random attractor by analyzing the corresponding random dynamical system and the Lyapunov exponents.
The rest of this paper is organized as follows. In Section 2, some preliminaries and notations for random dynamical systems are introduced. In Section 3, we present a continuous random dynamical system for the stochastic fractional Ginzburg-Landau equation. In Section 4, the existence of the random attractor for the stochastic Ginzburg-Landau equation is established. Section 5 is dedicated to the upper bound estimates of the Hausdorff dimension of the random attractor.

Preliminaries and notations.
In this section, we introduce some basic concepts related to random attractors of stochastic dynamical systems. For the detailed information and related applications, we refer to [1,2] and the references therein.

Definition 2.2. A set-valued map A(ω)
: Ω → 2 X taking values in the closed subsets of X is called to be measurable, if for all x ∈ X the mapping ω → d(A(ω), x) is measurable. Definition 2.3. The random omega limit set of a bounded set B ⊂ X at time t is given by Definition 2.4. Let S(t, s; ω) t≥s,ω∈Ω be a stochastic dynamical system. A random set A(ω) is called a random attractor if the following conditions are satisfied for P-a.e. ω ∈ Ω.
• It is the minimal closed set such that for t ∈ R and B ⊂ X it holds Namely, A(ω) attracts B (B is a deterministic set).
• A(ω) is the largest compact measurable set, which is invariant in the sense that S(t, s; ω)A(θ s ω) = A(θ t ω), s < t.
Following [2], we have the result regarding the existence of random attractors.

SHUJUAN LÜ, HONG LU AND ZHAOSHENG FENG
Theorem 2.1. Let S(t, s; ω) t≥s,ω∈Ω be a stochastic dynamical system satisfying the following conditions: 1. S(t, r; ω)S(r, s; ω) = S(t, s; ω)x, for all s ≤ r ≤ t and x ∈ X; 2. S(t, s; ω) is continuous in X, for all s ≤ t; 3. for all s < t and x ∈ X, the mapping ω → S(t, s; ω)x from (Ω, F) to (X, B(x)) is measurable; 4. for all t, x ∈ X and P-a.e. ω ∈ Ω, the mapping s → S(t, s; ω)x is right continuous at any point.
Assume that there exists a group θ t , t ∈ R, of measure preserving mapping, such that holds for P-a.e. ω ∈ Ω. There exists a compact attracting set K(ω) at time 0 for P-a.e. ω ∈ Ω. We set where the union is taken over all the bounded subsets of X, and A(B, ω) is given by Then, A(ω) is the random attractor.
Although the random attractor is not uniformly bounded, it is expected that the theory on the Hausdorff dimension of a global attractor of a deterministic dynamical system can be generalized to the stochastic case under some assumptions. Due to [3], we have Theorem 2.2. Let A(ω) be a compact measurable set which is invariant under a random map S(ω), ω ∈ Ω, for some ergodic metric dynamical system (Ω, F, P, (θ t ) t∈R ). Assume that the following conditions are satisfied.
(1) S(ω) is almost surely uniformly differentiable on A(ω), that is, for every u, u + h ∈ A(ω) there exists DS(ω, u) in L(X), the space of the bounded linear operators from X to X, such that (3) α 1 (DS(ω, u)) ≤ᾱ 1 (ω) holds when u ∈ A(ω) and there is a random variablē Then the Hausdorff dimension d H (A(ω)) of A(ω) is less than d almost surely.
For convenience, we redefine some notations related to fractional derivative equations and fractional Sobolev spaces. Firstly, we present the definition and properties of (− ) α through the Fourier series [7]. Since u is a periodic function, it can be expressed by a Fourier series u = k∈Z 2 u k e ik·x , and u xi = p (D) denote the complete Sobolev space of the order α under the norm: By virtue of the definition of (− ) α and integration by parts [7], we have where α 1 , α 2 are nonnegative constants and satisfy α 1 + α 2 = α.
In addition, the following Gagliardo-Nirenberg inequality [15] will be frequently used.
Lemma 2.2. Suppose that Λ ⊂ R n is a bounded domain and its boundary is smooth. Let u belong to L q (Λ) and its derivatives of the order m, D m u, belong to L r (Λ), where 1 ≤ q and r ≤ ∞. For the derivatives D j u, 0 ≤ j < m, it holds Here the constant c depends only on n, m, j, q, r and θ, with the two exceptional cases: 1. If j = 0, rm < n and q = ∞, then we make the additional assumption that either u tends to zero at infinity or u ∈ Lq for someq > 0.
Throughout the whole paper, we denote by (·, ·) the usual inner product of L 2 (D), by · H m the norm of the Sobolev space H m (D), In the forthcoming discussions, we use c and c j (j = 1, 2, · · · ) to denote different positive constants which depend only on the constants ρ, ν, µ, α and σ. Moreover, we denote D f dx by the notation f for simplicity.

Stochastic fractional Ginzburg-Landau equation.
In this section, we present the existence of a continuous random dynamical system for the stochastic fractional complex Ginzburg-Landau equation perturbed by a multiplicative white noise in the Itô sense. Thanks to the special linear multiplicative noise, the stochastic fractional Ginzburg-Landau equation can be reduced to an equation with random coefficients by a suitable transform of variables. Let us consider a set of continuous functions with the value 0 at 0: The process z(t) = e −βW (t) satisfies the stochastic differential equation: Furthermore, for any t and s it has z(t, θ s ω) = z(t + s, ω), P-a.s..
Here the exceptional set may be a priori depending on t and s. In fact, we suppose that z has a continuous modification and, once this modification is chosen, the exceptional set is independent of t.
We rewrite the unknown v(t) as v(t) = z(t)u(t) to obtain the following random differential equation with the initial data at time s and the periodic boundary condition We now construct a random dynamical system modeling the stochastic fractional Ginzburg-Landau equation. The existence and uniqueness of the solution of the problem (3.6)-(3.8) can be obtained [17], which defines a stochastic dynamical system (S(t, s; w)) t≥s,ω∈Ω by S(t, s; w)u s = u(t, ω; s, u s ) = v(t, ω; s, u s z(s, ω))z(t, ω).

4.
A priori estimates and existence of a random attractor. In this section, we discuss a priori estimates of the solution of equation (3.6), which can be used to prove the existence of a compact absorbing set. By virtue of Theorem 2.1, the existence of random attractor can be established.
Proof. Taking the inner product of equation (3.6) with v in L 2 and considering the real part, we obtain (4.10) By Young's inequality, we have . Rewrite (4.10) as We infer that for ε = ρ 4β , there exists s 1 (ω) ≤ −2 such that W (s) s < ε, as s < s 1 (ω), which implies that for ∀s < s 1 (ω) there holds

SHUJUAN LÜ, HONG LU AND ZHAOSHENG FENG
For any t ∈ [−2, 0] we deduce that (4.14) Consequently, we see that the proof is completed. There exists a random radius r 1 (ω) > 0 such that, for any given R > 0, there exists s 2 (ω) ≤ −2 such that for all s ≤s 2 (ω) and u s ∈ H 1 p (D) satisfying u s H 1 p (D) ≤ R, the following inequality holds for P-a.e. ω ∈ Ω, where r 2 1 (ω) = e 2ρ + c 1 · r 2 0 (ω). Proof. Taking the inner product of equation (3.6) with − v in L 2 and considering the real part, we obtain d dt Estimating the third term on the left-hand side by integration by parts gives and Y H represents the conjugate transpose of the matrix Y . We observe that condition (4.15) implies that the matrix M is nonnegative definite. Thus, we have On the other hand, by the Gagliardo-Nirenberg inequality we deduce that where
For all t ∈ [−2, 0], we derive that Hence, we complete the proof.
Proof. Taking the inner product of equation (3.6) with (− ) 2−α v in L 2 and considering the real part, we obtain We estimate the two terms on the right-hand side of equation (4.23), respectively. When σ ≥ 1 2 , using the Gagliardo-Nirenberg inequality, Young's inequality, as well where In addition, applying the Gagliardo-Nirenberg inequality again, we have where c 2 = c(3ρ + β 2 ) 2 + 1 2 .

Substituting (4.24) and (4.25) into equation (4.23) yields
Multiplying this inequality by e ρt and then integrating it over (s, t) with −2 ≤ s < t ≤ 0 leads to After integrating (4.27) with respect to s on [−2, −1] and applying the Gagliardo-Nirenberg inequality, by virtue of Lemma 4.1 and 4.2, for any −1 ≤ t ≤ 0 we derive Hence, the proof is completed.
By Lemma 4.3, we deduce that for any given R > 0 there exists ans 2 (ω) ≤ −2 such that for any s ≤s 2 (ω), it holds , for P-a.e. ω ∈ Ω. Let K(ω) be a ball in H 1 p (D) with the radius r 0 (ω) + r 1 (ω). It is shown that for any B bounded in H 1 p (D) there exists ans 2 (ω) such that for any s ≤s 2 (ω), there holds S(0, s; ω)B ⊂ K(ω), P-a.e. ω ∈ Ω. This implies that K(ω) is an attracting set at time 0, since H where K(ω) is a random variable such that

31)
and v(t) = e −βW (t) S(t, 0; ω)u 0 , where v j (t) (j = 1, 2) are two solutions of equation (3.6) with v j (0) = v 0 j , V (t) satisfies system (5.30)-(5.31) and h = v 0 1 − v 0 2 . Then E (t) satisfies the equation . By a similar discussion as described in Section 4, we have Applying Taylor's expansion for the function G(v 1 ,v 1 ) = |v 1 | 2σ v 1 at the point (v 2 ,v 2 ), we get Taking the inner product of equation (5.32) with E (t) in L 2 and considering the real part, we obtain For the first term on the right-side of equation (5.33), using Hölder's inequality and Young's inequality, we derive that (5.34) For the second term on the right-hand side of equation ( By Gronwall's inequality, we have Taking the inner product of equation (5.38) with (v 1 − v 2 ) in L 2 and considering the real part yields (5.39) Using Taylor's series, we get So the first term on the right-hand side of inequality (5.39) is bounded by Combining (5.39) and (5.40) leads to Hence, inequality (5.37) can be rewritten as Choose K(ω) = max{K 1 (ω), 1}. Then, we have E(ln K(ω)) < ∞.
From [3], we see that where Q d (s) is the orthogonal projector in L 2 onto the space spanned by V 1 (s), · · · , V d (s), and V i (s) is the solution of equation (5.30) with V i (0) = η i .