The Existence of Exponential Attractor for Discrete Ginzburg-Landau Equation

This paper studies the following discrete systems of the complex Ginzburg-Landau equation: . Under some conditions on the parameters , and , we prove the existence of exponential attractor for the semigroup associated with these discrete systems.


Introduction
In the study of infinite dynamical systems, attractors occupy a central position (see, e.g., Chepyzhov and Vishik [1], Hale [2], Ladyzhenskaya [3], and Temam [4]).Exponential attractors are realistic objects intermediate between the global attractors and the inertial manifolds.There are several approaches for proving the existence of exponential attractors for parabolic and hyperbolic partial differential equations (PDEs) arising from mathematical physics.For example, we can refer to [5][6][7] for the existence of the exponential attractors for general evolution equations in Banach spaces, to [8] for the exponential attractors for reaction diffusion equations in unbounded domains, to [9] for the exponential attractors of the nonlinear wave equations, and to [10] for the exponential attractor for the generalized 2D Ginzburg-Landau equations.Also there are some references investigating the exponential attractors for lattice dynamical systems (LDSs).We can see [11][12][13] for the exponential attractors for firstorder LDSs; see [14,15] for the pullback exponential attractors for first-and second-order LDSs; see [16,17] for second-order nonautonomous LDSs and discrete Zakharov equations for the uniform exponential attractors.
Lattice systems including coupled ordinary differential equations, coupled map lattices, and cellular automata are spatiotemporal systems with discretization in some variables.In some cases, lattice systems arise as the spatial discretization of partial differential equations on unbounded or bounded domains.This paper will study the following discrete systems (lattice systems): where  is the unit of imaginary numbers and , , , ,  are parameters.Equation (1) can be regarded as a discrete analogue of the following complex Ginzburg-Landau equation on the real line: The complex Ginzburg-Landau equation is a simplified mathematical model for various pattern formation systems in mechanics, physics, and chemistry.We can refer to [10,38,39] for the detailed significations of the complex Ginzburg-Landau equation.
The existence of the exponential attractors for continuous complex Ginzburg-Landau equation in two-dimensional space was proved in [10].Later, under some conditions on , , , , , and   , [37] established the existence of global attractor for the semigroup associated with discrete systems (1)- (2).The aim of this paper is to prove the existence of exponential attractors for discrete systems (1)- (2).To this end, we will establish the following three items: Compared with previous works such as [9], here we no longer require the compactness of the invariant set B (this fact was first noted by Babin and Nicolaenko [8] and then by Eden et al. [6]), which can usually be obtained by the compact embedding between Sobolev spaces when studying PDEs.Note that the compact embedding theorem of Sobolev spaces seems difficult to be applicable when studying LDSs.This is caused by the discrete characteristics of LDSs which restrict us to choose the phase spaces.Fortunately, the intrinsic characteristics of LDSs enable us to use the -contraction property to compensate the compactness of the invariant set.

Positively Invariant Set and Lipschitz Continuity
Set and equip it with the inner product and norm as where V  denotes the conjugate of V  .Then (ℓ 2 , ‖ ⋅ ‖, (⋅, ⋅)) is a separable Hilbert space.We now introduce the operators , , and  * on ℓ 2 as follows: In fact,  * is the adjoint operator of  and one can easily check that Using the notations introduced above, we can write problem (1)-( 2) as where and For the well-posedness of problem ( 1)-( 2), we have the following.

Lemma 4 .
Let assumption (H) hold.Then there exists a time  * such that the operator ( * ) :=  * : B  → B is an contraction on B.