Global Attractor of Two-Dimensional Strong Damping KDV Equation and Its Dimension Estimation

Firstly, a priori estimates are obtained for the existence and uniqueness of solutions of two dimensional KDV equations, and prove the existence of the global attractor, finally geting the upper bound estimation of the Hausdorff and fractal dimension of attractors.


Introduction
Studies on the infinite dimension system with high dimension have obtained many achievements in recent years, such as [1][2][3][4][5].In the paper [6,7].The authors study the estimates of global attractor for one-dimensional KDV equation and its dimension.Based on these work, this paper further studies the global attractor of twodimensional KDV equations and its upper bound estimation of the Hausdorff and fractal dimension of attractors.
The following form 2D-KDV equation is studied in this paper        (1.4) where , ,    are positive constants.When 0       , the equation is the KDV equation.The rest of this paper is organized as follows.In Section 2, we introduce basic concepts concerning global attractor.In Section 3, we obtain the existence of the uniqueness global attractor, which has fractal and Hausdorff dimension.
In this paper, C denotes a positive constant whose value may change in different positions of chains of inequalities.

Preliminaries
Now, we can do priori estimates for Equation (1.1) Certainly there exist Proof.We multiply u for both sides of Equation (1.1), we obtain , , , Using the Growall inequality, we can get where     , , (2.12) According to ( 12) and ( 13), Lemma 1 and Young inequality, we can obtain that , , Using (2.10), (2.14) and (2.15), we can get Using Growall inequality, we have , e 1 e , Thus there exists Proof.We multiply 2 u  for both sides of Equation (1.1), we obtain that , , According to Lemma 1, Lemma 2 and Young inequality, we get that , , Using the Growall inequality, we can get Proof.We multiply 2 3  t u  for both sides of Equation (1.1), we obtain that are the semigroup operators. :  , here I is unit operator.Set   S t satisfy the following conditions: S t is a completely continuous operator.
Then, the semigroup operators   S t exist a compact global attractor A .

The Existence and Uniqueness of Solution
Proof.By the Galerkin method, we can easily obtain the existence of solutions.Next, we prove the uniqueness of solutions.Set Take the inner product with  , we gets So, we have From Lemmas 1-3, we have Using Young inequality, we obtain   ■

Global Attractor
 there exists a compact global attractor A , such that S t are the semigroup operators.

Dimension Estimation
Considering the following first variation equations where be the solutions of the linear variational equations corresponding to the initial value here  represents the outer product, tr represents the trace, N Q means that the   2 L  to the orthogonal projection on the span . So, from (3.3.8)we can obtain where N  is called Secondary index, namely

L
 , 1 p    , for simplicity, we denote by  and   the norm in the case 2 p  and p   , respectively.Suppose that

3
The global attractor A of Theorem 3.2 has finite fractal and Hausdorff dimension in ,