Global Attractors and Dimension Estimation of the 2 D Generalized MHD System with Extra Force

In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a two dimensional generalized magnetohydrodynamic (MHD) system. Then the existence of the global attractor is proved. Finally, the upper bound estimation of the Hausdorff and fractal dimension of attractor is got.


Introduction
In this paper, we study the following magnetohydrodynamic system: is bounded set, ∂Ω is the bound of Ω , where u is the velocity vector field, v is the magnetic vector field, , 0, , 2 n γ η α β > > are the kinematic viscosity and diffusivity constants respectively.
( ) When 1 α β = = , problem (1.1) reduces to the MHD equations.In particular, if 0 γ η = = , problem (1.1) becomes the ideal MHD equations.It is therefore reasonable to call (1.1) a system of generalized MHD equations, or simply GMHD.Moreover, it has similar scaling properties and energy estimate as the Navier-Stokes and MHD equations.
The solvability of the MHD system was investigated in the beginning of 1960s.In particular in [1]- [4] the global existence of weak solutions and local in time well-posedness was proved for various initial boundary value problems.However, similar to the situation with the Navier-Stokes equations, the problem of the global smooth solvability for the MHD equations is still open.
Analogously to the case of the Navier-Stokes system (see [5]- [8]) we introduce the concept of suitable weak solutions.We prove the existence of the global attractor (see [9]) and getting the upper bound estimation of the Hausdorff and fractal dimension of attractor for the MHD system.

The Priori Estimate of Solution of Problem (1.1)
, , , u x t v x t of problem (1.1) We multiply v with both sides of the second equation of problem (1.1) and obtain According to (2.1) + (2.2), so we obtain According to Poincare and Young inequality, we obtain From (2.5)-(2.7),we obtain ( ) , according that we obtain Using the Gronwall's inequality, the Lemma 1 is proved. Lemma 2. Under the condition of Lemma 1, and ( ) ( ) ( ) Proof.For the problem (1.1) multiply the first equation by 2  A u α with both sides, for the problem (1.1) multiply the second equation by 2  A v β with both sides and obtain , d , According to the Sobolev's interpolation inequalities, , , From (2.12)-(2.17),we have ( ) Using the Gronwall's inequality, the Lemma 2 is proved.
By the method of Galerkin and Lemma 1-Lemma 2,we can easily obtain the existence of solutions.Next, we prove the uniqueness of solutions in detail.Assume ( ) ( ) , , , w u v w u v are two solutions of problem (1.1), let ( ) ( ) ( ) The two above formulae subtract and obtain For the problem (3.3) multiply the first equation by u with both sides and obtain ( ) For the problem (3.3) multiply the second equation by v with both sides and obtain ( ) , , , , According to (3.1) + (3.2), we have According to Sobolev inequality, when n < 4 , , According to (3.8)-(3.9),wecan get ( ) From (3.10)-(3.13),( ) According to the consistent Gronwall inequality, ( ) ( ) the uniqueness is proved. Theorem 2. [9] Let E be a Banach space, and it exists a constant t 0 , so that ( ) ( ) S t is a completely continuous operator A.
Therefore, the semigroup operators ( ) S t exist a compact global attractor.
, , 3) So the semigroup operator ( ) : S t E E → is completely continuous. In order to estimate the Hausdorff and fractal dimension of the global attractor A of problem (1.1), let problem (1.1) linearize and obtain is the solutions of the problem (3.14).We know , 0, ; It is easy to prove the problem (3.14) has the uniqueness of solutions , 0, ; To prove ( ) S t in ( ) , , , R R R R and T are constants, so it exists a constant ( ) , , , , , .
Proof.Meet the initial value problem (3.14) of respectively for ( ) For the problem (3.16) multiply the first equation by 2   1 A α θ with both sides and for the problem (3.16) mul- tiply the second equation by 2