In this paper, we study the following magnetohydrodynamic system:

here is bounded set, is the bound of, where u is the velocity vector field, v is the magnetic

vector field, are the kinematic viscosity and diffusivity constants respectively.. Let.

When, problem (1.1) reduces to the MHD equations. In particular, if, 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.

Lemma 1. Assume so the smooth solution of problem (1.1) satisfies

Proof. We multiply u with both sides of the first equation of problem (1.1) and obtain

We multiply v with both sides of the second equation of problem (1.1) and obtain

According to we 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

Let, according that we obtain

Using the Gronwall’s inequality, the Lemma 1 is proved.

Lemma 2. Under the condition of Lemma 1, and

, , , so the solution of problem (1.1) satisfies

Proof. For the problem (1.1) multiply the first equation by with both sides, for the problem (1.1) multiply the second equation by with both sides and obtain

According to the Sobolev’s interpolation inequalities,

According to (2.9)-(2.10), we have

Here

In a similar way, we can obtain

Here

Here

Here

According to the Poincare’s inequalities

From (2.12)-(2.17), we have

Here

So

We obtain

Using the Gronwall’s inequality, the Lemma 2 is proved.

Theorem 1. Assume that and so problem (1.1)

exist a unique solution

Proof. 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 are two solutions of problem (1.1), let, Here so the difference of the two solution satisfies

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

According to (3.1) + (3.2), we have

According to Sobolev inequality, when n < 4

According to (3.8)-(3.9),we can get

From (3.10)-(3.13),

Here

So, we have

Let, so we obtain

According to the consistent Gronwall inequality,

So we can get the uniqueness is proved.

Theorem 2. [9] Let E be a Banach space, and are the semigroup operators on E. here I is a unit operator. Set satisfy the follow conditions

1) is bounded. Namely, , it exists a constant, so that ;

2) It exists a bounded absorbing set namely it exists a constant t₀, so that ;

3) When, is a completely continuous operator A.

Therefore, the semigroup operators exist a compact global attractor.

Theorem 3. Assume ,. Problem (1.1) have global attractor

Proof.

1) When From Lemma 1,

So in E is uniformly bounded.

2) has E in a bounded absorbing set

From Lemma 2, when there is

Since is tightly embedded, so is in the tight absorbing set in E.

3) So the semigroup operator 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

Assume is the solutions of the problem (3.14). We know

. It is easy to prove the problem (3.14) has the uniqueness of solutions

.

To prove in has differential, let so there has

Theorem 4. Assume and T are constants, so it exists a constant and has so there is

Proof. Meet the initial value problem (3.14) of respectively for, solutions for, , let,. So, satisfies

Here

For the problem (3.16) multiply the first equation by with both sides and for the problem (3.16) multiply the second equation by with both sides and obtain

Then

Here.

For the problem (3.16) multiply the first equation by with both sides and for the problem (3.16) multiply the second equation by with both sides and obtain

According to the Sobolev’s interpolation inequalities

According to (3.22)-(3.23), we have

In a similar way, we can obtain

So, we can get

Here, we obtain

According to the Poincare’s inequalities

Let,

According to Gronwall’s inequalities, we obtain

Let be the solutions of the linear Equation (3.14), and satisfies, Assume

So, we can get

Here

For the problem (3.33) multiply the first equation by w₁ with both sides and for the problem (3.33) multiply the second equation by w₂ with both sides and obtain

According to (3.8)-(3.9), then

According to, we obtain

Here

We obtain

So

Let be the solutions of the linear Equation (3.33) correspond- ing to the initial value so there is

is linear mapping that is defined in the problem (3.34), represents the outer product, tr represents the trace, Q_N is the orthogonal projection from to the span

Theorem 5. Under the assume of Theorem 3, the global attractor A of problem (1.1) has finite Hausdorff and fractal dimension, and

Here J₀ is a minimal positive integer of the following inequality

Proof. By theorem [8] , we need to estimate the lower bound of Let be the orthogonal basis of subspace of

According to (3.8)-(3.9), we can get

Under the bounded condition, select is the standard eigenfunction of, and the corresponding eigenvalues are and

Let. Therefore, we can get

Let.

By and

So, we can obtain

We have

Therefore

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11161057.