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In this paper, we studied the solution existence and uniqueness and the attractors of the 2D Maxwell-Navier-Stokes with extra force equations.

In recent years, the Maxwell-Navier-Stokes equations have been studied extensively, and the studies have obtained many achievements [

2D Maxwell-Navier-Stokes equations for initial data

long time behaviors of the solutions of nonlinear partial differential equations also are seen in [

In this paper,we will study the 2D Maxwell-Navier-Stokes equations with extra force and dissipation in a bounded area under homogeneous Dirichlet boundary condition problems:

here

Let

Lemma 1. Assume

bound questions (1.1) satisfies

here

Proof. For the system (1.1) multiply the first equation by

For the system (1.1) multiply the second equation by

For the system (1.1) multiply the third equation by

Because

According to Poincare’s inequality, we obtain

According to

According to Young’s inequality, we obtain

From (2.4) (2.5) (2.6) (2.7) (2.8) (2.9), we obtain

so

Let

so

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

Lemma 2. Under the condition of Lemma 1, and

so the solution

here

Proof. For the system (1.1) multiply the first equation by

For the system (1.1) multiply the second equation by

For the system (1.1) multiply the third equation by

According

here

so

According to the Sobolev’s interpolation inequalities

so

According to the Sobolev’s interpolation inequalities and Young’s inequalities

According to the Holder’s inequalities and inequalities

and

According to the (2.13) (2.14) (2.15) (2.16) (2.17) (2.18), we obtain

here

According to the Poincare’s inequalities

According to the Young’s inequalities

In a similar way,we can obtain

From (2.19)-(2.23), we have

Let

So

According to the Gronwall’s inequality,we can get the Lemma 2.

Theorem 1. Assume that

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

rence of the two solution satisfies

The two above formulae subtract and obtain

For the system (3.1) multiply the first equation by

For the system (3.1) multiply the second equation by

For the system (3.1) multiply the third equation by

According to (3.2) + (3.3) + (3.4), we obtain

here

Notice that

From the (3.5), (3.6), (3.7) and (3.8), we can obtain

Let

so, we have

According to the consistent Gronwall inequality, the uniqueness is proved.

Theorem 2. [

conditions.

1)

2) It exists a bounded absorbing set

3) When

Therefor, the semigroup operators

Theorem 3. Under the assume of Theorem 1, questions (1.1) have global attractor

1)

2)

Proof. Under the conditions of Theorem 1 and Theorem 2, it exists the solution semigroup

tions (1.1),

From Lemma 1 - Lemma 2, to

This shows

Furthermore, when

therefore,

is the bounded absorbing set of semigroup

Since

If we want to estimate the Hausdorff and fractal dimension of the attractor A of question (1.1), we need proof of the solution of question (1.1) that is differentiable. We are studying the solution’s differentiability hardly and positively. Over a time, we will get some results.

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