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

Applied Mathematics
Vol.06 No.04(2015), Article ID:56037,12 pages
10.4236/am.2015.64068

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

Zhaoqin Yuan, Liang Guo, Guoguang Lin*

Department of Mathematics, Yunnan University, Kunming, China   Received 16 March 2015; accepted 26 April 2015; published 29 April 2015

ABSTRACT

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.

Keywords:

MHD System, Existence, Global Attractor, Dimension Estimation 1. Introduction

In this paper, we study the following magnetohydrodynamic system: (1.1)

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  - 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  - ) we introduce the concept of suitable weak solutions. We prove the existence of the global attractor (see  ) and getting the upper bound estimation of the Hausdorff and fractal dimension of attractor for the MHD system.

2. The Priori Estimate of Solution of Problem (1.1)

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 (2.1)

We multiply v with both sides of the second equation of problem (1.1) and obtain (2.2)

According to we obtain (2.3)

According to (2.1) + (2.2), so we obtain

(2.4)

According to Poincare and Young inequality, we obtain

(2.5)

(2.6)

(2.7)

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

(2.8)

According to the Sobolev’s interpolation inequalities,

(2.9)

(2.10)

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

(2.11)

Here

In a similar way, we can obtain

(2.12)

Here

(2.13)

Here

(2.14)

Here

According to the Poincare’s inequalities

(2.15)

(2.16)

(2.17)

From (2.12)-(2.17), we have

Here

So

We obtain

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

3. Global Attractor and Dimension Estimation

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

(3.1)

(3.2)

The two above formulae subtract and obtain

(3.3)

For the problem (3.3) multiply the first equation by u with both sides and obtain

(3.4)

For the problem (3.3) multiply the second equation by v with both sides and obtain

(3.5)

According to

(3.6)

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

(3.7)

According to Sobolev inequality, when n < 4

(3.8)

(3.9)

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

(3.10)

(3.11)

(3.12)

(3.13)

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.  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

(3.14)

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

(3.15)

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

(3.16)

Here

(3.17)

(3.18)

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

(3.19)

Then

(3.20)

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

(3.21)

According to the Sobolev’s interpolation inequalities

(3.22)

(3.23)

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

(3.24)

In a similar way, we can obtain

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

(3.31)

So, we can get

Here, we obtain

According to the Poincare’s inequalities

(3.32)

Let,

According to Gronwall’s inequalities, we obtain

(3.33)

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

(3.34)

So, we can get

(3.35)

Here

(3.36)

(3.37)

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

(3.38)

According to (3.8)-(3.9), then

(3.39)

(3.40)

(3.41)

(3.42)

(3.43)

(3.44)

(3.45)

(3.46)

(3.47)

(3.48)

According to, we obtain

Here

We obtain

So

(3.49)

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

(3.50)

is linear mapping that is defined in the problem (3.34), represents the outer product, tr represents the trace, QN 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  , we need to estimate the lower bound of Let be the orthogonal basis of subspace of

(3.51)

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

(3.52)

(3.53)

(3.54)

(3.55)

(3.56)

(3.57)

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

Funding

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

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NOTES

*Corresponding author.