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In this paper, we prove the existence of inertial manifolds for 2D generalized MHD system under the spectral gap condition.

In [

where u is the fluid velocity field, v is the magnetic field,

In this paper, we consider the following 2D generalized MHD system:

where u is the fluid velocity field, v is the magnetic field,

This paper is organized as follows. In Section 2, we introduce basic concepts concerning inertial manifolds. In Section 3, we obtain the existence of the inertial manifolds.

We rewrite the problem (1.2) as a first order differential equation, the problem (1.2) is equivalent to:

where

Let H is a Banach space,

Definition 2.1. Suppose

1. M is a finite dimensional Lipshitz manifold;

2. M is positively invariant under

3. M is attracts every trajectory exponentially, i.e., for every

We now recall some notions. Let

Standing Hypothesis 2.2. We suppose that

Let now

Let the bound absorbing set

Suppose that

Denote by

Lemma 2.3. Defined by

where

Proof. Assume

where

Lemma 2.3 is proved.

Lemma 2.4. Let

otherwise, there exist constants

and

for all

Proof. Let

Putting

For

From Equation (2.13) and Equation (2.15), we have

We notice that Equation (2.14)

so we have

By Equation (2.16) and Equation (2.17), and use the Cauchy-Schwarz inequality, we obtain

Then using Lemma 2.3,we have

For

where

By multiplying (2.13) by

Using Holder inequality, from Equation (2.20) we have

In Equation (2.19) setting

where

By Equation (2.21) and Equation (2.22), we have

Integrating Equation (2.24) between 0 and

To complete the proof of Lemma 2.4, we consider the following two cases,

and

We only consider Equation (2.26), in this case,

where

since

then we obtain

Integrating Equation (2.30) between 0 and

In this section we will prove the existence of the inertial manifolds for solutions to the problem (2.1). We suppose that

Let constants

Note that

is the distance of

For every

possesses a unique solution

where

by

We need to prove the following two conclusions:

1. For

2.

So we give the following Lemmas.

Lemma 3.1. Let

Proof. The proof is similar to Temam [

Lemma 3.2. Let

and

Proof. For any

and

which proves Equation (3.8). We now prove Equation (3.9), by the definition of

And we have

Substituting Equation (3.13) into Equation (3.11) we obtain Equation (3.9). Lemma 3.2 is proved.

Lemma 3.3. Let

Proof. Let

Let

So we obtain

Further more, for

Setting

Lemma 3.3 is proved.

Lemma 3.4. Let

so for every

here

Proof. For any given

and

here

Multiplying the first equation in Equation (3.25) by

So we have

For

By Lemma 2.3, to do the following estimate,using Equation (3.11) and Equation (3.28) we obtain

here

here

Hence,

Then from Equation (3.15) we have

Combining Equation (3.31) and Equation (3.32), we obtain

Substituting Equation (3.33) into Equation (3.29), we obtain

Lemma 3.4 is proved.

Lemma 3.5. Let

here

Proof. Let

From the first inequality of Equation (3.26) and the following estimate, we have

then from the last inequality of Equation (3.35), we obtain

From Equation (3.36), we have

Due to

From Equation (3.6), Equation (3.35) and Equation (3.38), we have

Then using Equation (3.16), Equation (3.33) and

Lemma 3.5 is proved.

Lemma 3.6. Suppose that

we have

Proof. From

where

To make

Equation (3.46) is equivalent to

If Equation (3.48) is satisfied, so Equation (3.47) is equivalent to

Suppose that Equation (3.41) is equivalent to

Hence,

Hence,

Therefore Equation (3.49) follows from Equation (3.52). From Equation (3.41) we conclude that

we notice that

From Lemma 3.1 to Lemma 3.6,we can obtain the following conclusions.

Theorem 3.1. Suppose that

Theorem 3.2. Suppose that

where

Proof. Let

Assume

Let

Substituting

If Equation (3.56) is satisfied, assume

In a word, for

Theorem 3.2 is proved.

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

ZhaoqinYuan,LiangGuo,GuoguangLin, (2015) Inertial Manifolds for 2D Generalized MHD System. International Journal of Modern Nonlinear Theory and Application,04,190-203. doi: 10.4236/ijmnta.2015.43014