IJMNTAInternational Journal of Modern Nonlinear Theory and Application2167-9479Scientific Research Publishing10.4236/ijmnta.2015.43014IJMNTA-58922ArticlesEngineering Physics&Mathematics Inertial Manifolds for 2D Generalized MHD System haoqinYuan1LiangGuo1GuoguangLin1*Department of Mathematics, Yunnan University, Kunming, China* E-mail:gglin@ynu.edu.cn(GL);0708201504031902035 June 2015accepted 16 August 20 August 2015© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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

MHD System Spectral Gap Inertial Manifolds
1. Introduction

In  , Yuan, Guo and Lin prove the existence of global attractors and dimension estimation of a 2D genera- lized magnetohydrodynamic (MHD) system:

where u is the fluid velocity field, v is the magnetic field, is the constant kinematic viscosity and is constant magnetic diffusivity. is a bounded domain with a sufficiently smooth boundary,. More results about inertial manifolds can be founded in  -  .

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

where u is the fluid velocity field, v is the magnetic field, is the constant kinematic viscosity and is the constant magnetic diffusivity. is a bounded domain with a sufficiently smooth boundary, .

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.

2. Preliminaries

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

where, , and

Let H is a Banach space, , is norm of H, is inner product of H, ;, for any solution of the problem (2.1), , is norm of.

Definition 2.1. Suppose denote the semi-group of solutions to the problem (2.l) in, subset M is an inertial manifolds of the problem (2.l), that is M satisfying the following properties:

1. M is a finite dimensional Lipshitz manifold;

2. M is positively invariant under, that is, for all;

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

We now recall some notions. Let is a closed linear operator on satisfying the following Standing Hypothesis 2.2.

Standing Hypothesis 2.2. We suppose that is a positive definite, self-adjoint operator with a discrete spectrum, compacts in. Assume is the orthonormal basis in consisting of the corresponding eigenfunctions of the operator. Say

each with finite multiplicity and.

Let now and be two successive and different eigenvalues with, let further P be the orthogonal projection onto the first N eigenvectors of the operator.

Let the bound absorbing set, we define a smooth truncated function by setting is defined as

Suppose that the problem (2.1) is equivalent to the following preliminary equation:

Denote by is the orthogonal projection of H onto, and. Set, then Equation (2.4) is equivalent to

Lemma 2.3. Defined by of the problem (2.1) on the bounded set of is a Lipschitz function, for every, there exist a constant such that

where.

Proof. Assume, and let, use the fact that and using Poincare inequality, we have

where, so we can get

Lemma 2.3 is proved.

Lemma 2.4. Let be fixed, for any and all, there exist such that

otherwise, there exist constants and are dependent on such that

and

for all

Proof. Let with initial values respectively, are two different solutions of the problem (2.1), we have the fact that,. Put, so we obtain that

Putting

For, taking the derivative of Equation (2.14) with respect to t,we have

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, integrating the above inequality over, we obtain

where is given as in Lemma 2.3.

By multiplying (2.13) by, using Cauchy-Schwarz inequality and Lemma 2.3, we have

Using Holder inequality, from Equation (2.20) we have

In Equation (2.19) setting, we obtain

where

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

Integrating Equation (2.24) between 0 and, we obtain

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 is N + 1 eigenvector of the operator. By Equation (2.25) and Equation (2.28), we obtain

since, in Equation (2.29) setting, which proves Equation (2.11), where and. Using again Equation (2.20), we have

then we obtain

Integrating Equation (2.30) between 0 and, which proves Equation (2.12). Lemma 2.4 is proved.

3. Inertial Manifolds

In this section we will prove the existence of the inertial manifolds for solutions to the problem (2.1). We suppose that satisfies Standing Hypothesis 2.2 and recall that P is the orthogonal projection onto the first N orthonormal eigenvectors of.

Let constants be fixed, we define and denote the collection of all functions satisfies

Note that

is the distance of. So is completely space.

For every and the initial data, the initial value problem

possesses a unique solution.

where and the unique solution in Equation (3.4) is a successive bounded mapping acts from into. Particularly, the function

by, note that, we have

We need to prove the following two conclusions:

1. For and are sufficiently large, is a contraction.

2. is a unique fixed point in T, is a inertial manifold of 2D generalized MHD system.

So we give the following Lemmas.

Lemma 3.1. Let, so we have

Proof. The proof is similar to Temam  .

Lemma 3.2. Let, for, there exists constant such that

and

Proof. For any and, we denote, using Lemma 2.3 and see ( , Chapter 8: Lemma 2.1 and Lemma 2.2), we derive that there exists constant such that

and

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

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, one has and where for is sufficiently large one has.

Proof. Let, according to the definition of, we have, from Equation (3.6) and Equation (3.10), we have

Let and, suppose that and

So we obtain

Further more, for, we have

Setting in, then substituting into Equation (3.15) and Equation (3.16), and from Equation (3.14) we can derive that

Lemma 3.3 is proved.

Lemma 3.4. Let

so for every, one has

here

Proof. For any given, let are the solutions of the following initial value problem,

and

here Suppose that, so we have

Multiplying the first equation in Equation (3.25) by, using Equation (3.9) in Lemma 3.2, we obtain

So we have

For, from Equation (3.27) we have

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

here. From Equation (3.15), we have

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 is defined as in Lemma 3.4, for all,

here is defined by Equation (3.20), is defined by Equation (3.2).

Proof. Let and let is the solution of the initial value problem (3.25), then by the same way as in Lemma 3.2 we can prove that

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, integrating Equation (3.37) over, we have

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

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

Lemma 3.5 is proved.

Lemma 3.6. Suppose that

we have and, where is defined as in Lemma 3.5,

Proof. From is equivalent to

where and are defined as in Lemma 3.4. To find a sufficient condition of Equation (3.44), suppose that Equation (3.44) hold, so we have

To make, if and only if it satisfies

Equation (3.46) is equivalent to

If Equation (3.48) is satisfied, so Equation (3.47) is equivalent to or 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, Equation (3.48) follows from Equation (3.41), Equation (3.46) follows from Equation (3.48), Equation (3.46) follows from Equation (3.49), and from Equation (3.46) and Equation (3.47) we have. The last we need to prove is, from Lemma 3.5, we obtain

we notice that. Lemma 3.6 is proved.

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

Theorem 3.1. Suppose that is Lipschitz mapping space. satisfy Equation (3.1) and Equation (3.2), and is the unique solution of Equation (3.3) and Equation (3.4) for, respectively. Hence the transformation is a contraction, and exists a unique fixed point, is inertial manifolds of the problem (2.1).

Theorem 3.2. Suppose that is the mapping of, for any, there exists such that, for,

where, is defined as in Lemma 2.3.

Proof. Let with initial value, respectively, be two solutions of the problem (2.1). For any arbitrary and for, and use the fact there exists a constant

such that Equation (2.10) or Equation (2.11) is satisfied. From Equation (2.12), we have

Assume and for, therefore Equation (2.10) and Equation (2.11) can rewrite

Let is absorbing set, the orbital solution satisfies. Let such that

Substituting and into Equation (3.56) and Equation (3.57), we have

If Equation (3.56) is satisfied, assume, so we have the cone property

In a word, for, whenever By the properties of semigroups, for, we have

Theorem 3.2 is proved.

Supported

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

Cite this paper

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

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