Inertial Manifolds for 2 D Generalized MHD System

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


Introduction
In [1], Yuan, Guo and Lin prove the existence of global attractors and dimension estimation of a 2D generalized magnetohydrodynamic (MHD) system: where u is the fluid velocity field, v is the magnetic field, γ is the constant kinematic viscosity and η is constant magnetic diffusivity.More results about inertial manifolds can be founded in [2]- [11].
In this paper, we consider the following 2D generalized MHD system: where u is the fluid velocity field, v is the magnetic field, γ is the constant kinematic viscosity and η is the constant magnetic diffusivity.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.

Preliminaries
We rewrite the problem (1.2) as a first order differential equation, the problem (1.2) is equivalent to: Let H is a Banach space, ( ) ( ) > , 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 ( ) S t , that is, ( ) M is attracts every trajectory exponentially, i.e., for every 0 We now recall some notions.Let A is a closed linear operator on H satisfying the following Standing Hypothesis 2.2.
Standing Hypothesis 2.2.We suppose that A is a positive definite, self-adjoint operator with a discrete spectrum, is the orthonormal basis in H consisting of the corresponding eigenfunctions of the operator A .Say , 1, 2, , the problem (2.1) is equivalent to the following preliminary equation: ( ) V is a Lipschitz function, for every where otherwise, there exist constants ( ) ( ) respectively, are two different solutions of the problem (2.1), we have the fact that , so we obtain that ( ) ( ) . , , 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) Then using Lemma 2.3,we have where C is given as in Lemma 2.3.By multiplying (2.13) by U , using Cauchy-Schwarz inequality and Lemma 2.3, we have Using Holder inequality, from Equation (2.20) we have where ( ) ( ) (2.28) where 1 N λ + is N + 1 eigenvector of the operator A .By Equation (2.25) and Equation (2.28), we obtain 29) setting 0 t t = , which proves Equation (2.11), where ( ) ( ) . Using again Equation (2.20), we have Integrating Equation (2.30) between 0 and 0 t , which proves Equation (2.12).Lemma 2.4 is proved.

Inertial Manifolds
In this section we will prove the existence of the inertial manifolds for solutions to the problem (2.1).We suppose that A satisfies Standing Hypothesis 2.2 and recall that P is the orthogonal projection onto the first N orthonormal eigenvectors of A .
Let constants , 0 b l > be fixed, we define : For every ( ) ( ) ) and the unique solution , note that ( ) We need to prove the following two conclusions: 1.For , so we have Proof.The proof is similar to Temam [3].
Proof.For any , 0 M M > such that ( ) which proves Equation (3.8).We now prove Equation (3.9), by the definition of And we have , according to the definition of T , we have ( ) ( ) ( ) ( ) ( ) Proof.For any given , p p t p p t = = are the solutions of the following initial value problem, ( p t p t p t = − , so we have Multiplying the first equation in Equation (3.25) by Ap , using Equation (3.9) in Lemma 3.2, we obtain So we have ( ) By Lemma 2.3, to do the following estimate,using Equation (3.11) and Equation (3.28) we obtain ( here ( ) Hence, ( ) Combining Equation (3.31) and Equation (3.32), we obtain ( ) ( ) ( ) ( ) p p p = − is the solution of the initial 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 ( ) ( ) , , .To make 1 l l < , if and only if it satisfies ( ) is a bounded domain with a sufficiently smooth boundary ∂Ω , , inertial manifold of 2D generalized MHD system.So we give the following Lemmas.