The Global Attractors of the Solution for 2 D Maxwell-Navier-Stokes with Extra Force Equations

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


Introduction
In recent years, the Maxwell-Navier-Stokes equations have been studied extensively, and the studies have obtained many achievements [1] [2].The Maxwell-Navier-Stokes equations are a coupled system of equations consisting of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism.The coupling comes from the Lorentz force in the fluid equation and the electric current in the Maxwell equations.In [1], the authors studied the non-resistive limit of the 2D Maxwell-Navier-Stokes equations and established the convergence rate of the non-resistive limit for vanishing resistance by using the Fourier localization technique.In [2], the author has proved the existence and uniqueness of global strong solutions to the non-resistive of the 2D Maxwell-Navier-Stokes equations for initial data ( ) with 0 s > .The long time behaviors of the solutions of nonlinear partial differential equations also are seen in [3]- [10].
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: is bounded set, ∂Ω is the bound of Ω , v is the velocity of the fluid, γ is the viscosity, ε and η are resistive constants, j is the electric current which is given by Ohm's law, E is the electric field, B is the magnetic field and j B

×
is the Lorentz force. Let

The Priori Estimate of Solution of Questions (1.1)
Lemma 1. Assume ( ) Proof.For the system (1.1) multiply the first equation by v with both sides and obtain For the system (1.1) multiply the second equation by E with both sides and obtain For the system (1.1) multiply the third equation by B with both sides and obtain According to Poincare's inequality, we obtain According to Young's inequality, we obtain From (2.4) (2.5) (2.6) (2.7) (2.8) (2.9), we obtain ( ) Using the Gronwall's inequality, the Lemma 1 is proved.
Lemma 2. Under the condition of Lemma 1, and For the system (1.1) multiply the first equation by v −∆ with both sides and obtain For the system (1.1) multiply the second equation by E −∆ with both sides and obtain For the system (1.1) multiply the third equation by B −∆ with both sides and obtain According to the Sobolev's interpolation inequalities According to the Sobolev's interpolation inequalities and Young's inequalities According to the Holder's inequalities and inequalities According to the Young's inequalities In a similar way,we can obtain The two above formulae subtract and obtain ( ) For the system (3.1)multiply the first equation by v with both sides and obtain ( ) ( ) For the system (3.1)multiply the second equation by E with both sides and obtain ( ) ( ) For the system (3.1)multiply the third equation by B with both sides and obtain ( ) According to (3.2) + (3.3) + (3.4), we obtain here ( ) ( ) ( ) , and ( ) ( ) ( ) ) ( ) From the (3.5), (3.6), (3.7) and (3.8), we can obtain ( ) According to the consistent Gronwall inequality, the uniqueness is proved.Theorem 2. [8] Let X be a Banach space, and = here I is a unit operator.Set ( ) S t satisfy the follow conditions.

Discussion
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.