Global existence and convergence rates for the smooth solutions to the compressible magnetohydrodynamic equations in the half space

Qing Chen,Zhong Tan

doi:10.1186/s40068-014-0020-z

Abstract

BackgroundWith the characteristics of low pollution and low energy consumption, the magnetohydrodynamics has made widely attention. This paper provides the standard energy method to solve the magnetohydrodynamic equations (MHD) in the half space ℝ + 3 . It proves the global existence for the compressible (MHD) by combining the careful a priori estimates and the local existence result. This study also considers the large time behaviors of the solutions.ResultsThe interactions between the viscous, compressible fluid motion and the magnetic field are modeled by the magnetohydrodynamic system which describes the coupling between the compressible Navier-Stokes equations and the magnetic equations. This study has applied the analytical method to obtain the solutions to (MHD) in ℝ + 3 . It proves that under the assumption that the initial data are close to the constant state, the global existence of smooth solutions can be established. Moreover, the various decay rates of such solutions in Lp-norm with 2≤p≤+∞ and their derivatives in L2-norm can also be derived from combining the decay estimates of the linearized system and the energy method.ConclusionsThis study demonstrates that the global existence and the decay rates for the compressible (MHD) can be established under the similar initial assumptions as for the compressible Navier-Stokes equations. Especially, the results suggest that if the initial velocity is small, the velocity decays at a certain rate. This implies that only under the initial assumption that the data are large, it may reach the requirements of (MHD) power generation, which can be used to achieve the value of industrial application and environmental protection.

Highlights

With the characteristics of low pollution and low energy consumption, the magnetohydrodynamics has made widely attention
To complete the (MHD) generation process, which is of high industrial application value, a conductive gas will be directed through a magnetic field with a large velocity, under a high temperature condition
In this paper, we demonstrate that the global existence and the decay rates for the compressible (MHD) in R3+ can be established under the similar initial assumptions as for the compressible Navier-Stokes equations which can be seen in (Matsumura and Nishida 1983)

Summary

Introduction

With the characteristics of low pollution and low energy consumption, the magnetohydrodynamics has made widely attention. This paper provides the standard energy method to solve the magnetohydrodynamic equations (MHD) in the half space R3+. It proves the global existence for the compressible (MHD) by combining the careful a priori estimates and the local existence result. Magnetohydrodynamics, which combines the environmental fluid mechanics and electrodynamics theories to study the interaction discipline between the conduction fluid and electromagnetic, is the theory of the macroscopic, and it has spanned a very large range of applications (Gerebeau et al 2006). In (Chen and Tan 2010, 2012) we established the global existence and decay rates of the smooth solutions for the Cauchy problem. We will extend our results (Chen and Tan 2010) to the initial boundary problem in the half space

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