Decay Rate Of The Solution To The Compressible Magnetohydrodynamic Equations In R~3

The present paper is dedicated to the decay rate of the compress-ible magnetohydrodynamic system in R~3,under the condition of the initial perturbation at the constant steady state is small.Combining the linear decay rate of spectral analysis and the elaborate ener-gy estimates for the nonlinear system,we prove the solution of the magnetohydrodynamic equations converges to its constant equilibri-um state at the same decay rate as the linearized one,which shows that the convergence rate is optimal.This paper is divided into five chapters:In Chapter 1,we summarizes the background,significance and progress of the research on the magnetohydrodynamic equations.We briefly describes the main conclusions of this study and the d-ifficulties encountered in the process of proof,and introduces some preliminary knowledge in the end.In Chapter 2,we prove the existence of the solution.In Chapter 3,we obtain the decay rate of the solution for the linear system,through the spectral analysis on the semigroup.In Chapter 4,we prove the decay rate of the solution in the nonlinear system by using the method of energy estimates and the estimate results of the solutions of the linearized equations,we obtain the optimal decay rate estimates.In Chapter 5,we summarize and prospect this study.