Asymptotic Stability Of Viscous Shock Profiles For Compressible Planar Magnetohydrodynamics System

Magnetohydrodynamic system(MHD)is an important branch of physics that studies the interaction of plasma and other conductive fluids with electromagnetic fields.As far as the semiconductor magnetohydrodynamic model which is a kind of symmetrical quasi-linear hyperbolic equations.But generally speaking,even if the initial value of the equations is sufficiently smooth,the classical solutions of the quasi-linear hyperbolic equations will still blow up and produce shock waves in a finite time.Therefore,the study of shock wave solutions of magnetohydrodynamic systems is particularly important.In this article we will study the asymptotic stability of the viscous shock solution of the compressible planar magnetohydrodynamic system.The research content of the article is mainly divided into the following two parts.In the second chapter of the thesis,inspired by the special relationship between the magnetohydrodynamic system and the Navier-Stokes equation,we use coordinate transformation and primitive function methods to improve the problem.First find the shock wave velocity,then construct the traveling wave solution,and use the phase plane analysis method to get the existence,uniqueness,monotonicity and boundedness of the traveling wave solution and other important properties.Finally,the disturbance equation is analyzed to prove the stability of the shock solution.Specifically,we first put forward a reasonable a priori hypothesis,then analyze the disturbance equation on [0,T] through the method of energy estimation,and finally close the a priori hypothesis we proposed.In the third chapter of the paper,based on the results obtained on the time [0,T]in the second chapter,we use an iterative method to transform the local solution into the global solution,thereby obtaining the existence of the global solution to the Cauchy problem and the asymptotic stability of the viscous shock solution when the time tends to infinity.