The Asymptotic Stability Of The Viscous Shock Wave Solutions Of Hyperbolic Conservation Law Equations With Positive Definite Viscosity

In this paper,we study the asymptotic stability of viscous shock solutions of hyperbolic conservation law equations with positive definite viscosity of divergence type.The results show that when the corresponding inviscid equation has a shock solution,the solution of the viscous equation is the shock solution plus a small disturbance,and when the time tends to positive infinity,the solution of the viscosity equation approaches shock solution for inviscid equations.This paper is arranged as follows.In the first chapter,we briefly introduce the research history of the Riemann problem of the conservation law equations related to this paper,the research status of asymptotic stability of shock waves,rarefaction waves,and contact discontinuities,and the main content of this paper.In the second chapter,we introduce the stability theorem of hyperbolic conservation laws with positive definite viscosity and the properties of weak shock.In the third chapter,the proof of stability theorem is given.The method of energy estimation is used to estimate the main wave first,then to estimate the equation of higher order,and finally to estimate vertically,so as to obtain the stability theorem.Finally we prove some priori estimates by energy method and then give the proof of our main results by the continuous induction.In the forth chapter,two examples of solutions to systems of equations with positive definite viscosity are given.