Characteristic Decompositions And Interaction Of Rarefaction Waves To The 2-D Magnetohydrodynamic Equations

This paper mainly study simple waves and characteristic decompositions of two-dimensional magnetohydrodynamic(MHD)system and the expansion problem of a wedge of gas into vacuum to the two dimensional isentropic irrotational pseudo-steady MHD equations for the polytropic perfect gas.In the last section of this paper,we also study the limit behaviors of the Riemann solutions for generalized Chaplygin gas as magnetic field vanishes,This paper is constructed as follows.In Chapter 1,we mainly introduce the background and some developments of the MHD equations.Furthermore,the derivation and simplification of the original equations are given.In Chapter 2,we present some basic concepts of the hyperbolic conservation equations and the definitions of the 1-D and 2-D Riemann problems.In Chapter 3,we obtain the characteristic decompositions of the two dimen-sional compressible magnetohydrodynamics system for a polytropic van der Waals gas and a poly tropic perfect gas.We use these characteristic decompositions to establish a proof that any wave adjacent to a constant state is a simple wave.This paper extends the the well-known result on the reducible equations in Courant and Friedrichs' book "supersonic flow and shock waves",that any hyperbolic state ad-jacent to a constant state must be a simple wave,and the results on the two-dimensional compressible Euler equations.In Chapter 4,we study the expansion of a wedge of magnetic fluid into vacuum.The magnetic fluid away from the sharp corner of the wedge expands into the vacuum as two symmetrical planar rarefaction waves.The problem can be reduced to the interaction of the two rarefaction waves.In order to determine the flow in the interaction zone,we consider a Goursat problem for a two-dimensional(2D)self-similar magnetohydrodynamic(MHD)system.The 2D self-similar MHD system is a mixed type system,and the type in each point is determined by the local fluid velocity and the local magneto-acoustic speed.We prove that the system is uniform hyperbolic in the interaction zone when the half-angle of the wedge is between 0 and 2?(?0).The existence of global classical solution to the Goursat problem is obtained by using characteristic decomposition method.In Chapter 5,we concern about the Euler equations in the magnetogasdynamics for generalized Chaplygin gas.The global solutions to the Riemann problem are obtained constructively by using phase plane analysis method.The limit behaviors of the Riemann solutions as magnetic field vanishes are also obtained.