The Existence Of Global Solutions To The Two Dimensional Riemann Problems For Pressure Gradient System

This dissertation is devoted to the study of the global existence of the solutions to the two-dimensional Riemann problems for the pressure gradient system.The Riemann problem is to consider the intersection between two shocks and two contact discontinuities.High-dimensional conservation laws have broad applications in physics,whose typical example is the Euler system.However,due to the strong non-linearity of the system,there is still no general mathematical theory for two-dimensional Euler system.Hence the study of the pressure gradient system,one of the approximated models of Euler system,is of important theoretical value.This dissertation is organized as follows.In Chapter One,we introduce the previous research of the two-dimensional Riemann problem for the pressure gradient system and the main problem of this dissertation: the global existence of the solutions to the interaction between two shock waves and two contact discontinuities.In Chapter Two,we restate the above problem as a free boundary problem in polar coordinates,and then derive the boundary conditions on each boundary.We also illustrate the main result of this dissertation.In Chapter Three,we prove the global existence of the solutions of the free boundary problem.We also show that the diffracted shock wave as a free boundary would not intersect with the sonic circle corresponding to the internal state.