The Existence Of Solutions For Shock Diffraction Problems Of The Pressure Gradient System

In gas dynamics if we only consider the effect of the pressure and ignore the inertial,the Euler system can be simplified into the pressure gradient system.This dissertation is devoted to the study of the global existence of the solutions for the shock diffraction problem by a two-dimensional convex wedge,modeled by the pressure gradient system.Mathematically this shock diffraction problem can be described as a problem of a nonlinear degenerate elliptic equation with a free boundary.The dissertation is organized as follows.In Chapter One,we introduce the physical background of the pressure gradient system and describe the shock diffraction problem.In Chapter Two,we restate the shock diffraction problem as a free boundary problem in the self-similar coordinates,and illustrate the main results in this dissertation.In Chapter Three,we prove the global existence of the solutions to the shock diffraction problem.The proof is divided into three steeps.First,we add the regularized differential operator ?? to make sure that the governing equation is uniformly elliptic.We obtain the global existence of the solutions to the fixed boundary value problem by Perron method.Second,we prove the global existence and uniform estimates of the solutions to the free boundary problem by Schauder fixed point theorem.Finally,we show that the estimates of the solutions are independent of ?,then let ? ? 0 we get the global solution of the original problem.