Regularity Of Solutions To Shock Diffraction For The Two-dimensional Pressure Gradient System

The pressure gradient system,as a subsystem of the Euler equations,is considered to be a good approximate model of the Euler equations.The shock diffraction problem,which occurs in many important physical problems,is an important cornerstone of the study of multi-dimensional conservation laws.It also plays a significant role in the study of mathematical fluid dynamics and other related problems.Therefore,the study of shock diffraction problem for the pressure gradient system is of great significance both in theory and in practice.In this thesis,the regularity of solution near the degenerate boundary to shock diffraction problem of two-dimensional pressure gradient equations in gas dynamics is studied.The thesis is organized as follows.Chapter One is devoted to introducing the derivation of the pressure gradient system and its research progress.The mathematical formulation of the shock diffraction problem and the main research methods involved in this thesis are also illustrated.In Chapter Two,we study the shock diffraction problem of the two-dimensional pres-sure gradient system and transform it into a free boundary value problem of nonlinear degenerate elliptic partial differential equations of second order.Meanwhile,we give the main theorem of this thesis:the solution is C^1,?up to the degenerate boundary in the subsonic domain.Moreover,the C^0,1regularity is optimal for the solution across the de-generate boundary and at the intersection of the degenerate boundary and the diffraction shock.In Chapter Three,we obtain some estimates of the solution by the maximum princi-ple,and establish the regularity theory of the solution near the degenerate boundary.