A Degenerate Boundary Value Problem To The 2-D Pseudo-steady Euler Equations

In this thesis,we mainly consider the existence of supersonic solutions near the sonic curves for the two-dimensional Riemann problems in gas dynamics.Given a smooth curve as a sonic curve.In the self-similar plane,we study the degenerate boundary value problems to the two-dimensional pseudo-steady Euler equations under the isentropic and irrotational assumptions.This system is degenerate hyperbolic on the sonic curve.We introduce a suitable partial hodograph transformation to deal with the singularity,to transform the original problem into a new degenerate initial value problem.We establish the local existence and uniqueness of the classical solution of the new problem.The work of this thesis is mainly divided into four chapters.The first chapter describes the research background of the two-dimensional Euler equations,presents the research content and structural arrangement of this paper.In the second chapter,we introduce the idea of characteristic decomposition.By introducing the pseudo flow angle and the pseudo Mach angle as dependent variables,we derive the characteristic decompositions of the two-dimensional isentropic irrotational pseudo-steady Euler equations.Then we reconstruct the problem under the angle variables.In the third chapter,we introduce a partial hodograph transformation to transform the Euler equations into a new degenerate hyperbolic system with a clear singularity structure.Then we restate the problem and the main conclusion.Using the iterative method,we prove the existence and uniqueness of the classical solutions of the new system.In the fourth chapter,we summarize the main contents and conclusions of this thesis,and present some discussions on future works.