The Sonic-supersonic Structures For The Two-dimensional Nonlinear Wave System

This paper studies a kind of degenerate boundary value problems for the twodimensional self-similar nonlinear wave system,and explores the structure of solutions near the degenerate line.We describe the method of characteristic decomposition in chapter 2,which is the method used in this paper.We first introduce the basic idea of characteristic decomposition and than present the existence conditions of characteristic decomposition for general 2 × 2 hyperbolic systems.Finally,the characteristic decomposition of the two-dimensional self-similar nonlinear wave system is derived.We study a kind of degenerate boundary value problems widely existing in the two-dimensional Riemann problem of nonlinear wave system in Chapter 3.Due to the hyperbolic degeneracy on the boundary,we introduce a partial hodograph transformation to deal with the possible singularity in the system.In terms of the new coordinates,the nonlinear wave system is transformed to a hyperbolic equations,which display clear regular-singularity structures.The local existence of classical solutions for the new system is established in a weighted metric space.Returning the solution to the original variables,we obtain the existence of classical solutions to the degenerate boundary value problem for the nonlinear wave system.