On A Degenerate Mixed-type Boundary Value Problem To The 2-D Steady Euler Equations

In this paper,we study the existence of the supersonic solution near the sonic curves arising from the transonic channel flow problems in gas dynamics.Given two smooth curves,one of which is a sonic curve and the other is a characteristic curve,we construct a local classical supersonic solution for the two-dimensional steady Euler equations in the angular region.In the second chapter,we introduce the idea of characteristic decomposition.The existence conditions of characteristic decomposition for the general 2 × 2 quasilinear hyperbolic equations are provided.Then we derive the system of the two-dimensional Euler equations under the isentropic and irrotational assumptions.By introducing characteristic tilt angles as variables,we derive the characteristic decompositions of the Euler equations.In the third chapter,we study the degenerate boundary value problem of twodimensional steady isentropic irrotational Euler equations.Firstly,we describe the problem and present the main result of this paper under the given boundary conditions.Secondly,we divide the proof of the conclusion into three steps.The first step is to introduce partial hodograph transformation and unknown function transformation,and to transform the equations into a linear system with clear singularity-regularity structures.The second step is to prove that the iterative sequence of integral equations satisfies a series of estimates,The third step is to establish the existence and uniqueness of the classical solution of the linear system by using the iterative method.Finally,we return the solution under the plane of partial hodograph to the original physical plane to obtain the sonic-supersonic classical solution of the original Euler equations.Finally,we summarize the conclusion of this paper and give some discussions about the future work in the fourth chapter.