Sonic-supersonic solutions for the two-dimensional steady full Euler equations

TL;DR

This paper develops a new mathematical framework to analyze classical sonic-supersonic solutions near sonic curves for the two-dimensional full Euler equations, addressing degeneracy issues and establishing local smooth solutions.

Contribution

Introduces a novel variable transformation and analytical approach to construct classical solutions near sonic curves for the full Euler equations, a first in this field.

Findings

Established local smooth solutions near sonic curves

Revealed regularity-singularity structure of the equations

Provided a new method for analyzing degenerate hyperbolic systems

Abstract

This paper focuses on the structure of classical sonic-supersonic solutions near sonic curves for the two-dimensional full Euler equations in gas dynamics. In order to deal with the parabolic degeneracy near the sonic curve, a novel set of dependent and independent variables are introduced to transform the Euler equations into a new system of governing equations which displays a clear regularity-singularity structure. With the help of technical characteristic decompositions, the existence of a local smooth solution for the new system is first established in a weighted metric space by using the iteration method and then expressed in terms of the original physical variables. This is the first time to construct a classical sonic-supersonic solution near a sonic curve for the full Euler equations.