Existence and singularity formation for the supersonic expanding wave of radially symmetric non-isentropic compressible Euler equations

TL;DR

This paper investigates the existence and formation of singularities in supersonic expanding waves for radially symmetric non-isentropic compressible Euler equations, providing conditions for smoothness and singularity development.

Contribution

It introduces gradient variables and invariant domains to analyze solution behavior, establishing criteria for smoothness and finite-time singularity formation.

Findings

Solutions are smooth if initial gradient variables are non-negative.

Singularities form in finite time if initial gradient variables are very negative.

Invariant domain construction aids in a priori estimates of solutions.

Abstract

This paper studies the existence and singularity formation of supersonic expanding waves for the radially symmetric non-isentropic compressible Euler equations of polytropic gases. We introduce a suitable pair of gradient variables to characterize the rarefaction and compression properties of the solutions. Based on their Riccati equations, we construct several useful invariant domains to establish a series of priori estimates of solutions under some assumptions on the initial data. We show that the solution is smooth in the characteristic triangle or quadrangle domain if both of these two gradient variables are non-negative at the initial time. On the other hand, when one of these two variables is very negative at some initial point, the solution forms a singularity in finite time.