Singularity formation for the supersonic inward wave of compressible Euler equations with radial symmetry

TL;DR

This paper investigates how smooth solutions to the radially symmetric compressible Euler equations develop singularities in finite time, focusing on initial supersonic inward waves for gases with specific adiabatic indices.

Contribution

It demonstrates finite-time singularity formation for smooth solutions in radially symmetric Euler equations with certain initial conditions and gas parameters, using characteristic and invariant domain methods.

Findings

Singularity forms in finite time for $\gamma \geq 3$

Smooth solutions develop singularities from initial supersonic inward waves

Applicable to polytropic ideal gases with radial symmetry

Abstract

In this paper, we consider the singularity formation of smooth solutions for the compressible radially symmetric Euler equations. By applying the characteristic method and the invariant domain idea, we show that, for polytropic ideal gases with $\gamma\geq3$, the smooth solution develops a singularity in finite time for a class of initial supersonic inward waves.