Gradient blowup of smooth vacuum solutions to 1D compressible Euler equations

TL;DR

This paper studies the formation of gradient blowup at vacuum boundaries in 1D compressible Euler equations, showing solutions transition from smooth to less regular in finite time.

Contribution

It introduces a class of solutions that evolve from smooth initial data to gradient blowup near the vacuum boundary, based on stability analysis of self-similar solutions.

Findings

Solutions become $C^{1- ext{mu}}$ near the boundary in finite time.

Gradient blowup occurs at the vacuum boundary.

The analysis is based on stability of self-similar waiting time solutions.

Abstract

We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and square-integrable, and which, in finite time, transition to $C^{1-\mu}$ regularity for $\mu \in [1/2,1)$ near the boundary, leading to the gradient blowup at the boundary. It is based on stability analysis of self-similar waiting time solutions \cite{JLN2025} recently constructed by the authors.