Waiting Time Solutions in gas dynamics

TL;DR

This paper constructs a family of self-similar solutions to the 1D compressible Euler equations with vacuum boundary conditions, revealing a transition from smooth to Hölder continuous behavior near singular points and the emergence of weak discontinuities.

Contribution

It introduces a continuum of self-similar waiting time solutions for the Euler equations with vacuum boundary, detailing their regularity transition and boundary behavior.

Findings

Solutions are confined by a stationary vacuum interface for finite time.

Solutions transition from $C^1$ regularity to Hölder continuity near singular points.

A weak discontinuity emerges along the sonic curve when the boundary starts moving.

Abstract

In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent $\ga\in(1,3)$ in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least $C^1$ regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only H\"{o}lder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.