Global existence of near-affine solutions to the compressible Euler equations

TL;DR

This paper proves the global existence and stability of near-affine solutions to the compressible Euler equations with vacuum states for all adiabatic indices greater than one, extending previous results beyond the prior gamma threshold.

Contribution

It establishes the global stability of affine solutions to the compressible Euler equations for all gamma > 1, surpassing previous limitations and including shallow water equations.

Findings

Global existence of solutions for all gamma > 1.

Stability of affine flows in vacuum states.

Extension to shallow water equations at gamma=2.

Abstract

We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris found a class of expanding finite degree-of-freedom global-in-time affine solutions, obtained by solving nonlinear ODEs. In three space dimensions, the stability of these affine solutions, and hence global existence of solutions, was established by Had\v{z}i\'{c} \& Jang with the pressure-density relation $p = \rho^\gamma$ with the constraint that $1< \gamma\le {\frac{5}{3}} $. They asked if a different approach could go beyond the $\gamma > {\frac{5}{3}} $ threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all $\gamma >1$, thus also establishing global existence for the shallow water equations when $\gamma=2$.