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

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 Hadžić \& 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 $\gamma =2$.

arXiv comment51 pages, details added to Section 4.7, to appear in Arch. Rational Mech. Anal