Global Existence of Near-Affine Solutions to the Compressible Euler Equations

Abstract

We establish the global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into a 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 (Arch Ration Mech Anal 225(1):141–176, 2017) 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 the global existence of solutions, was established by Hadžic and Jang (Expanding large global solutions of the equations of compressible fluid mechanics, 2016) with the pressure-density relation $$p = \rho ^\gamma $$ with the constraint that $$1< \gamma \leqslant {\frac{5}{3}} $$ ; 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 the stability of affine flows and global existence for all $$\gamma >1$$ , thus also establishing global existence for the shallow water equations when $$\gamma =2$$ .