Global existence of the nonisentropic compressible Euler equations with vacuum boundary surrounding a variable entropy state

Abstract

Global existence for the nonisentropic compressible Euler equations with vacuum boundary for all adiabatic constants γ > 1 is shown through perturbations around a rich class of background nonisentropic affine motions. The notable feature of the nonisentropic motion lies in the presence of non-constant entropies, and it brings a new mathematical challenge to the stability analysis of nonisentropic affine motions. In particular, the estimation of the curl terms requires a careful use of algebraic, nonlinear structure of the pressure. With suitable regularity of the underlying affine entropy, we are able to adapt the weighted energy method developed for the isentropic Euler Hadžić and Jang (2018 Inventiones Mathematicae 214 1205–1266) to the nonisentropic problem. For large γ values, inspired by Shkoller and Sideris (2019 Arch. Ration. Mech. Anal. 234 115), we use time-dependent weights that allow some of the top-order norms to potentially grow as the time variable tends to infinity. We also exploit coercivity estimates here via the fundamental theorem of calculus in time variable for norms which are not top-order.