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

## TL;DR

This paper proves the global existence of solutions to the nonisentropic compressible Euler equations with vacuum boundary conditions around a class of affine motions, addressing challenges posed by variable entropy.

## Contribution

It extends the stability analysis of affine motions to nonisentropic cases with variable entropy, adapting energy methods to handle new mathematical complexities.

## Key findings

- Global solutions exist for all adiabatic constants $b3 > 1$.
- The energy method is successfully adapted to nonisentropic flows.
- Time-dependent weights help control norm growth for large $b3$.

## Abstract

Global existence for the nonisentropic compressible Euler equations with vacuum boundary for all adiabatic constants $\gamma > 1$ 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 by Had\v{z}i\'c and Jang to the nonisentropic problem. For large $\gamma$ values, inspired by Shkoller and Sideris, 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.