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 $\gamma >1$ 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 Hadžić 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.

arXiv comment52 pages, to appear in Nonlinearity. Please see arXiv:2007.03849 for a recent development in this direction

References in corpus (2)

• Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation
• Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum

Cited by in corpus (4)

• Recent developments in mathematical aspects of relativistic fluids
• Global solutions to the compressible Euler equations with heat transport by convection around Dyson's isothermal affine solutions
• The vacuum boundary problem for the spherically symmetric compressible Euler equations with positive density and unbounded entropy
• Non-isentropic cavity flow for the multi-d compressible Euler system