The compressible Euler system with nonlocal pressure: Global existence and relaxation

TL;DR

This paper studies a modified compressible Euler system with a nonlocal pressure term, proving global existence of solutions near equilibrium, their convergence to classical solutions, and the limit behavior as friction increases, all in an unbounded domain.

Contribution

It introduces a nonlocal pressure modification to the Euler system, establishes global existence and uniqueness near equilibrium, and proves convergence to classical solutions and porous media equations as parameters vary.

Findings

Global existence and uniqueness of solutions near equilibrium

Convergence of solutions to classical Euler system as nonlocal parameter vanishes

Asymptotic behavior showing convergence to porous media equations with increasing friction

Abstract

We here investigate a modification of the compressible barotropic Euler system with friction, involving a fuzzy nonlocal pressure term in place of the conventional one. This nonlocal term is parameterized by $\epsilon$ > 0 and formally tends to the classical pressure when $\epsilon$ approaches zero. The central challenge is to establish that this system is a reliable approximation of the classical compressible Euler system. We establish the global existence and uniqueness of regular solutions in the neighborhood of the static state with density 1 and null velocity. Our results are demonstrated independently of the parameter $\epsilon$, which enable us to prove the convergence of solutions to those of the classical Euler system. Another consequence is the rigorous justification of the convergence of the mass equation to various versions of the porous media equation in the asymptotic limit where the friction tends to infinity. Note that our results are demonstrated in the whole space, which necessitates the use of a suitable L^1-in-time space framework.