On the low Mach number limit for the compressible Euler system

TL;DR

This paper introduces a new approach using dissipative measure-valued solutions to analyze the low Mach number limit of the compressible Euler equations, demonstrating convergence to incompressible solutions even with complex initial data.

Contribution

It presents a novel framework for studying singular limits in fluid dynamics, accommodating both well-prepared and ill-prepared initial conditions.

Findings

Dissipative measure-valued solutions converge to incompressible Euler solutions as Mach number approaches zero.

The approach handles ill-prepared data with acoustic waves on unbounded domains.

Dispersion effects eliminate acoustic wave difficulties in the low Mach limit.

Abstract

In this paper, we propose a new approach to singular limits of inviscid fluid flows based on the concept of dissipative measure-valued solutions. We show that dissipative measure-valued solutions of the compressible Euler equations converge to the smooth solution of the incompressible Euler system when the Mach number tends to zero. This holds both for well-prepared and ill-prepared initial data, where in the latter case the presence of acoustic waves causes difficulties. However this effect is eliminated on unbounded domains thanks to dispersion.