Measure-valued low Mach number limits of ideal fluids

TL;DR

This paper develops a measure-valued solution framework for the low Mach number limit of ideal fluids, providing conditions for convergence and revealing additional information beyond classical weak solutions.

Contribution

It introduces a measure-valued solution concept explicitly incorporating pressure dependence and establishes criteria for low Mach limit convergence.

Findings

Sufficient conditions for measure-valued solutions to converge in low Mach limit

A Jensen-type inequality as a necessary condition for limits

Measure-valued solutions contain more information than classical weak solutions

Abstract

As a framework for handling low Mach number limits we consider a notion of measure-valued solution of the incompressible Euler system which explicitly formulates the usually suppressed dependence on the pressure variable. We clarify in which sense such a measure-valued solution is generated as a low Mach limit and state sufficient conditions for this convergence. For the special case of pressure-free solutions we are able to give such sufficient conditions only on the incompressible solution itself. As a necessary condition for low Mach limits we obtain a Jensen-type inequality. The same necessary condition actually holds true for limits of weak solutions or vanishing viscosity limits. We illustrate that this Jensen inequality is not trivially fulfilled. Since measure-valued solutions in the classical sense are always generated by weak solutions, this shows that our solution concept contains more information and low Mach limits lead to a partial selection criterion for incompressible solutions.