Measure-valued solutions to the stochastic compressible Euler equations and incompressible limits

TL;DR

This paper introduces a new class of solutions for stochastic compressible Euler equations, establishes a relative energy inequality, and studies the incompressible limit under general stochastic perturbations.

Contribution

It develops dissipative measure-valued martingale solutions that incorporate the probability space as part of the solution, extending the theory to nonlinear multiplicative stochastic perturbations.

Findings

Established pathwise weak-strong uniqueness principle.

Derived a relative energy inequality for the stochastic system.

Analyzed the low Mach (incompressible) limit under stochastic perturbations.

Abstract

We introduce a new concept of dissipative measure-valued martingale solutions to the stochastic compressible Euler equations. These solutions are weak in the probabilistic sense i.e., the probability space and the driving Wiener process are an integral part of the solution. We derive the relative energy inequality for the stochastic compressible Euler equations and, as a corollary, we exhibit pathwise weak-strong uniqueness principle. Moreover, making use of the relative energy inequality, we investigate the low Mach limit (incompressible limit) of the underlying system of equations. As a main novelty with respect to the related literature, our results apply to general nonlinear multiplicative stochastic perturbations of Nemytskij type.