Incompressible limit for compressible fluids with stochastic forcing

TL;DR

This paper investigates the behavior of stochastic compressible Navier-Stokes equations as the Mach number approaches zero, establishing the convergence to an incompressible stochastic Navier-Stokes system using advanced probabilistic and analytical techniques.

Contribution

It introduces a novel approach combining weak martingale solutions and stochastic energy inequalities to rigorously derive the incompressible limit under stochastic forcing.

Findings

Convergence of compressible to incompressible stochastic Navier-Stokes equations

Identification of the limit system as a stochastic incompressible Navier-Stokes equation

Development of uniform bounds using stochastic energy inequalities

Abstract

We study the asymptotic behavior of the isentropic Navier-Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic incompressible Navier-Stokes system is identified as the limit problem.