Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space

TL;DR

This paper proves the existence of weak martingale solutions for stochastic compressible Navier-Stokes equations and demonstrates their convergence to incompressible flows as the Mach number tends to zero, under random forcing.

Contribution

It establishes the existence of solutions and the incompressible limit for stochastic compressible flows with general nonlinear noise, extending previous deterministic results.

Findings

Existence of finite energy weak martingale solutions.

Convergence to incompressible system as Mach number approaches zero.

Handles general nonlinear multiplicative noise.

Abstract

We give an existence and asymptotic result for the so-called finite energy weak martingale solution of the compressible isentropic Navier--Stokes system driven by some random force in the whole spatial region. In particular, given a general nonlinear multiplicative noise, we establish the convergence to the incompressible system as the Mach number, representing the ratio between the average flow velocity and the speed of sound, approaches zero.