The Global Existence of Martingale Solutions to Stochastic Compressible Navier-Stokes Equations with Density-dependent Viscosity

TL;DR

This paper proves the global existence of martingale solutions for stochastic compressible Navier-Stokes equations with density-dependent viscosity and vacuum, extending deterministic results to stochastic settings with new approximation techniques.

Contribution

It introduces a novel regularization approach and artificial damping terms to establish existence of solutions in the stochastic case with lower regularity.

Findings

Established global martingale solutions for stochastic equations with density-dependent viscosity.

Developed a regularized system to handle low regularity in stochastic setting.

Implemented a new limiting process for artificial terms in the approximation.

Abstract

The global existence of martingale solutions to the compressible Navier-Stokes equations driven by stochastic external forces, with density-dependent viscosity and vacuum, is established in this paper. This work can be regarded as a stochastic version of the deterministic Navier-Stokes equations \cite{Vasseur-Yu2016} (Vasseur-Yu, Invent. Math., 206:935--974, 2016.), in which the global existence of weak solutions was established for adiabatic exponent $\gamma > 1$. For the stochastic case, the regularity of density and velocity is even worse for passing the limit in nonlinear terms. We design a regularized system to approximate the original system. To make up for the lack of regularity of velocity, we need to add an artificial Rayleigh damping term besides the artificial viscosity and damping forces in \cite{Vasseur-Yu-q2016,Vasseur-Yu2016}. Moreover, we have to send the artificial terms to $0$ in a different order.