Random Perturbations of Viscous Compressible Fluids: Global Existence of Weak Solutions

TL;DR

This paper proves the global existence of weak solutions for stochastic compressible Navier-Stokes equations with large initial data, using energy methods, martingale solutions, and layered approximation schemes within a bounded domain.

Contribution

It develops stochastic analogues of Lions' weak compactness techniques and combines multiple approximation schemes to establish well-posedness for stochastic compressible fluids.

Findings

Proved global existence of weak solutions for stochastic compressible Navier-Stokes equations.

Established energy-based methods for stochastic fluid systems.

Integrated deterministic and stochastic approximation techniques.

Abstract

This article is devoted to the well-posedness of the stochastic compressible Navier Stokes equations. We establish the global existence of an appropriate class of weak solutions emanating from large inital data, set within a bounded domain. The stochastic forcing is of multiplicative type, white in time and colored in space. Energy methods are used to merge techniques of P.L. Lions for the deterministic, compressible system with the theory of martingale solutions to the incompressible, stochastic system. Namely, we develop stochastic analogues of the weak compactness program of Lions, and use them to implement a martingale method. The existence proof involves four layers of approximating schemes. We combine the three layer scheme of Feiresil/Novotny/Petzeltova for the deterministic, compressible system with a time splitting method used by Berthelin/Vovelle for the one dimensional stochastic compressible Euler equations.