Local strong solutions to the stochastic compressible Navier-Stokes system

TL;DR

This paper proves the local existence and uniqueness of strong solutions to the stochastic compressible Navier-Stokes equations, using advanced probabilistic and PDE techniques to handle the nonlinear stochastic forces.

Contribution

It introduces a novel approach combining symmetric hyperbolic reformulation and stochastic compactness to establish local strong solutions for the stochastic compressible Navier-Stokes system.

Findings

Established local in time existence of unique strong solutions

Developed a new approximation method using stochastic compactness

Proved pathwise uniqueness and solution regularity

Abstract

We study the Navier-Stokes system describing the motion of a compressible viscous fluid driven by a nonlinear multiplicative stochastic force. We establish local in time existence (up to a positive stopping time) of a unique solution, which is strong in both PDE and probabilistic sense. Our approach relies on rewriting the problem as a symmetric hyperbolic system augmented by partial diffusion, which is solved via a suitable approximation procedure using the stochastic compactness method and the Yamada-Watanabe type argument based on the Gy\"ongy-Krylov characterization of convergence in probability. This leads to the existence of a strong (in the PDE sense) pathwise solution. Finally, we use various stopping time arguments to establish the local existence of a unique strong solution to the original problem.