Stochastically forced compressible Navier-Stokes equations with slip boundary conditions of friction type

TL;DR

This paper proves the existence of weak solutions for a stochastic compressible Navier-Stokes model with slip boundary conditions, combining advanced approximation schemes and convex analysis.

Contribution

It introduces a new weak solution concept for stochastic compressible Navier-Stokes equations with slip boundaries and proves their existence on smooth bounded domains.

Findings

Existence of weak solutions under slip boundary conditions.

Extension of approximation schemes to stochastic setting.

Integration of convex approximation methods for absolute values.

Abstract

We study a mathematical model of a compressible viscous fluid driven by stochastic forces under slip boundary conditions of friction type. We introduce a notion of a weak solution that is analytically and probabilistically consistent with this model. Our main result establishes the existence of such weak solutions under slip boundary conditions on bounded domains with $C^{2+\nu}$-boundary ($\nu>0$). The proof of this result combines an extended version of the four-layer approximation scheme on the torus by Breit/Feireisl/Hofmanov\'{a} (2018) with the convex approximation method for absolute value functions studied by Ne\v{c}asov\'{a}/Ogorzaly/Scherz (2023).