Compressible Navier--Stokes system with transport noise

TL;DR

This paper investigates the compressible Navier--Stokes equations with transport noise, exploring different noise regularities and interpretations, including classical, rough path, and stochastic frameworks, to understand their effects on solutions.

Contribution

It introduces a comprehensive analysis of the Navier--Stokes system with transport noise across multiple noise regimes, connecting physical modeling and mathematical regularization.

Findings

Existence of solutions in classical weak sense for smooth noise

Extension to rough noise using rough path theory

Martingale solutions for Stratonovich Brownian noise

Abstract

We consider the barotropic Navier--Stokes system driven by a physically well-motivated transport noise in both continuity as well as momentum equation. We focus on three different situations: (i) the noise is smooth in time and the equations are understood as in the sense of the classical weak deterministic theory, (ii) the noise is rough in time and we interpret the equations in the framework of rough paths with unbounded rough drivers and (iii) we have a Brownian noise of Stratonovich type and study the existence of martingale solutions. The first situation serves as an approximation for (ii) and (iii), while (ii) and (iii) are motivated by recent results on the incompressible Navier--Stokes system concerning the physical modeling as well as regularization by noise.