Markov selection for the stochastic compressible Navier--Stokes system

TL;DR

This paper establishes the existence of a Markov selection for solutions to the stochastic compressible Navier--Stokes system, addressing unique challenges posed by the non-continuous velocity field and auxiliary variables.

Contribution

It introduces a novel Markov selection method for the stochastic compressible Navier--Stokes equations, overcoming difficulties due to non-continuous velocity fields and variable recovery.

Findings

Existence of an almost sure Markov selection for the system

Overcoming the non-continuity of the velocity field in time

Introduction of an auxiliary variable to facilitate the Markov selection

Abstract

We analyze the Markov property of solutions to the compressible Navier--Stokes system perturbed by a general multiplicative stochastic forcing. We show the existence of an almost sure Markov selection to the associated martingale problem. Our proof is based on the abstract framework introduced in [F. Flandoli, M. Romito: Markov selections for the 3D stochastic Navier--Stokes equations. Probab. Theory Relat. Fields 140, 407--458. (2008)]. A major difficulty arises from the fact, different from the incompressible case, that the velocity field is not continuous in time. In addition, it cannot be recovered from the variables whose time evolution is described by the Navier--Stokes system, namely, the density and the momentum. We overcome this issue by introducing an auxiliary variable into the Markov selection procedure.