Invariant measures for the stochastic one-dimensional compressible Navier-Stokes equations

TL;DR

This paper establishes the existence of invariant measures for the long-term behavior of solutions to a stochastically forced one-dimensional compressible Navier-Stokes system, using advanced probabilistic and analytical techniques.

Contribution

It introduces a novel approach to prove invariant measures for non-Feller Markov semigroups on non-complete spaces in the context of stochastic fluid dynamics.

Findings

Existence of invariant measures for the stochastic compressible Navier-Stokes equations.

Development of generalized Krylov-Bogoliubov method for non-Feller semigroups.

Polynomial and exponential moment bounds for solutions.

Abstract

We investigate the long-time behavior of solutions to a stochastically forced one-dimensional Navier-Stokes system, describing the motion of a compressible viscous fluid, in the case of linear pressure law. We prove existence of an invariant measure for the Markov process generated by strong solutions. We overcome the difficulties of working with non-Feller Markov semigroups on non-complete metric spaces by generalizing the classical Krylov-Bogoliubov method, and by providing suitable polynomial and exponential moment bounds on the solution, together with pathwise estimates.