Stochastic Compressible Euler Equations with Frictional Damping: Existence of $L^\infty$ Martingale Solutions and Asymptotic Porous Medium-Like Behavior

TL;DR

This paper proves the existence of global solutions for stochastic compressible Euler equations with damping and shows these solutions converge exponentially to a steady state, resembling porous medium behavior, under stochastic forcing.

Contribution

It establishes the first rigorous pathwise convergence result for long-time behavior of stochastic Euler equations with damping, including existence of solutions and asymptotic analysis.

Findings

Solutions exist globally in time with entropy inequalities.

Solutions converge exponentially to a steady state almost surely.

The density approximates the porous medium equation asymptotically.

Abstract

We study the one-dimensional isentropic compressible Euler equations with linear (frictional) damping, subject to multiplicative, white-in-time stochastic forcing. The system is posed on a bounded interval with $L^\infty$ initial data and Dirichlet boundary conditions imposed on the momentum. We establish the global-in-time existence of $L^\infty$ martingale solutions that satisfy an appropriate entropy inequality. Then, we analyze the long-time behavior of these solutions and show that, under suitable assumptions on the noise, they converge almost surely and exponentially fast to a constant steady state of the system. The limiting density is well-approximated by the asymptotic solution of the deterministic porous medium equation, while the momentum exhibits the asymptotic behavior predicted by Darcy's law. The analysis in the stochastic setting is delicate, as temporal white-noise perturbations can significantly influence the long-time statistics of the solution. Our approach hinges on deriving sharp moment estimates for the entropy, which enable us to quantify and ultimately prove the decay of stochastic effects. To the best of our knowledge, this work provides the first rigorous pathwise convergence result for the long-time behavior of solutions to the stochastic isentropic compressible Euler equations with linear damping.