Statistically stationary solutions to the stochastic isentropic compressible Euler equations with linear damping

TL;DR

This work establishes the existence of statistically stationary weak martingale entropy solutions for the one-dimensional stochastic isentropic Euler equations with linear damping on the torus. It constructs approximate dynamics using a two-parameter scheme with truncation ${R}$ and artificial viscosity ${\varepsilon}$, together with regularized noise, and proves uniform-in-time bounds, invariant regions, and the existence of invariant measures for the approximate system. By passing to the limits ${R\to\infty}$ and ${\varepsilon\to0}$ through Skorohod representations and Young-measure techniques, the paper obtains a stationary solution to the original stochastic problem that satisfies an entropy inequality for a broad class of entropies. The results advance the understanding of long-time statistical behavior of stochastic inviscid fluids in 1D and provide a rigorous framework for invariant measures and stationary statistics in damped stochastic Euler dynamics. The methodology blends multi-level approximation, entropy methods, invariant-region techniques, and measure-valued compactness to address the nonlinearity and lack of compactness inherent in inviscid stochastic compressible flows.

Abstract

We study the long time behavior of isentropic compressible Euler equations with linear damping driven by a white-in-time noise, on a one-dimensional torus. We prove the existence of a statistically stationary solution in the class of weak martingale entropy solutions for any adiabatic constant $γ>1$, which satisfies an associated entropy inequality. To establish this result, we use a multi-level approximation scheme consisting of a truncation parameter $R$ and an artificial viscosity parameter $ε$. The truncated system preserves the structure of the regularized system with the artificial viscosity, thereby providing key properties such as an invariant region and non-existence of vacuum at the approximate level. These properties allow us to construct an invariant measure for the approximate system in both $R$ and $ε$ associated to a Feller semigroup for the well-posed dynamics of the approximate system for any $γ> 1$. This gives us a statistically stationary solution for the approximate problem, which we then successively pass to the limit as $R \to \infty$ and as $ε\to 0$ to obtain a statistically stationary solution to the original stochastic system. Our analysis is novel, using new techniques for establishing uniform bounds on entropies of all orders, which allow us to pass to the limit in the parameters. We believe that this result is a valuable step towards further understanding the long-time statistical behavior of the stochastic Euler equations in one spatial dimension.

Theorems & Definitions (105)

• Definition 1.1
• Definition 1.2
• Remark 1.1: Remarks on the definition of a weak martingale entropy solution
• Remark 1.2: Remark on admissible entropies
• Definition 1.3
• Theorem 1.1: Main theorem
• Remark 1.3
• Remark 1.4
• Proposition 1.1
• proof