Stochastic Euler Equations with Pseudo-differential Noise: Continuous and Discontinuous Perturbations in Compressible and Incompressible Flows

TL;DR

This paper develops a new analytical framework for stochastic Euler equations with mixed continuous and discontinuous pseudo-differential noise, covering both compressible and incompressible flows, and establishes existence, long-term behavior, and invariant measures.

Contribution

It introduces novel tools to handle the interaction of jump discontinuities and nonlocal operators, extending the analysis to broad classes of pressure laws and solving open problems in stochastic fluid dynamics.

Findings

Established local-in-time classical solutions for stochastic Euler equations.

Generalized the Makino transform for a wide class of pressure laws.

Provided the first positive answer to Shirikyan's open problem on damped Euler equations with mixed noise.

Abstract

We study stochastic Euler equations in both compressible and incompressible regimes, on the whole space and on the torus, driven by genuinely mixed multiplicative noise: continuous Stratonovich/It\^o components and a discontinuous Marcus component. The Stratonovich and Marcus noise amplitudes are pseudo-differential operators. We develop a local-in-time theory of classical solutions for both regimes. The presence of pseudo-differential Marcus noise necessitates new analytical tools, which we develop to control the delicate interaction between jump discontinuities and nonlocal operators. We establish a transformation principle for the compressible barotropic case that generalizes the Makino transform beyond the polytropic setting and covers a broad class of physically relevant pressure laws outside the standard polytropic $\gamma$-law. This class includes (piecewise-defined) Chaplygin-type laws, the pressure law for white dwarf stars, etc. Most of these equations of state have not been analyzed in the stochastic compressible setting. For the incompressible damped case, we obtain further long-time behavior. We develop a criterion for the existence of invariant probability measures for a general Markov process accommodating mismatched topologies, extending the classical Krylov--Bogoliubov approach. This abstract criterion allows us to study invariant probability measures for a broad class of singular stochastic evolution systems in Hilbert spaces. Notably, this application gives what appears to be the first positive answer to Shirikyan's open problem on the damped Euler equations on $\mathbb T^2$ under genuinely mixed multiplicative noise. Furthermore, our framework goes beyond the original formulation of the problem: it resolves a substantially strengthened version in every dimension $d\ge 2$, on both $\mathbb T^d$ and $\mathbb R^d$.