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Abstract:We study stochastic Euler equations in compressible and incompressible regimes, on the whole space and the torus, driven by mixed multiplicative noise: continuous Stratonovich/Itô components and a discontinuous Marcus component. The noise amplitudes are pseudo-differential operators including the transport operator. We develop a local-in-time theory of classical solutions, establishing existence, uniqueness, and a blow-up criterion. Discontinuous Marcus noise requires new analytical tools to control interactions between jump discontinuities and nonlocal operators.
For compressible equations, we formulate a transformation-and-extension principle recovering the Makino variable and its symmetric quasilinear formulation. Using transformed variables, compatible nonlinear Sobolev estimates are developed to close high-order stochastic energy bounds. This accommodates broad physically relevant state equations, including piecewise $\gamma$-laws, Chaplygin laws, and the white dwarf pressure law. Many equations remain unexplored in multidimensional stochastic compressible settings, even under pure Itô forcing.
For the incompressible damped case, we identify damping--noise regimes guaranteeing global existence, uniform bounds, and decay. To study statistical behavior, we establish a novel existence criterion for invariant probability measures tailored to Markov semigroups satisfying a \emph{restricted Feller property under mismatched metrics}. Bypassing single-topology Feller continuity robustly extends the Krylov--Bogoliubov theory. We use this to construct invariant measures for singular stochastic evolution systems in Hilbert spaces. For mixed multiplicative noise and $d\ge 2$, we prove existence of invariant measures for stochastic damped Euler equations on $\mathbb{T}^d$ under moderate damping--noise, and uniqueness under strong damping--noise on $\mathbb{T}^d$ and $\mathbb{R}^d$.

Submission history

From: Hao Tang [view email]
[v1] Sat, 16 May 2026 12:36:43 UTC (106 KB)
[v2] Mon, 22 Jun 2026 14:38:41 UTC (111 KB)
[v3] Tue, 18 Aug 2026 01:23:28 UTC (112 KB)