| 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 comprising a continuous component (in both the Stratonovich and It\^o senses) and a discontinuous component (in the Marcus sense). The Stratonovich/Marcus noise amplitudes are (nonlocal) pseudo-differential operators that include, as special cases, the classical transport operator. Within this setting, we develop a local-in-time theory of classical solutions for both regimes, establishing existence, uniqueness, and a blow-up criterion. The inclusion of discontinuous pseudo-differential Marcus noise is novel and necessitates additional analytical techniques to control the interaction between jumps and nonlocal operators.
In the compressible barotropic case, we consider a broad class of physically relevant pressure laws extending far beyond the polytropic $\gamma$-law. This class includes piecewise-defined $\gamma$-laws, (piecewise-defined) Chaplygin-type laws, the pressure of white dwarf stars, and other astrophysically relevant regimes. These pressure laws have not been analyzed in the literature on stochastic compressible fluids, even under purely It\^o-type forcing.
For the incompressible damped equations, we specify a hierarchy of damping-noise conditions of increasing strength that yield global-in-time existence, uniform-in-time bounds, and decay estimates, respectively. To analyze invariant measures, we formulate an abstract singular stochastic evolution system that captures Euler-specific features, notably the mismatch between the topology in which solutions are constructed and the topology in which the Feller property holds. We extend the Krylov--Bogoliubov argument to accommodate this mismatch, establishing existence and uniqueness of invariant measures for the stochastic incompressible damped Euler equations. This provides the first positive answer to a strengthened version of Shirikyan`s problem. |
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