H\"{o}lder continuous solutions to stochastic 3D Euler equations via stochastic convex integration

Abstract

In this paper, we are concerned with the three dimensional Euler equations driven by an additive stochastic forcing. First, we construct global H\"{o}lder continuous (stationary) solutions in $C(\mathbb{R};C^{\vartheta})$ space for some $\vartheta>0$ via a different method from \cite{LZ24}. Our approach is based on applying stochastic convex integration to the construction of Euler flows in \cite{DelSze13} to derive uniform moment estimates independent of time. Second, for any divergence-free H\"{o}lder continuous initial condition, we show the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in $L^p_{\rm{loc}}([0,\infty);C^{\vartheta'}) \cap C_{\rm{loc}}([0,\infty);H^{-1})$ for all $p\in [1,\infty)$ and some $\vartheta'>0$.