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

TL;DR

This paper constructs global Hölder continuous solutions to stochastic 3D Euler equations using stochastic convex integration, demonstrating existence of infinitely many solutions with specific regularity properties.

Contribution

It introduces a novel application of stochastic convex integration to the 3D Euler equations, producing global Hölder continuous solutions and establishing infinite solution multiplicity.

Findings

Existence of global Hölder continuous solutions via stochastic convex integration.

Construction of infinitely many solutions with specified regularity.

Uniform moment estimates independent of time.

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 $\vartheta'>0$.