A proof of Onsager's conjecture for the stochastic 3D Euler equations

This paper investigates the stochastic 3D Euler equations on a periodic domain ${\mathbb{T}}^3$, driven by a $G{G}^{\ast }$-Wiener process $B$ of trace class: \begin{align*} $\mathrm{d}$ u+$\mathrm{d}\mathrm{i}\mathrm{v}$(u\otimes u),$\mathrm{d}$ t+\nabla p,$\mathrm{d}$t=$\mathrm{d}$B, \quad $\mathrm{d}\mathrm{i}\mathrm{v}$ u=0. \end{align*} First, for any $\vartheta <1\mathrm{/}3$, we construct infinitely many global-in-time probabilistically strong and analytically weak solutions $u\in C([0,\mathrm{\infty }),C^{\vartheta }({\mathbb{T}}^3,{\mathbb{R}}^3))$. These solutions dissipate the energy pathwisely up to a stopping time $\mathfrak{t}$, which can be chosen arbitrarily large with high probability, i.e. it holds almost surely \begin{align*} |u(t\wedge\mathfrak{t})|{L^2}^2< |u(s\wedge\mathfrak{t})|{L^2}^2 +2 \int_{s\wedge\mathfrak{t}}^{t\wedge\mathfrak{t}} \big\langle u®, $\mathrm{d}$ B® \big\rangle +$\mathrm{T}\mathrm{r}$\big(GG^\big) (t\wedge\mathfrak{t}-s\wedge\mathfrak{t}), \end{align} for any $0\le s<t<\mathrm{\infty }$. We also provide a brief proof of energy conservation for $\vartheta >1\mathrm{/}3$ based on \cite{CET94}, thereby confirming the Onsager theorem for the stochastic 3D Euler equations. Second, let $0<\bar{\vartheta }<\bar{\beta }<1\mathrm{/}3$, we construct infinitely many global-in-time probabilistically strong and analytically weak solutions in $C([0,\mathrm{\infty }),C^{\bar{\vartheta }}({\mathbb{T}}^3,{\mathbb{R}}^3))$ for arbitrary divergence-free initial data in $C^{\bar{\beta }}({\mathbb{T}}^3,{\mathbb{R}}^3)$. Our construction relies on the convex integration method developed in the deterministic setting by \cite{Ise18}, adapting it to the stochastic context by introducing a novel energy inequality into the convex integration scheme and combining stochastic analysis arguments with a Wong–Zakai type estimate.