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

Abstract

This paper investigates the stochastic 3D Euler equations on a periodic domain $\mathbb{T}^3$, driven by a $GG^*$-Wiener process $B$ of trace class: \begin{align*} \mathrm{d} u+\mathrm{div}(u\otimes u)\,\mathrm{d} t+\nabla p\,\mathrm{d}t=\mathrm{d}B, \quad \mathrm{div} u=0. \end{align*} First, for any $\vartheta<1/3$, we construct infinitely many global-in-time probabilistically strong and analytically weak solutions $u\in C([0,\infty),C^{\vartheta}(\mathbb{T}^3,\mathbb{R}^3))$. 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(r), \mathrm{d} B(r) \big\rangle +\mathrm{Tr}\big(GG^*\big) (t\wedge\mathfrak{t}-s\wedge\mathfrak{t}), \end{align*} for any $0\leq s < t<\infty$. We also provide a brief proof of energy conservation for $\vartheta>1/3$ based on \cite{CET94}, thereby confirming the Onsager theorem for the stochastic 3D Euler equations. Second, let $0<\bar{\vartheta}<\bar{\beta}<1/3$, we construct infinitely many global-in-time probabilistically strong and analytically weak solutions in $C([0,\infty),C^{\bar{\vartheta}}(\mathbb{T}^3,\mathbb{R}^3))$ 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.