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

TL;DR

This work proves Onsager’s conjecture for the stochastic 3D Euler equations driven by a trace-class $GG^*$-Wiener noise on the torus. By decomposing the solution into a stochastic part $z=B$ and a convective part $v$, the authors develop a stochastic convex integration scheme with a novel energy inequality, enabling pathwise energy dissipation for Hölder exponents below the Onsager threshold and energy conservation above it. The construction yields infinitely many global, probabilistically strong and analytically weak solutions with prescribed energy profiles and initial data, with a Wong–Zakai type estimate linking the stochastic integral to the energy increment. This advances the stochastic Onsager program by achieving quantitative control of energy in a pathwise sense and establishing nonuniqueness at the critical regularity in a stochastic setting, thus bridging deterministic convex integration methods with stochastic analysis and turbulence theory.

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 $0<\bar{\vartheta}<\barβ<1/3$, 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 $C^{\barβ}(\mathbb{T}^3,\mathbb{R}^3)$. 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.

Theorems & Definitions (67)

• Conjecture 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Corollary 1.7
• Proposition 3.1
• Proposition 3.2
• Proposition 4.1