A proof of Onsager's conjecture for the stochastic 3D Euler equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914152050589696 |
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| author | Lü, Huaxiang Lü, Lin Zhu, Rongchan |
| author_facet | Lü, Huaxiang Lü, Lin Zhu, Rongchan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06915 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A proof of Onsager's conjecture for the stochastic 3D Euler equations Lü, Huaxiang Lü, Lin Zhu, Rongchan Probability Analysis of PDEs 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. |
| title | A proof of Onsager's conjecture for the stochastic 3D Euler equations |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2505.06915 |