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Stationary solutions to stochastic 3D Euler equations in Hölder space

Lin Lü, Rongchan Zhu

TL;DR

This work addresses the existence and non-uniqueness of global stationary Hölder continuous solutions to the 3D Euler equations on $\mathbb{T}^3$ driven by additive noise. It develops a stochastic convex integration framework with pathwise estimates and a noise-cutoff mechanism that yields a sequence of adapted approximate solutions converging to a limit in $C(\mathbb{R};C^{\vartheta})$, while controlling the Reynolds stress uniformly in time. The main contributions are the construction of infinitely many stationary, analytically weak solutions with Hölder regularity and finite energy bounds in expectation, together with a Krylov–Bogoliubov argument to obtain shift-invariant laws, establishing non-uniqueness in the stochastic setting. The results advance understanding of stochastic fluid dynamics under additive forcing and connect to Onsager-type regularity thresholds by showing Hölder regularity can persist in the stationary, probabilistic regime, aided by a novel combination of stochastic convex integration and pathwise estimates.

Abstract

We establish the existence of infinitely many global and stationary solutions in $C(\mathbb{R};C^{\vartheta})$ space for some $\vartheta>0$ to the three dimensional Euler equations driven by an additive noise. The result is based on a new stochastic version of the convex integration method, incorporating the stochastic convex integration method developed in \cite{HZZ22b} and pathwise estimates to derive uniform moment estimates independent of time.

Stationary solutions to stochastic 3D Euler equations in Hölder space

TL;DR

This work addresses the existence and non-uniqueness of global stationary Hölder continuous solutions to the 3D Euler equations on driven by additive noise. It develops a stochastic convex integration framework with pathwise estimates and a noise-cutoff mechanism that yields a sequence of adapted approximate solutions converging to a limit in , while controlling the Reynolds stress uniformly in time. The main contributions are the construction of infinitely many stationary, analytically weak solutions with Hölder regularity and finite energy bounds in expectation, together with a Krylov–Bogoliubov argument to obtain shift-invariant laws, establishing non-uniqueness in the stochastic setting. The results advance understanding of stochastic fluid dynamics under additive forcing and connect to Onsager-type regularity thresholds by showing Hölder regularity can persist in the stationary, probabilistic regime, aided by a novel combination of stochastic convex integration and pathwise estimates.

Abstract

We establish the existence of infinitely many global and stationary solutions in space for some to the three dimensional Euler equations driven by an additive noise. The result is based on a new stochastic version of the convex integration method, incorporating the stochastic convex integration method developed in \cite{HZZ22b} and pathwise estimates to derive uniform moment estimates independent of time.
Paper Structure (30 sections, 12 theorems, 207 equations)

This paper contains 30 sections, 12 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Function spaces
  5. Probabilistic elements
  6. Stochastic convex integration with pathwise estimates and results
  7. Stochastic convex integration set-up and iterative proposition
  8. Proof of main result
  9. Convex integration scheme
  10. Choice of parameters
  11. Mollification
  12. Perturbation of the velocity
  13. Flow maps and cutoffs
  14. Construction of $w_{q+1}$
  15. Definition of the Reynolds stress $\mathring{R}_{q+1}$
  16. ...and 15 more sections

Key Result

Theorem 1.2

Suppose that $\mathrm {Tr}((-\Delta)^{\frac{3}{2}+\sigma}GG^*)<\infty$ for some $\sigma>0$, then there exists an analytically weak solution to eul1 in the sense of Definition d:sol with $(\mathcal{F}_{t})_{t\in{\mathbb R}}$ being the normal filtration generated by the Wiener process $B$, which belon

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 3.1
  • proof
  • Remark 3.2: On the choice of cut-off
  • Proposition 3.3
  • ...and 13 more