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Hölder continuous solutions to stochastic 3D Euler equations via stochastic convex integration

Lin Lü

TL;DR

The paper tackles the 3D Euler equations with additive stochastic forcing on the torus and achieves global Hölder regularity through stochastic convex integration. By decomposing the solution into a stochastic convolution $z$ and a nonlinear velocity component $v$, and by evolving through a controlled frequency-boosting perturbation built from Beltrami waves, the authors obtain a sequence of approximate solutions with vanishing Reynolds stress and uniform moment bounds. They establish the existence of a global adapted Hölder solution $u=v+z$ and a stationary counterpart, while proving non-uniqueness via freedom in the energy functional. The second main result extends the framework to the Cauchy problem with divergence-free Hölder initial data, yielding infinitely many probabilistically strong and analytically weak global solutions with prescribed initial data through a stopping-time strategy, without requiring transport-estimate pathwise control. Overall, the work broadens stochastic convex integration to yield Hölder continuous, globally defined, non-unique solutions for stochastic 3D Euler and addresses well-posedness questions for prescribed initial data in a robust probabilistic setting.

Abstract

In this paper, we are concerned with the three dimensional Euler equations driven by an additive stochastic forcing. First, we construct global Hölder continuous (stationary) solutions in $C(\mathbb{R};C^{\vartheta})$ space for some $\vartheta>0$ via a different method from \cite{LZ24}. Our approach is based on applying stochastic convex integration to the construction of Euler flows in \cite{DelSze13} to derive uniform moment estimates independent of time. Second, for any divergence-free Hölder continuous initial condition, we show the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in $L^p_{\rm{loc}}([0,\infty);C^{\vartheta'}) \cap C_{\rm{loc}}([0,\infty);H^{-1})$ for all $p\in [1,\infty)$ and some $\vartheta'>0$.

Hölder continuous solutions to stochastic 3D Euler equations via stochastic convex integration

TL;DR

The paper tackles the 3D Euler equations with additive stochastic forcing on the torus and achieves global Hölder regularity through stochastic convex integration. By decomposing the solution into a stochastic convolution and a nonlinear velocity component , and by evolving through a controlled frequency-boosting perturbation built from Beltrami waves, the authors obtain a sequence of approximate solutions with vanishing Reynolds stress and uniform moment bounds. They establish the existence of a global adapted Hölder solution and a stationary counterpart, while proving non-uniqueness via freedom in the energy functional. The second main result extends the framework to the Cauchy problem with divergence-free Hölder initial data, yielding infinitely many probabilistically strong and analytically weak global solutions with prescribed initial data through a stopping-time strategy, without requiring transport-estimate pathwise control. Overall, the work broadens stochastic convex integration to yield Hölder continuous, globally defined, non-unique solutions for stochastic 3D Euler and addresses well-posedness questions for prescribed initial data in a robust probabilistic setting.

Abstract

In this paper, we are concerned with the three dimensional Euler equations driven by an additive stochastic forcing. First, we construct global Hölder continuous (stationary) solutions in space for some via a different method from \cite{LZ24}. Our approach is based on applying stochastic convex integration to the construction of Euler flows in \cite{DelSze13} to derive uniform moment estimates independent of time. Second, for any divergence-free Hölder continuous initial condition, we show the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in for all and some .
Paper Structure (29 sections, 13 theorems, 231 equations)

This paper contains 29 sections, 13 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Probabilistic elements
  5. Construction of Hölder continuous solutions via stochastic convex integration
  6. Proof of the main result
  7. Choice of parameters and Mollification
  8. Perturbation of the velocity
  9. Construction of amplitude functions $a_k$
  10. Construction of $v_{q+1}$
  11. Inductive estimates for $v_{q+1}$
  12. Definition of the Reynolds stress $\mathring{R}_{q+1}$
  13. Inductive estimates for $\mathring{R}_{q+1}$
  14. Estimates on the transport error
  15. Estimates on the oscillation error
  16. ...and 14 more sections

Key Result

Theorem 1.1

Suppose that $\mathrm {Tr}((-\Delta)^{3/2+\varkappa}GG^*)<\infty$ for some $\varkappa>0$, then Moreover, the solutions are non-unique, as their energy can be determined by various energy functions.

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 3.1
  • Proposition 3.2
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • Lemma 3.6
  • ...and 10 more