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Non-uniqueness of Hölder continuous solutions for stochastic Euler and Hypodissipative Navier-Stokes equations

Kush Kinra, Ujjwal Koley

Abstract

We construct infinitely many Hölder continuous, global-in-time, and stationary solutions to the stochastic Euler equations and the hypodissipative Navier-Stokes equations, taking values in the space $C(\mathbb{R};C^{\vartheta})$. For the Euler case, the Hölder exponent $\vartheta$ satisfies $0<\vartheta<\frac{5}{7}β$ with $0<β< \frac{1}{200}$, while for the hypodissipative Navier-Stokes equations, $β$ must additionally satisfy $0<β< \min\left\{ \frac{2(1-2α)}{21}, \frac{1}{200}\right\}$. The construction relies on a modified stochastic convex integration scheme, which is central to the analysis. This scheme incorporates Beltrami flows as building blocks and carefully tracks inductive estimates, both pathwise and in expectation. These refinements allow us to achieve improved Hölder regularity for solutions to the underlying stochastic equations, advancing the scope of convex integration techniques in the stochastic setting.

Non-uniqueness of Hölder continuous solutions for stochastic Euler and Hypodissipative Navier-Stokes equations

Abstract

We construct infinitely many Hölder continuous, global-in-time, and stationary solutions to the stochastic Euler equations and the hypodissipative Navier-Stokes equations, taking values in the space . For the Euler case, the Hölder exponent satisfies with , while for the hypodissipative Navier-Stokes equations, must additionally satisfy . The construction relies on a modified stochastic convex integration scheme, which is central to the analysis. This scheme incorporates Beltrami flows as building blocks and carefully tracks inductive estimates, both pathwise and in expectation. These refinements allow us to achieve improved Hölder regularity for solutions to the underlying stochastic equations, advancing the scope of convex integration techniques in the stochastic setting.
Paper Structure (40 sections, 22 theorems, 244 equations)

This paper contains 40 sections, 22 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. The equations
  3. Convex integration related results
  4. Scope of the paper
  5. Organization of the article
  6. Preliminaries
  7. Function spaces
  8. Inverse divergence operator $\mathcal{R}$
  9. Probabilistic elements
  10. Stochastic convex integration set-up and first main result for \ref{['eqn_stochatic_u-SHNSE']}
  11. Iterative proposition
  12. Main result
  13. Convex integration scheme for \ref{['eqn_stochatic_u-SHNSE']}
  14. Choice of parameters
  15. Mollification
  16. ...and 25 more sections

Key Result

Theorem 1.3

Assume that $\{\mathcal{F}_{t}\}_{t\in\mathbb{R}}$ is the normal filtration generated by the Wiener process $\mathrm{W}$, $\alpha\in(0,\frac{1}{2})$ and $\mathop{\mathrm{Tr}}((-\Delta)^{\frac{5}{2}}GG^{\ast})<\infty$, then there exist infinitely many analytically weak solutions to eqn_stochatic_u-SH

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 2.1
  • Proposition 3.1
  • proof
  • ...and 26 more