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Non-Leray-Hopf solutions to 3D stochastic hyper-viscous Navier-stokes equations: beyond the Lions exponents

Wenping Cao, Zirong Zeng, Deng Zhang

Abstract

We consider the 3D stochastic Navier-Stokes equations (NSE) on torus where the viscosity exponent can be larger than the Lions exponent 5/4. For arbitrarily prescribed divergence-free initial data in $L^{2}_x$, we construct infinitely many probabilistically strong and analytically weak solutions in the class $L^{r}_ΩL_{t}^γW_{x}^{s,p}$, where $r\geq1$ and $(s, γ, p)$ lie in two supercritical regimes with respect to the Ladyžhenskaya-Prodi-Serrin (LPS) criteria.It shows that even in the high viscosity regime beyond the Lions exponent, though solutions are unique in the Leray-Hopf class, the uniqueness fails in the mixed Lebesgue spaces and, actually, there exist infinitely manly non-Leray-Hopf solutions which can be very close to the Leray-Hopf solutions. Furthermore, we prove the vanishing noise limit result, which relates together the stochastic solutions and the deterministic solutions constructed by Buckmaster-Vicol [4] and the recent work [23].

Non-Leray-Hopf solutions to 3D stochastic hyper-viscous Navier-stokes equations: beyond the Lions exponents

Abstract

We consider the 3D stochastic Navier-Stokes equations (NSE) on torus where the viscosity exponent can be larger than the Lions exponent 5/4. For arbitrarily prescribed divergence-free initial data in , we construct infinitely many probabilistically strong and analytically weak solutions in the class , where and lie in two supercritical regimes with respect to the Ladyžhenskaya-Prodi-Serrin (LPS) criteria.It shows that even in the high viscosity regime beyond the Lions exponent, though solutions are unique in the Leray-Hopf class, the uniqueness fails in the mixed Lebesgue spaces and, actually, there exist infinitely manly non-Leray-Hopf solutions which can be very close to the Leray-Hopf solutions. Furthermore, we prove the vanishing noise limit result, which relates together the stochastic solutions and the deterministic solutions constructed by Buckmaster-Vicol [4] and the recent work [23].

Paper Structure

This paper contains 68 sections, 15 theorems, 281 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Background
  3. Formulation of main results
  4. Uniqueness:
  5. Strategy of the proof
  6. Organization
  7. Notations
  8. Main iteration
  9. Proof of Theorem \ref{['Thm-Nonuniq-Hyper']}
  10. Proof of Theorem \ref{['Theorem Vanishing noise']}
  11. Proof of Theorem \ref{['Theorem Vanishing noise']} (continued).
  12. Mollification
  13. Velocity perturbations in the supercritical regime S1
  14. Intermittent jets
  15. $\bullet$ Spatial building blocks.
  16. ...and 53 more sections

Key Result

Theorem 1.2

Let $\widetilde{u}\in C_{0}^{\infty}([0,T]\times{\mathbb{T}}^{3})$ be any deterministic divergence-free and mean-free smooth vector fields with $\widetilde{u}(0)=0$. Then, for any $r\geq 1$, $\epsilon>0$, for any $(s,p,\gamma)\in \mathcal{S}_{1}$ when ${\alpha}\in [\frac{5}{4},2)$ or $(s,p,\gamma)\i Moreover, $u$ is close to $\widetilde{u}+z:$

Figures (1)

  • Figure 1: The case $\alpha \in [\frac{5}{4},2), s=0$.

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 2.1: HZZ-2CLZ
  • Proposition 2.2: Main iteration
  • Lemma 2.3
  • proof
  • ...and 9 more