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On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations

Alexey Cheskidov, Hedong Hou

Abstract

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We also establish uniqueness of stationary weak solutions in an endpoint critical space. Similar results are proved for the fractional Navier-Stokes equations with arbitrarily large power of the Laplacian in both Lebesgue and Besov spaces.

On non-uniqueness of mild solutions and stationary singular solutions to the Navier-Stokes equations

Abstract

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We also establish uniqueness of stationary weak solutions in an endpoint critical space. Similar results are proved for the fractional Navier-Stokes equations with arbitrarily large power of the Laplacian in both Lebesgue and Besov spaces.
Paper Structure (26 sections, 13 theorems, 237 equations)

This paper contains 26 sections, 13 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Failure of unconditional uniqueness
  3. Fractional Navier-Stokes equations
  4. Stationary singular solutions
  5. Notation
  6. Organization
  7. Local well-posedness of the fractional Navier-Stokes equations
  8. Stationary singular solutions and mild solutions
  9. Iterative procedure
  10. Proofs of Main Theorems
  11. Proof of Theorem \ref{['thm:main']} admitting Propositions \ref{['prop:iteration']} and \ref{['prop:L2-iteration']}
  12. Proof of \ref{['item:main-Lp-Besov']}
  13. Proof of \ref{['item:main-L2']}
  14. Proofs of Theorems \ref{['thm:nonunique-mild-NSE']} and \ref{['thm:nonunique-mild-NSEa']} admitting Proposition \ref{['prop:iteration']}
  15. Proof of Theorem \ref{['thm:unique-NS-L2B-1']}
  16. ...and 11 more sections

Key Result

theorem 1

Let $d \ge 2$, $\theta>0$, and $q,r \in [1,\infty]$. Then for any $\epsilon>0$, there exists initial data $u_{\initial} \in B^{-\theta}_{q,r}(\bT^d)$ with such that there exist $T>0$ and two mild solutions to the Navier-Stokes equations e:NSE with

Theorems & Definitions (32)

  • theorem 1: Failure of unconditional uniqueness of \ref{['e:NSE']}
  • theorem 2: Failure of unconditional uniqueness of \ref{['e:NSEa']}
  • definition 1: Singular solutions
  • theorem 3: Existence of non-trivial stationary singular solutions
  • theorem 4: Uniqueness of stationary solutions
  • remark 1: Local well-posedness in endpoint spaces
  • remark 2
  • lemma 1: Smoothing effects
  • proof : Proof of Proposition \ref{['prop:lwp-NSEa']}
  • proof
  • ...and 22 more