Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data

Matei P. Coiculescu, Stan Palasek

Abstract

We consider the Cauchy problem for the incompressible Navier-Stokes equations in dimension three and construct initial data in the critical space $BMO^{-1}$ from which there exist two distinct global solutions, both smooth for all $t>0$. One consequence of this construction is the sharpness of the celebrated small data global well-posedness result of Koch and Tataru. This appears to be the first example of non-uniqueness for the Navier-Stokes equations with data at the critical regularity. The proof is based on a non-uniqueness mechanism proposed by the second author in the context of the dyadic Navier-Stokes equations.

Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data

Abstract

We consider the Cauchy problem for the incompressible Navier-Stokes equations in dimension three and construct initial data in the critical space from which there exist two distinct global solutions, both smooth for all . One consequence of this construction is the sharpness of the celebrated small data global well-posedness result of Koch and Tataru. This appears to be the first example of non-uniqueness for the Navier-Stokes equations with data at the critical regularity. The proof is based on a non-uniqueness mechanism proposed by the second author in the context of the dyadic Navier-Stokes equations.

Paper Structure

This paper contains 32 sections, 21 theorems, 231 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Well-posedness and Ill-posedness at the Critical Regularity
  3. Known Approaches towards Non-Uniqueness
  4. Convex Integration
  5. The Jia--Šverák Program
  6. Strategy of the Proof
  7. Non-Uniqueness Mechanism
  8. Structure of the Principal Part
  9. Construction of the Perturbation
  10. Plan of the Paper
  11. Preliminaries
  12. Notation
  13. Littlewood--Paley Estimates
  14. Function Spaces
  15. Parameters
  16. ...and 17 more sections

Key Result

Theorem 1.1

There exists $\epsilon>0$ such that if $U^0$ is divergence-free withSee §wellandillposednesssubsection for the definition of $BMO^{-1}$ and further discussion on why this is a natural setting for these questions.$\|u\|_{BMO^{-1}}<\epsilon$, then there exists a unique global-in-time solution that is

Figures (1)

  • Figure 1: For initial data in various critical spaces, one can ask whether the Navier--Stokes equations are locally or globally well-posed. In some cases, the answer depends on the size of the data. For the "open" case, the answer may depend on the exact space; for instance a negative answer for $B_{p,\infty}^{-1+3/p}$ is suggested by jia2014localjia2015incompressibleguillod2023numerical, while a positive or negative answer in $H^{1/2}$ or $L^3$ would resolve the Clay problem.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1: Bernstein inequalities
  • Lemma 2.2: Estimates on the heat propagator
  • Definition 3.1
  • ...and 38 more