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Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equation

Thomas Hou, Yixuan Wang, Changhe Yang

Abstract

The nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations is one of the central open problems in mathematical fluid dynamics. In this paper, we provide, to our knowledge, the first rigorous computer-assisted proof demonstrating such nonuniqueness. Inspired by earlier works in this area, we construct a Leray-Hopf solution in the self-similar setting and then establish the existence of a second solution by analyzing the stability of the linearized operator around this profile, showing that it corresponds to an unstable perturbation. To achieve this, we develop an innovative numerical method that computes candidate solutions with high precision and propose a framework for rigorously establishing exact solutions in a neighborhood of these candidates. A key step is to decompose the linearized operator into a coercive part plus a compact perturbation, which is further approximated by a finite-rank operator up to a small error. The invertibility of the linearized operator restricted to the image of this finite-rank approximation is then rigorously verified using computer-assisted proofs. This certifies the existence of an unstable eigenpair and, consequently, yields a second solution - indeed, infinitely many Leray-Hopf solutions.

Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equation

Abstract

The nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations is one of the central open problems in mathematical fluid dynamics. In this paper, we provide, to our knowledge, the first rigorous computer-assisted proof demonstrating such nonuniqueness. Inspired by earlier works in this area, we construct a Leray-Hopf solution in the self-similar setting and then establish the existence of a second solution by analyzing the stability of the linearized operator around this profile, showing that it corresponds to an unstable perturbation. To achieve this, we develop an innovative numerical method that computes candidate solutions with high precision and propose a framework for rigorously establishing exact solutions in a neighborhood of these candidates. A key step is to decompose the linearized operator into a coercive part plus a compact perturbation, which is further approximated by a finite-rank operator up to a small error. The invertibility of the linearized operator restricted to the image of this finite-rank approximation is then rigorously verified using computer-assisted proofs. This certifies the existence of an unstable eigenpair and, consequently, yields a second solution - indeed, infinitely many Leray-Hopf solutions.

Paper Structure

This paper contains 99 sections, 13 theorems, 501 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Leray--Hopf solution and strong-weak uniqueness
  3. Self-similar setting
  4. Eigenvalue of the linearized operator and nonuniqueness
  5. Verification of eigenvalues.
  6. Outline of the nonuniqueness proof (Section \ref{['Sec2:nonuniqueness']}).
  7. Computer-assisted proofs
  8. Organization of the paper
  9. Literature review
  10. Nonuniqueness.
  11. Computer-assisted proofs.
  12. Notations
  13. Nonuniqueness Based on Theorem \ref{['thm:stationary solu main']}
  14. Regularity of the profile
  15. Nonuniqueness of Leray--Hopf solutions
  16. ...and 84 more sections

Key Result

Theorem 1

There exist infinitely many distinct suitable Leray--Hopf solutions to the Navier--Stokes equation eq:NS eq on $\mathbb{R}^3 \times [0, 1]$ with the same divergence-free initial condition $u_{\operatorname{in}} =u_{\operatorname{loc}}$ of compact support, with $u_{\operatorname{loc}}\in L^q$ for any

Figures (4)

  • Figure 1: Figures of $\overline{U}$
  • Figure 2: Figures of $\overline{v}$
  • Figure 3: Figures of $\overline{U} + \sigma \overline{v}$ with different $\sigma$
  • Figure 4: Vorticity of $\overline{U}, \overline{v}$ and $\overline{U} + 10 \overline{v}$

Theorems & Definitions (24)

  • Theorem 1: Nonuniqueness of Leray--Hopf solutions
  • Definition 1: Leray--Hopf solution
  • Remark 1: Suitability
  • Remark 2
  • Theorem 2
  • Proposition 1: Exact self-similar profile
  • Proposition 2: Exact unstable eigenpair
  • Proposition 3: Smoothness
  • Proposition 4: Far field asymptotics
  • Remark 3
  • ...and 14 more