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Nonuniqueness of weak solutions to the Navier-Stokes equation

Tristan Buckmaster, Vlad Vicol

Abstract

For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Holder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

Nonuniqueness of weak solutions to the Navier-Stokes equation

Abstract

For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Holder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

Paper Structure

This paper contains 26 sections, 20 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Previous works
  3. Main results
  4. Outline of the convex integration scheme
  5. Parameters
  6. Inductive estimates
  7. The main proposition and iterative procedure
  8. Proof of Theorem \ref{['thm:compact:support']}
  9. Proof of Theorem \ref{['thm:NSE:Euler']}
  10. Intermittent Beltrami Waves
  11. Beltrami waves
  12. Intermittent Beltrami waves
  13. $L^p$ decorrelation
  14. The perturbation
  15. Mollification of $v_q$
  16. ...and 11 more sections

Key Result

Theorem 1.2

There exists $\beta>0$, such that for any nonnegative smooth function $e(t) \colon [0,T] \to \mathbb{R}_{\geq 0}$, there exists $v \in C^0_t ([0,T]; H^\beta_x(\mathbb T^3))$ a weak solution of the Navier-Stokes equations, such that $\int_{\mathbb{T}^3} |v(x,t)|^2 \, dx = e(t)$ for all $t \in [0,T]$.

Theorems & Definitions (35)

  • Definition 1.1
  • Theorem 1.2: Nonuniqueness of weak solutions
  • Theorem 1.3: Dissipative Euler solutions arise in the vanishing viscosity limit
  • Proposition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof : Proof of Proposition \ref{['p:gamma_def']}
  • Proposition 3.5
  • ...and 25 more