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Non-uniqueness of weak solutions to the Navier-Stokes equations in R^3

Changxing Miao, Yao Nie, Weikui Ye

Abstract

To our knowledge, the convex integration method has been widely applied to the study of non-uniqueness of solutions to the Naiver-Stokes equations in the periodic region, but there are few works on applying this method to the corresponding problems in the whole space or other regions. In this paper, we prove that weak solutions of the Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy in the whole space, which extends the non uniqueness result for the Navier-Stokes equations on torus T3in the groundbreaking work (Buckmaster and Vicol, Ann. of Math., 189 (2019), pp.101-144) to R3. The critical ingredients of the proof include developing an iterative scheme in which the approximation solution is refined by decomposing it into local and non-local parts. For the non-local part, we introduce the localized corrector which plays a crucial role in balancing the compact support of the Reynolds stress error with the non-compact support of the solution. As applications of this argument, we first prove that there exist infinitely many weak solutions that dissipate the kinetic energy in smooth bounded domain. Moreover, we show the instability of the Navier-Stokes equations near Couette flow in L2(R3).

Non-uniqueness of weak solutions to the Navier-Stokes equations in R^3

Abstract

To our knowledge, the convex integration method has been widely applied to the study of non-uniqueness of solutions to the Naiver-Stokes equations in the periodic region, but there are few works on applying this method to the corresponding problems in the whole space or other regions. In this paper, we prove that weak solutions of the Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy in the whole space, which extends the non uniqueness result for the Navier-Stokes equations on torus T3in the groundbreaking work (Buckmaster and Vicol, Ann. of Math., 189 (2019), pp.101-144) to R3. The critical ingredients of the proof include developing an iterative scheme in which the approximation solution is refined by decomposing it into local and non-local parts. For the non-local part, we introduce the localized corrector which plays a crucial role in balancing the compact support of the Reynolds stress error with the non-compact support of the solution. As applications of this argument, we first prove that there exist infinitely many weak solutions that dissipate the kinetic energy in smooth bounded domain. Moreover, we show the instability of the Navier-Stokes equations near Couette flow in L2(R3).

Paper Structure

This paper contains 24 sections, 27 theorems, 273 equations, 1 table.

Table of Contents

  1. Introduction
  2. Previous works and our main results
  3. Main ideas
  4. Applications to other problems
  5. Application I: The non-uniqueness result in smooth bounded domain
  6. Application II: The instability of the system \ref{['NS']} near Couette flow
  7. Organization of the paper
  8. Induction scheme
  9. Parameters
  10. Iterative procedure
  11. Proof of Theorem \ref{['t:main']}
  12. Proof of Proposition \ref{['iteration']}
  13. Mollification
  14. Perturbation
  15. Estimates for amplitudes $a_{(k,q)}$
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $T>0$, $e(t),\widetilde{e}(t): [0,T]\to (0,\infty)$ be two nonnegative smooth functions with $e(t)=\widetilde{e}(t)$ for $t\in[0, \tfrac{T}{2}]$. Then there exist weak solutions $u, \widetilde{u} \in C([0,T];L^2(\mathbb R^3))$ in the sense of Definition def-weak for NS with $u(0,x)=\widetilde{u

Theorems & Definitions (42)

  • Definition 1.1: Weak solution
  • Theorem 1.2: Non-uniqueness of weak solutions
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1: Estimates for $(u_{\ell_q}, \mathring R_{\ell_q}, R^{\text{rem}}_q)$
  • Proposition 3.2: Estimates for $(\chi_q\rho_q)^{1/2}$ and $(\chi_q\rho_q)^{-1}$
  • proof
  • ...and 32 more