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Sharp non-uniqueness for the Navier-Stokes equations in R^3

Changxing Miao, Yao Nie, Weikui Ye

Abstract

In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the incompressible Navier-Stokes equations in $\R^3$. To be more precise, we exhibit the non-uniqueness result in a strong sense, that is, any weak solution is non-unique in L^p([0,T];L^\infty(\R^3)) with 1\le p<2. Moreover, this non-uniqueness result is sharp with regard to the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint (2, \infty), which extends the sharp nonuniqueness for the Navier-Stokes equations on torus $\TTT^3$ in the recent groundbreaking work (Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987-1054) to the setting of the whole space. The key ingredient is developing a new iterative scheme that balances the compact support of the Reynolds stress error with the non-compact support of the solution via introducing incompressible perturbation fluid.

Sharp non-uniqueness for the Navier-Stokes equations in R^3

Abstract

In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the incompressible Navier-Stokes equations in . To be more precise, we exhibit the non-uniqueness result in a strong sense, that is, any weak solution is non-unique in L^p([0,T];L^\infty(\R^3)) with 1\le p<2. Moreover, this non-uniqueness result is sharp with regard to the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint (2, \infty), which extends the sharp nonuniqueness for the Navier-Stokes equations on torus in the recent groundbreaking work (Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987-1054) to the setting of the whole space. The key ingredient is developing a new iterative scheme that balances the compact support of the Reynolds stress error with the non-compact support of the solution via introducing incompressible perturbation fluid.

Paper Structure

This paper contains 14 sections, 18 theorems, 211 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Induction scheme
  4. Parameters
  5. Iterative procedure
  6. Proof of Theorem \ref{['t:main0']}
  7. Proof of Proposition \ref{['iteration']}
  8. Mollification
  9. Gluing procedure
  10. Perturbation
  11. Construction the perturbation
  12. Estimates for the perturbation
  13. Estimates for the Reynolds stress $\mathring R_{q+1}$
  14. Iterative estimates at $q+1$ level

Key Result

Theorem 1.2

Let $1\le p<2$ and $T_0>0$. Any weak solution $u$ of the equations NS in $L^p([0, T_0]; L^{\infty}(\mathbb R^3))$ is non-unique.

Theorems & Definitions (34)

  • Definition 1.1: Very weak solution
  • Theorem 1.2: Sharp and strong non-uniqueness
  • Remark 1.3
  • Lemma 2.1: Geometric Lemma 2Beekie
  • Remark 2.2
  • Lemma 2.3: 1CheskidovMoS
  • Proposition 2.4
  • proof
  • Definition 2.5: Mollifiers
  • Definition 2.6: BCD11MWZ
  • ...and 24 more