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A Beale-Kato-Majda criterion with optimal frequency and temporal localization

Xiaoyutao Luo

Abstract

We obtain a Beale-Kato-Majda-type criterion with optimal frequency and temporal localization for the 3D Navier-Stokes equations. Compared to previous results our condition only requires the control of Fourier modes below a critical frequency, whose value is explicit in terms of time scales. As applications it yields a strongly frequency-localized condition for regularity in the space $B^{-1}_{\infty,\infty}$ and also a lower bound on the decaying rate of $L^p$ norms $2\leq p <3$ for possible blowup solutions. The proof relies on new estimates for the cutoff dissipation and energy at small time scales which might be of independent interest.

A Beale-Kato-Majda criterion with optimal frequency and temporal localization

Abstract

We obtain a Beale-Kato-Majda-type criterion with optimal frequency and temporal localization for the 3D Navier-Stokes equations. Compared to previous results our condition only requires the control of Fourier modes below a critical frequency, whose value is explicit in terms of time scales. As applications it yields a strongly frequency-localized condition for regularity in the space and also a lower bound on the decaying rate of norms for possible blowup solutions. The proof relies on new estimates for the cutoff dissipation and energy at small time scales which might be of independent interest.

Paper Structure

This paper contains 13 sections, 9 theorems, 94 equations.

Table of Contents

  1. introduction
  2. Main results
  3. Some remarks on the main results
  4. Organization of the paper
  5. Preliminaries
  6. Notations
  7. Littlewood-Paley decomposition
  8. Energy, dissipation and flux
  9. Cutoff Energy inequality
  10. Energy and dissipation
  11. Bound the flux
  12. Dissipation at small time scales
  13. Proof of main theorems

Key Result

Theorem 1.1

There exist universal constants $c$ and $\delta_{BKM}$ such that if a regular solution $u$ of eq:3dNSE on $(0,T)$ satisfies that then $u$ is regular on $(0,T]$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['prop:cutoff-e']}
  • ...and 11 more