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Quantitative estimates for the forced Navier-Stokes equations and applications

Tobias Barker, Henry Popkin

Abstract

In this paper, we prove a localisation of a slightly supercritical (Orlicz) regularity criterion for the 3D incompressible Navier-Stokes equations. This is a refinement to the recent partial positive answer to Tao's conjecture [Tao21] as given in [BP21b]. The proof requires new quantitative estimates for critically bounded solutions of the forced Navier-Stokes equations, where the forcing is induced by the localisation. A by-product of these new estimates is an application to the Boussinesq equations, where we prove a quantitative blow-up rate for the critical $L^3$ norm of the velocity. We prove these quantitative estimates using Carleman inequalities as in [Tao21], and subsequently in [BP21a], with an additional forcing term. An obstacle to doing this is that, in the Carleman inequalities, the forcing term is amplified on large scales. Additionally, the low regularity of the forcing requires the addition of Caccioppoli-type estimates to deal with the Carleman inequalities appropriately.

Quantitative estimates for the forced Navier-Stokes equations and applications

Abstract

In this paper, we prove a localisation of a slightly supercritical (Orlicz) regularity criterion for the 3D incompressible Navier-Stokes equations. This is a refinement to the recent partial positive answer to Tao's conjecture [Tao21] as given in [BP21b]. The proof requires new quantitative estimates for critically bounded solutions of the forced Navier-Stokes equations, where the forcing is induced by the localisation. A by-product of these new estimates is an application to the Boussinesq equations, where we prove a quantitative blow-up rate for the critical norm of the velocity. We prove these quantitative estimates using Carleman inequalities as in [Tao21], and subsequently in [BP21a], with an additional forcing term. An obstacle to doing this is that, in the Carleman inequalities, the forcing term is amplified on large scales. Additionally, the low regularity of the forcing requires the addition of Caccioppoli-type estimates to deal with the Carleman inequalities appropriately.
Paper Structure (17 sections, 20 theorems, 283 equations)

This paper contains 17 sections, 20 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Comparison to previous literature and strategy
  3. Outline of the paper
  4. Notation and definitions
  5. Notation
  6. Ingredients for the proof of the main quantitative estimate
  7. Serrin-type bootstrapping with forcing
  8. Local-in-space smoothing with incorporated forcing term
  9. Backwards propagation of vorticity concentration
  10. Epoch of Regularity
  11. Annulus of Regularity
  12. Carleman inequalities with forcing
  13. Proof of the main quantitative estimate via the spatial concentration method
  14. Application of the quantitative estimate I: localized Orlicz-type quantity blow-up
  15. Global mild criticality breaking for the forced Navier-Stokes equations
  16. ...and 2 more sections

Key Result

Theorem A

There exists a universal constant $\theta\in (0,1)$ such that the following holds. Suppose that $v:\mathbb{R}^3\times (0,\infty)\to\mathbb{R}^3$ is a suitable Leray-Hopf weak solution of the Navier-Stokes equations on $\mathbb{R}^3\times (0,\infty)$ which first blows up at time $T^*>0,$ and let $(x_

Theorems & Definitions (48)

  • Theorem A
  • Theorem B
  • Remark 1.0.1
  • Definition 1.3.1
  • Definition 1.3.2
  • Definition 1.3.3
  • Lemma 2.1.1
  • proof
  • Lemma 2.1.2
  • proof
  • ...and 38 more