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Localisation and compactness properties of the Navier-Stokes global regularity problem

Terence Tao

Abstract

In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth $H^1$, Schwartz, or mild $H^1$ categories, and with or without a forcing term. In particular, we show that if one has global well-posedness in $H^1$ for the periodic Navier-Stokes problem with a forcing term, then one can obtain global regularity both for periodic and for Schwartz initial data (thus yielding a positive answer to both official formulations of the problem for the Clay Millennium Prize), and can also obtain global smooth solutions from smooth $H^1$ data, and global almost smooth solutions from smooth finite energy data. Our main new tools are localised energy and enstrophy estimates to the Navier-Stokes equation that are applicable for large data or long times, and which may be of independent interest.

Localisation and compactness properties of the Navier-Stokes global regularity problem

Abstract

In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth , Schwartz, or mild categories, and with or without a forcing term. In particular, we show that if one has global well-posedness in for the periodic Navier-Stokes problem with a forcing term, then one can obtain global regularity both for periodic and for Schwartz initial data (thus yielding a positive answer to both official formulations of the problem for the Clay Millennium Prize), and can also obtain global smooth solutions from smooth data, and global almost smooth solutions from smooth finite energy data. Our main new tools are localised energy and enstrophy estimates to the Navier-Stokes equation that are applicable for large data or long times, and which may be of independent interest.

Paper Structure

This paper contains 16 sections, 32 theorems, 472 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notation and basic estimates
  4. Symmetries of the equation
  5. Pressure normalisation
  6. Local well-posedness theory in $H^1$
  7. Homogenisation
  8. Compactness
  9. Energy localisation
  10. Bounded total speed
  11. Enstrophy localisation
  12. Consequences of enstrophy localisation
  13. Smooth $H^1$ solutions
  14. Smooth finite energy solutions
  15. Quantitative $H^1$ bounds
  16. ...and 1 more sections

Key Result

Proposition 1.7

Conjecture global-periodic is equivalent to Conjecture global-periodic-homog.

Figures (1)

  • Figure 1: Known implications between the various conjectures described here (i.e. existence of smooth or mild solutions, or local or global quantitative bounds in the periodic, Schwartz, $H^1$, or finite energy categories, with or without normalised pressure, and with or without the $f=0$ condition) and also in tao-quantitative (the latter conjectures and implications occupy the far left column). A positive solution to the red problems, or a negative solution to the blue problems, qualify for the Clay Millennium prize as stated in feff.

Theorems & Definitions (107)

  • Definition 1.1: Smooth solutions to the Navier-Stokes system
  • Remark 1.2
  • Conjecture 1.3: Global regularity for homogeneous Schwartz data
  • Conjecture 1.4: Global regularity for homogeneous periodic data
  • Conjecture 1.5: Global regularity for Schwartz data
  • Conjecture 1.6: Global regularity for periodic data
  • Proposition 1.7: Elimination of forcing term
  • Conjecture 1.8: Global regularity for periodic data with normalised pressure
  • Conjecture 1.9: Global well-posedness in periodic $H^1$
  • Conjecture 1.10: A priori periodic $H^1$ bound
  • ...and 97 more