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 $H^1$, 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.

95 pages, one figure, to apepar, APDE. This is the final version

References in corpus (1)

• A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation

Cited by in corpus (5)

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• Some infinite matrix analysis, a Trotter product formula for dissipative operators, and an algorithm for the incompressible Navier-Stokes equation
• Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions
• On the existence of smooth periodic solutions of 3D homogeneous Navier-Stokes equations
• On global schemes for highly degenerate Navier Stokes equation systems