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

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

2011-08-04

arxiv.org/abs/1108.1165v3

Mathematics

Analysis of PDEs

92 pages, one figure, submitted, APDE. More typos fixed

Scientific paper

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.

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