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, and 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 almost smooth solutions from smooth $H^1$ data or smooth finite energy data, although we show in this category that fully smooth solutions are not always possible. 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.