2007/10/31 by Terence Tao · 1 citation
Mathematics · #math.AP #msc:35Q30
published as Dynamics of PDE 4 (2007), 293--302 · 12 pages, no figures. More minor corrections (not appearing in the published version)
arxiv created 2009/05/21 · arxiv updated 2009/12/01
The global regularity problem for the periodic Navier-Stokes system asks whether to every smooth divergence-free initial datum u0: (\R/\Z)3 → \R3 there exists a global smooth solution u. In this note we observe (using a simple compactness argument) that this qualitative question is equivalent to the more quantitative assertion that there exists a non-decreasing function F: \R+ → \R+ for which one has a local-in-time a priori bound ‖ u(T) ‖H1x((\R/\Z)3) ≤ F(‖u0‖H1x((\R/\Z)3)) for all 0 < T ≤ 1 and all smooth solutions u: [0,T] × (\R/\Z)3 → \R3 to the Navier-Stokes system. We also show that this local-in-time bound is equivalent to the corresponding global-in-time bound.