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

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

2007-10-08

arxiv.org/abs/0710.1604v5

Dynamics of PDE 4 (2007), 293--302

Mathematics

Analysis of PDEs

12 pages, no figures. More minor corrections (not appearing in the published version)

Scientific paper

The global regularity problem for the periodic Navier-Stokes system asks whether to every smooth divergence-free initial datum $u_0: (\R/\Z)^3 \to \R^3$ 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 $F: \R^+ \to \R^+$ for which one has a local-in-time \emph{a priori} bound $$ \| u(T) \|_{H^1_x((\R/\Z)^3)} \leq F(\|u_0\|_{H^1_x((\R/\Z)^3)})$$ for all $0 < T \leq 1$ and all smooth solutions $u: [0,T] \times (\R/\Z)^3 \to \R^3$ to the Navier-Stokes system. We also show that this local-in-time bound is equivalent to the corresponding global-in-time bound.

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