A Quantitative Formulation of the Global Regularity Problem for the Periodic Navier-stokes Equation

The global regularity problem for the periodic NavierStokes system ∂tu+ (u · ∇)u = ∆u−∇p ∇ · u = 0 u(0, x) = u0(x) for u : R×(R/Z) → R and p : R×(R/Z) → R asks whether to every smooth divergence-free initial datum u0 : (R/Z) 3 → R there exists a global smooth solution. 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 localin-time a priori bound ‖u(T )‖H1 x ((R/Z)) ≤ F (‖u0‖H1 x ((R/Z)3)) for all 0 < T ≤ 1 and all smooth solutions u : [0, T ]×(R/Z) → R to the Navier-Stokes system. We also show that this local-in-time bound is equivalent to the corresponding global-in-time bound.