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

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