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

Abstract

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

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

References in corpus (1)

• Ill-posedness for nonlinear Schrodinger and wave equations

Cited by in corpus (2)

• Localisation and compactness properties of the Navier-Stokes global regularity problem
• Global existence for a translating near-circular Hele-Shaw bubble with surface tension