A Geometrical Regularity Criterion in Terms of Velocity Profiles for the Three-Dimensional Navier–Stokes Equations

Abstract

Summary In this article, we present a new kind of regularity criteria for the global well-posedness problem of the three-dimensional Navier–Stokes equations in the whole space. The novelty of the new results is that they involve only the profiles of the magnitude of the velocity. One particular consequence of our theorem is as follows. If for every fixed $t\in (0,T)$, the ‘large velocity’ region $\Omega:=\{(x,t)\mid |u(x,t)|>C(q)\left|\mkern-2mu\left|{u}\right|\mkern-2mu\right|_{L^{3q-6}}\}$, for some $C(q)$ appropriately defined, shrinks fast enough as $q\nearrow \infty$, then the solution remains regular beyond $T$. We examine and discuss velocity profiles satisfying our criterion. It remains to be seen whether these profiles are typical of general Navier–Stokes flows.