A Regularity Criterion for the Navier–Stokes Equations

Abstract

We prove that a weak solution u = (u 1, u 2, u 3) to the Navier–Stokes equations is strong, if any two components of u satisfy Prodi–Ohyama–Serrin's criterion. As a local regularity criterion, we prove u is bounded locally if any two components of the velocity lie in L 6, ∞.