On the Serrin-type condition on one velocity component for the Navier-Stokes equations

In this paper we consider the regularity problem of the Navier-Stokes equations in ${\mathbb{R}}^3$. We show that the Serrin-type condition imposed on one component of the velocity $u_3\in L^p(0,T;L^q({\mathbb{R}}^3))$ satisfying $\frac{2}{p}+\frac{3}{q}<1$, $3<q\le +\mathrm{\infty }$ implies the regularity of the weak Leray solution $u:{\mathbb{R}}^3\times (0,T)\to {\mathbb{R}}^3$ with the initial data belonging to $L^2({\mathbb{R}}^3)\cap L^3({\mathbb{R}}^3)$. The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.