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