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

Abstract: 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.