On the Serrin-Type Condition on One Velocity Component for the Navier–Stokes Equations

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