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 $$ {\mathbb {R}}^{3} $$ . We show that the Serrin-type condition imposed on one component of the velocity $$ u_3\in L^p(0,T; L^q({\mathbb {R}}^{3} ))$$ with $$ \frac{2}{p}+ \frac{3}{q} <1$$ , $$ 3<q \le +\infty $$ implies the regularity of the weak Leray solution $$ u: {\mathbb {R}}^{3} \times (0,T) \rightarrow {\mathbb {R}}^{3} $$ , 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.