Regularity criterion of weak solutions to the 3D Navier–Stokes equations via two velocity components

This paper is concerned with the regularity criterion of Leray–Hopf weak solutions to the 3D Navier–Stokes equations with respect to Serrin type condition on two velocity filed components. It is shown that the weak solution u=(u1,u2,u3)u=(u1,u2,u3) is regular on (0,T](0,T] if there exist two solution components, for example, u2u2 and u3u3, satisfying the condition∇u2,∇u3∈Lp1(0,T;Lq1(R3)),for 2p1+3q1⩽2,32