A regularity criterion of Serrin-type for the Navier–Stokes equations involving the gradient of one velocity component

Abstract

Abstract In the present paper we prove that a weak Leray solution to the Navier–Stokes equations in ℝ 3 × ( 0 , T ] $\mathbb {R}^3\times (0,T]$ is regular provided the gradient of one component belongs to the Prodi–Serrin class ∇ u 3 ∈ L 4 ( 0 , T ; L 2 ) $\nabla u^3 \in L^4(0,T; L^2)$ .