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In the present paper we prove that a weak Leray solution to the Navier–Stokes equations inℝ  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   )$\nabla u^3 \in L^4(0,T; L^2)$   .

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