On the regularity to the solutions of the Navier–Stokes equations via one velocity component Milan Pokorn ´ y 1 and Yong Zhou 2 1. Mathematical Institute of Charles University, Sokolovsk´ a 83, 186 75 Praha 8, Czech Republic E-mail: pokorny@karlin.mff.cuni.cz 2. Zhejiang Normal University, E-mail: Abstract We consider the regularity criteria for the incompressible Navier–Stokes equa- tions connected with one velocity component. Based on the method from [4] we prove that the weak solution is regular, provided u 3 ∈ L t (0,T ; L s (R 3 )), 2 t + 3 s ≤ 3 4 + 1 2s , s> 10 3 or provided ∇u 3 ∈ L t (0,T ; L s (R 3 )), 2 t + 3 s ≤ 19 12 + 1 2s if s ∈ ( 30 19 , 3] or 2 t + 3 s ≤ 3 2 + 3 4s if s ∈ (3, ∞]. As a corollary, we also improve the regularity criteria expressed by the regularity of ∂p ∂x 3 or ∂u 3 ∂x 3 . 1 Introduction We consider the incompressible Navier–Stokes equations in the full three-dimensional space, i.e. 1.1 (1.1) ∂ u ∂t + u ·∇u − ν Δu + ∇p = f div u =0  in (0,T ) × R 3 , u(0,x)= u 0 (x) in R 3 , where u : (0,T ) × R 3 → R 3 is the velocity field, p : (0,T ) × R 3 → R is the pressure, f : (0,T ) × R 3 → R 3 is the given external force, ν> 0 is the viscosity. In what follows, we consider ν = 1 and f ≡ 0. The value of the viscosity does not play any role in our further considerations. We could also easily formulate suitable regularity assumptions on f so that the main results remain true. However, it would partially complicate the calculations, thus we skip it. The existence of a weak solution to ( 1.1 1.1) (provided u 0 and f satisfy certain regularity assumptions) is well known since the famous paper by Leray [11]. Its regularity and Mathematics Subject Classification (2000). 35Q30 Keywords. Incompressible Navier–Stokes equations, regularity of solution, regularity criteria 1