Regularity for Solutions to the Navier-Stokes Equations with one Velocity Component Regular

Abstract

In this paper, we establish a regularity criterion for solutions to the Navier-stokes equations, which is only related to one component of the velocity field. Let (u, p) be a weak solution to the Navier-Stokes equations. We show that if any one component of the velocity field u, for example u3, satisfies either u3 ∈ L∞ (ℝ3 x (0, T)) or ∇u3 ∈ Lp(0, T; Lq(ℝ3)) with 1/p + 3/2q = 1/2 and q ≥ 3 for some T > 0, then u is regular on [0, T].