One component regularity for the Navier–Stokes equations

We establish sufficient conditions for the regularity of solutions of the Navier–Stokes system based on conditions on one component of the velocity. The first result states that if , where and 54/23 ⩽ r ⩽ 18/5, then the solution is regular. The second result is that if , where and 24/5 ⩽ r ⩽ ∞, then the solution is regular. These statements improve earlier results on one component regularity.