Regularity criterion for solutions to the three-dimensional Navier–Stokes equations in the turbulent channel

In this paper, we establish a regularity criterion of solutions to the incompressible Navier–Stokes equations in three-dimensional infinite channel in terms of the derivatives of the vertical component of the velocity field. More precisely, we show that the strong solution exists on the time interval [ 0 , T ] provided that one of the following conditions holds: ∂ u ˜ 3 ∂ x i ∈ L q ( [ 0 , T ] ; L p ( Ω ) ) , 1 ≤ i ≤ 3 where p > 3 , 1 ≤ q < ∞ , 3 p + 2 q ≤ 1 2 + 3 2 p for i = 1 , 2 , and p > 2 , 1 ≤ q < ∞ , 3 p + 2 q ≤ 3 4 + 3 2 p for i = 3 .