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∂xi∈Lq([0,T];Lp(Ω)), 1≤i≤3 where p>3, 1≤q<∞, 3p+2q≤12+32p for i=1,2, and p>2, 1≤q<∞, 3p+2q≤34+32p for i=3.