Regularity criteria via one directional derivative of the velocity in anisotropic Lebesgue spaces to the 3D Navier-Stokes equations

In this paper, we consider the regularity criterion for 3D incompressible Navier-Stokes equations in terms of one directional derivative of the velocity in anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a regular solution if the partial derivative(3)u satisfies integral(T)(0) parallel to parallel to parallel to partial derivative(3)u(t)parallel to(Lx1p)parallel to(Lx2q)parallel to(beta)(Lx3r)/1+ln (parallel to partial derivative(3)u (t) parallel to (L2) +e) dt < infinity, where 2/beta + 1/p + 1/q + 1/r = 1 and 2 < p, q, r <= infinity, 1 - (1/p+ 1/q+ 1/r) >= 0. (C) 2021 Published by Elsevier Inc.