On 3D MHD equations with regularity in one directional derivative of the velocity

This work establishes a new regularity criterion for the 3D incompressible magneto hydrodynamical (MHD) equations in terms of one directional derivative of the velocity (i.e., partial derivative(3)u) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if partial derivative(3)u satisfies integral(T)(0) parallel to vertical bar vertical bar partial derivative(3)u(tau)vertical bar vertical bar L-x3(alpha)parallel to L-x1x2(beta q) d tau < +infinity, where 2/q + 1/alpha + 2/beta = k is an element of [1, 3/2] and 3/k < alpha <= beta <= 1/k-1, for some T > 0, then the corresponding solution (u, b) to the 3D MHD equations is regular on (0, T) x R-3. (C) 2018 Elsevier Ltd. All rights reserved.