Abstract
In this paper we continue studying regularity criteria for the 3D MHD equations via partial derivatives of the velocity or the pressure. We obtain some new regularity criteria which improve the related results in [1, 3, 9, 11, 17]. Precisely, we first prove that if for any i, j, k is an element of {1, 2, 3} there holds (partial derivative u(1)/partial derivative x(i), partial derivative u(2)/partial derivative x(j), partial derivative u(3)/partial derivative x(k)) is an element of L-T(alpha,gamma) with 2/alpha + 3/gamma <= 1 + 1/gamma, 2 <= gamma <= infinity, then the solution (u, b) is smooth on R-3 x (0, T]. Secondly, we show that any component (resp. components) of (partial derivative u(1)/partial derivative x(i), partial derivative u(2)/partial derivative x(j), partial derivative u(3)/partial derivative x(k)) in the criterion above can be replaced by the corresponding velocity component (resp. components) which is (resp. are) in the space L-T(alpha',gamma') with 2/alpha' + 3/gamma' <= 1, 3 < gamma' <= infinity. Fianlly, we obtain a Ladyzhenskaya-Prodi-Serrin type regularity condition involving two components of the gradient of pressure, which in fact partially answers an open question proposed in [9] and improves Theorem 3.3 in Berselli and Galdi's article [1].