A regularity criterion for the 3D incompressible magneto-hydrodynamics equations

The paper is dedicated to study of the regularity criterion for weak solutions to the 3D incompressible MHD equations. Employing the Littlewood Paley decomposition, we show that if (del) over bar(u) over bar = (partial derivative(1)(u) over bar, partial derivative(2)(u) over bar) is an element of L-partial derivative 1 ([0, T); B-r1,2r/3(0) (R-3)), 2/r(1) + 3/r(1) = 2,3/2 <r(1) <= infinity and <(del)over bar>(b) over bar = (partial derivative(1)(b) over bar, partial derivative(2)(b) over bar) is an element of L-partial derivative 1 ([0, T); B-r1,2r/3(0) (R-3)), 2/r(1) + 3/r(1) = 2,3/2 <r(1) <= infinity, then the solutions to the MHD actually is smooth on (0,T). (C) 2017 Elsevier Inc. All rights reserved.