THE REGULARITY CRITERIA ON THE MAGNETIC FIELD TO THE 3D INCOMPRESSIBLE MHD EQUATIONS

This note is devoted to studying the regularity conditions of the mild solution (u,B) to the 3D incompressible MHD equations. More precisely, for the 3D incompressible MHD equations, [He and Xin, J. Diff. Eqs., 213(2):235-254, 2005] (see also [Zhou, Discrete Contin. Dyn. Syst., 12:881-886, 2005]) proved that the velocity field is dominant in the MHD fluids; meanwhile, the effect of the magnetic field B is vague. In this note, we shall establish the regularity criteria for the MHD equations in terms of integral(T)(0)* parallel to u(3)parallel to(p)((H) over dot1/2+2/p(R3)) + parallel to vertical bar partial derivative(3)vertical bar(-1/2-delta)Bh parallel to(p)((H) over dot 1+2/p+delta(R3)) ds < infinity, with p is an element of (2, infinity), delta = 3(1/r - 1/2) > 0, here r sufficiently close to 2. This result follows along the lines of [Chemin and Zhang, Ann. Sci. Ec Norm Super, 49:131-167, 2016], [Chemin et al., Arch. Ration Mech. Anal., 224(3):871-905, 2017] and [Han et al., Arch. Ration. Mech. Anal., 231:939-970, 2019], which partially improved the works of [Yamazaki, Bull. Sci. Math., 140:575-614, 2016] and [Liu, J. Diff. Eqs., 260:6989-7019, 2016].