Remarks on the regularity criteria of three-dimensional magnetohydrodynamics system in terms of two velocity field components

Abstract

We study the three-dimensional magnetohydrodynamics system and obtain its regularity criteria in terms of only two velocity vector field components eliminating the condition on the third component completely. The proof consists of a new decomposition of the four nonlinear terms of the system and estimating a component of the magnetic vector field in terms of the same component of the velocity vector field. This result may be seen as a component reduction result of many previous works [C. He and Z. Xin, “On the regularity of weak solutions to the magnetohydrodynamic equations,” J. Differ. Equ. 213(2), 234–254 (2005); Y. Zhou, “Remarks on regularities for the 3D MHD equations,” Discrete Contin. Dyn. Syst. 12(5), 881–886 (2005)].