On the Critical Blow Up Criterion with One Velocity Component for 3D Incompressible MHD System

This work establishes a sharp regularity criterion for the 3D incompressible MHD equations. Let (v, b) be a smooth enough solution of the MHD system. In Liu (2016), Y. Liu proved the regularity of (v, b) on (0, T*] if there exists p is an element of (4, infinity) such that integral(T)(0)* (parallel to v(3)(t)parallel to(p)((H)over dot1/2 + 2/p) + parallel to b(t)parallel to(p)((H)over dot1/2 + 2/p))dt < infinity. In this article, we settle the case p is an element of [2,4] and our proof also works for the case p is an element of (4, infinity). (C) 2019 Elsevier Ltd. All rights reserved.