On the critical blow up criterion with one velocity component for 3D incompressible MHD system

Abstract

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∈(4,∞) such that ∫0T∗(‖v3(t)‖Ḣ12+2pp+‖b(t)‖Ḣ12+2pp)dt<∞. In this article, we settle the case p∈[2,4] and our proof also works for the case p∈(4,∞).