On the critical one-component velocity regularity criteria to 3-D incompressible MHD system

Let (u,b)(u,b) be a smooth enough solution of 3-D incompressible MHD system. We prove that if (u,b)(u,b) blows up at a finite time T⁎T⁎, then for any p∈]4,∞[p∈]4,∞[, there holds ∫0T⁎(‖u3(t′)‖H˙12+2pp+‖b(t′)‖H˙12+2pp)dt′=∞. We remark that all these quantities are in the critical regularity of the MHD system.