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

Abstract: Let $(u,b)$ be a smooth enough solution of 3-D incompressible MHD system. We prove that if $(u,b)$ blows up at a finite time $T^*$, then for any $p\in]4,\infty[$, there holds $\int_0^{T^*}\bigl(|u^3(t')|^p_{\dH^{\frac 12+\frac 2p}}+|b(t')|^p_{\dH^{\frac 12+\frac 2p}}\bigr)dt'=\infty$. We remark that all these quantities are in the critical regularity of the MHD system.