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 $T^{\ast }$, 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.

22 pages, 7 sections

References in corpus (2)

• Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable
• On the critical one component regularity for 3-D Navier-Stokes system: general case