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

TL;DR

This paper establishes that finite-time blow-up of solutions to the 3-D incompressible MHD system is characterized by the divergence of certain critical regularity norms of the velocity component and magnetic field, specifically involving the third velocity component.

Contribution

It proves a new regularity criterion linking blow-up to the divergence of critical Sobolev norms of specific solution components in the MHD system.

Findings

Finite-time blow-up implies divergence of critical norms of $u^3$ and $b$.

The criterion involves integrals of these norms in the critical Sobolev space.

Results apply for any $p$ in the range (4, ∞).

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.