Non blow-up criterion for the 3-D Magneto-hydrodynamics equations in the limiting case

TL;DR

This paper establishes a criterion under which solutions to the 3-D MHD equations do not blow up, based on specific integrability conditions of the velocity and magnetic field components, extending the solution beyond a certain time.

Contribution

It provides a new non blow-up criterion for 3-D MHD equations involving conditions on the velocity and magnetic field components, expanding understanding of solution regularity.

Findings

Solutions extend beyond time T under specified conditions.

Conditions involve velocity in L^∞(0,T;L^3) and magnetic field in Ladyzhenskaya-Prodi-Serrin class.

No blow-up occurs if criteria are satisfied.

Abstract

In this paper, we prove that suitable weak solution $(u,b)$ of the 3-D MHD equations can be extended beyond $T$ if $u\in L^\infty(0,T;L^3(\R^3))$ and the horizontal components $b_h$ of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition.