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

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.