The local characterizations of the singularity formation for the MHD equations

Abstract

This paper characterizes the possible blow-up of solutions for the 3D magneto-hydrodynamics (MHD for short) equations. We first establish some ǫ-regularity criteria in L q,∞ spaces for suitable weak solutions, and then together with an embedding theorem from L p,∞ space into a Morrey type space to characterize the local behaviors of solutions near a potential singular point. More precisely, we show that if z 0 = (t 0 , x 0 ) is a singular point, then for any r > 0 it holds that where δ * is a positive constant independent on ν and p.