The local characterizations of the singularity formation for the MHD equations

This paper characterizes the possible blow-up of solutions for 3D magneto-hydrodynamics (MHD short) equations. We first establish some ϵ-regularity criteria in Lq,∞ spaces suitable weak solutions, and then together with an embedding theorem from Lp,∞ space into a Morrey type we characterize local behaviors near potential singular point. More precisely, show that if z0=(t0,x0) is point, any r>0 it holds lim supt→t0−(‖u(t,x)−u(t)x0,r‖L3,∞(Br(x0))+‖b(t,x)−b(t)x0,r‖L3,∞(Br(x0)))>δ∗;lim supt→t0−(t0−t)1μr2ν−3p‖(u,b)(t)‖Lp,∞(Br(x0))>δ∗for1μ+1ν=12, 2≤ν≤2p3, 3δ∗for1μ+1ν=12, ν∈{[2,∞],p≥3[2,2p3−p],32≤p<3 where δ∗ positive constant independent on ν p.