Regularity for 3-D MHD Equations in Lorentz Space $L^{3,∞}$

Abstract

The regularity for 3-D MHD equations is considered in this paper. It is proved that the solutions $(v,B,p)$ are Hölder continuous if the velocity field $v\in L^∞(0,T;L^{3,∞}_x (\mathbb{R}^3))$ with local small condition $$r^{−3}|\{ x∈B_r(x_0):|v(x,t_0)|>εr^{−1}\}|≤\varepsilon$$ and the magnetic field $B∈L^ ∞(0,T;VMO^{−1} (\mathbb{R}^3))$.