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 $\epsilon$-regularity criteria in $L^{q,\infty}$ spaces for suitable weak solutions, and then together with an embedding theorem from $L^{p,\infty}$ space into a Morrey type space to characterize the local behaviors of solutions near a potential singular point. More precisely, we show that if $z_{0}=\left(t_{0}, x_{0}\right)$ is a singular point, then for any $r>0$ it holds that $$ \limsup _{t \rightarrow t_{0}^{-}}\left(\left\|u(t, x)-u(t)_{x_{0}, r}\right\|_{L^{3, \infty}\left(B_{r}\left(x_{0}\right)\right)}+\left\|b(t, x)-b(t)_{x_{0}, r}\right\|_{L^{3, \infty}\left(B_{r}\left(x_{0}\right)\right)}\right)>\delta^{*}; $$ $$ \limsup\limits _{t \rightarrow t_{0}^{-}}\left(t_{0}-t\right)^{\frac{1}{\mu}} r^{\frac{2}{\nu}-\frac{3}{p}}\|(u,b)(t)\|_{L^{p, \infty}\left(B_{r}\left(x_{0}\right)\right)}>\delta^{*} \text { for } \frac{1}{\mu}+\frac{1}{\nu}=\frac{1}{2},\,2 \leq \nu \leq \frac{2 p}{3},\, 3<p\leq\infty;$$ $$ \limsup\limits _{t \rightarrow t_{0}^{-}}\left(t_{0}-t\right)^{\frac{1}{\mu}} r^{\frac{2}{\nu}-\frac{3}{p}+1}\|(\nabla u,\nabla b)(t)\|_{L^{p}\left(B_{r}\left(x_{0}\right)\right)}>\delta^{*} \text { for } \frac{1}{\mu}+\frac{1}{\nu}=\frac{1}{2},\, \nu \in\left\{\begin{array}{ll} {[2, \infty],} & p\geq 3 {[2, \frac{2p}{3-p}],} & \frac{3}{2}\leq p<3 \end{array}\right. $$ where $\delta^{*}$ is a positive constant independent on $\nu$ and $p$.