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 ϵ-regularity criteria in $L^{q,\mathrm{\infty }}$ spaces for suitable weak solutions, and then together with an embedding theorem from $L^{p,\mathrm{\infty }}$ space into a Morrey type space we 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 $\begin{array}{rl}& \underset{t\to t_0^{-}}{lim sup}({\Vert u(t,x)-u(t{)}_{x_0,r}\Vert }_{L^{3,\mathrm{\infty }}\left(B_r\left(x_0\right)\right)}\\ & +{\Vert b(t,x)-b(t{)}_{x_0,r}\Vert }_{L^{3,\mathrm{\infty }}\left(B_r\left(x_0\right)\right)})>{\delta }^{\ast };\\ & \underset{t\to t_0^{-}}{lim sup}{(t_0-t)}^{\frac{1}{\mu }}{r}^{\frac{2}{\nu }-\frac{3}{p}}\Vert (u,b)\left(t\right){\Vert }_{L^{p,\mathrm{\infty }}\left(B_r\left(x_0\right)\right)}>{\delta }^{\ast }\mathrm{f}\mathrm{o}\mathrm{r}\\ & \frac{1}{\mu }+\frac{1}{\nu }=\frac{1}{2},2\le \nu \le \frac{2p}{3},3<p\le \mathrm{\infty };\\ & \underset{t\to t_0^{-}}{lim sup}{(t_0-t)}^{\frac{1}{\mu }}{r}^{\frac{2}{\nu }-\frac{3}{p}+1}\Vert (\mathrm{\nabla }u,\mathrm{\nabla }b)\left(t\right){\Vert }_{L^p\left(B_r\left(x_0\right)\right)}>{\delta }^{\ast }\mathrm{f}\mathrm{o}\mathrm{r}\\ & \frac{1}{\mu }+\frac{1}{\nu }=\frac{1}{2},\nu \in \{\begin{array}{ll}[2,\mathrm{\infty }],& p\ge 3\\ [2,\frac{2p}{3-p}],& \frac{3}{2}\le p<3\end{array}\end{array}$ where ${\delta }^{\ast }$ is a positive constant independent on ν and p.

Keywords:

• Magneto-hydrodynamics equations
• Navier–Stokes equations
• weak solutions
• singularities
• regularity
• concentration rate

Mathematics Subject Classification (2010):

• 76D05
• 35B65
• 35Q30

Acknowledgments

The authors also thank the reviewers for careful reading and constructive comments.

Disclosure statement

The authors declared that they have no conflicts of interest to this work.

Additional information

Funding

The authors are supported by the Construct Program of the Key Discipline in Hunan Province and NSFC (National Natural Science Foundation of China) [grant number 11871209].