Energy conservation and regularity for the 3D magneto-hydrodynamics equations

Abstract

<p style='text-indent:20px;'>This paper studies the energy conservation and regularity problems for the 3D magneto-hydrodynamics (MHD) equations. We first establish some uniform bounds on some invariant quantities in terms of suitable weak solution <inline-formula><tex-math id="M1">\begin{document}$ (u,b)\in L^{2,\infty}(0,T;BMO(\Omega)) $\end{document}</tex-math></inline-formula>. As the applications, first, we show that as the solution <inline-formula><tex-math id="M2">\begin{document}$ (u,b) $\end{document}</tex-math></inline-formula> approaches a finite blowup time <inline-formula><tex-math id="M3">\begin{document}$ T $\end{document}</tex-math></inline-formula>, the <inline-formula><tex-math id="M4">\begin{document}$ BMO $\end{document}</tex-math></inline-formula> norm must blow up at a rate <inline-formula><tex-math id="M5">\begin{document}$ \frac{c}{\sqrt{T-t}} $\end{document}</tex-math></inline-formula> with some absolute constant <inline-formula><tex-math id="M6">\begin{document}$ c>0 $\end{document}</tex-math></inline-formula>. Then, a regularity criteria for suitable weak solutions is proved which allows the vertical part of the velocity and magnetic to be large under the norm of <inline-formula><tex-math id="M7">\begin{document}$ L^{2,\infty}\left([-1,0]; BMO(\mathbb{R}^3)\right) $\end{document}</tex-math></inline-formula>. Finally, we prove that any suitable weak solution of the MHD equations in <inline-formula><tex-math id="M8">\begin{document}$ L^{2,\infty}(0, T; BMO (\Omega)) $\end{document}</tex-math></inline-formula> satisfies the local energy equality for any bounded Lipschitz domain <inline-formula><tex-math id="M9">\begin{document}$ \Omega\subseteq\mathbb{R}^3 $\end{document}</tex-math></inline-formula>. As a corollary, we prove that any suitable weak solution of MHD equations in <inline-formula><tex-math id="M10">\begin{document}$ L^{2,\infty}(0, T; BMO_{loc} (\mathbb{R}^3)) $\end{document}</tex-math></inline-formula> satisfies the energy equality.</p>