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• W4289883927 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&gt;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)) $end{document}</tex-math></inline-formula> satisfies the local energy equality for any bounded Lipschitz domain <inline-formula><tex-math id=M9>begin{document}$ Omegasubseteqmathbb{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>" @default.
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• W4289883927 date "2022-01-01" @default.
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• W4289883927 title "Energy conservation and regularity for the 3D magneto-hydrodynamics equations" @default.
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• W4289883927 doi "https://doi.org/10.3934/dcds.2022110" @default.
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