Energy conservation and regularity for the 3D magneto-hydrodynamics equations
Energy conservation and regularity for the 3D magneto-hydrodynamics equations
复制标题
3D 磁流体动力学方程的能量守恒和规律性
DOI：
10.3934/dcds.2022110
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发表时间：
2022
影响因子：
1.1
通讯作者：
Fan Wu
中科院分区：
文献类型：
--
作者：
Wenke Tan;Fan Wu
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 begin{document}$ (u,b)in L^{2,infty}(0,T;BMO(Omega)) $end{document}. As the applications, first, we show that as the solution begin{document}$ (u,b) $end{document} approaches a finite blowup time begin{document}$ T $end{document}, the begin{document}$ BMO $end{document} norm must blow up at a rate begin{document}$ frac{c}{sqrt{T-t}} $end{document} with some absolute constant begin{document}$ c>0 $end{document}. Then, a regularity criteria for suitable weak solutions is proved which allows the vertical part of the velocity and magnetic to be large under the norm of begin{document}$ L^{2,infty}left([-1,0; BMO(mathbb{R}^3)right) $end{document}. Finally, we prove that any suitable weak solution of the MHD equations in begin{document}$ L^{2,infty}(0, T; BMO (Omega)) $end{document} satisfies the local energy equality for any bounded Lipschitz domain begin{document}$ Omegasubseteqmathbb{R}^3 $end{document}. As a corollary, we prove that any suitable weak solution of MHD equations in begin{document}$ L^{2,infty}(0, T; BMO_{loc} (mathbb{R}^3)) $end{document} satisfies the energy equality.
This paper studies the energy conservation and regularity problems for the 3D magneto-hydrodynamics (MHD) equations. We first establish some uniform bounds on some invariant quantities in terms of suitable weak solution begin{document}$ (u,b)in L^{2,infty}(0,T;BMO(Omega)) $end{document}. As the applications, first, we show that as the solution begin{document}$ (u,b) $end{document} approaches a finite blowup time begin{document}$ T $end{document}, the begin{document}$ BMO $end{document} norm must blow up at a rate begin{document}$ frac{c}{sqrt{T-t}} $end{document} with some absolute constant begin{document}$ c>0 $end{document}. Then, a regularity criteria for suitable weak solutions is proved which allows the vertical part of the velocity and magnetic to be large under the norm of begin{document}$ L^{2,infty}left([-1,0; BMO(mathbb{R}^3)right) $end{document}. Finally, we prove that any suitable weak solution of the MHD equations in begin{document}$ L^{2,infty}(0, T; BMO (Omega)) $end{document} satisfies the local energy equality for any bounded Lipschitz domain begin{document}$ Omegasubseteqmathbb{R}^3 $end{document}. As a corollary, we prove that any suitable weak solution of MHD equations in begin{document}$ L^{2,infty}(0, T; BMO_{loc} (mathbb{R}^3)) $end{document} satisfies the energy equality.
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影响因子：
2
作者：
M. Shinbrot
通讯作者：
M. Shinbrot
影响因子：
1.7
作者：
A. Cheskidov;Xiaoyutao Luo
通讯作者：
A. Cheskidov;Xiaoyutao Luo
DOI：
10.1016/j.jfa.2008.07.008
发表时间：
2008-07
期刊：
arXiv: Analysis of PDEs
影响因子：
--
作者：
J. Bourgain;Natavsa Pavlovi'c
通讯作者：
J. Bourgain;Natavsa Pavlovi'c
影响因子：
2.4
作者：
Caflisch, RE;Klapper, I;Steele, G
通讯作者：
Steele, G
影响因子：
4.9
作者：
Isett, Philip
通讯作者：
Isett, Philip