论文标题
具有3D MHD方程的磁场近似的原始方程
The primitive equations with magnetic field approximation of the 3D MHD equations
论文作者
论文摘要
在我们较早的工作\ cite {dll}中，我们已经显示了薄域上具有磁场（PEM）的三维原始方程的强溶液的全局良好性。本文的核心是为PEM的推导提供严格的理由，因为在各向异性水平粘度和磁场状态下，不可压缩的三维缩放型磁流失动力学（SMHD）方程的小长宽比极限（SMHD）方程。对于$ h^1 $ - 天然数据案例，我们证明了三维SMHD方程的全球Leray-Hopf弱解决方案强烈融合到PEM的全球强解决方案。在$ h^2 $ - 激进数据案例中，SMHD的强大解决方案可以扩展为小$ \ v $的全球。结果，我们观察到，SMHD强的全球强大解决方案融合了PEM的全球强解决方案。作为副产品，收敛速率与纵横比参数的顺序相同。
In our earlier work \cite{DLL}, we have shown the global well-posedness of strong solutions to the three-dimensional primitive equations with the magnetic field (PEM) on a thin domain. The heart of this paper is to provide a rigorous justification of the derivation of the PEM as the small aspect ratio limit of the incompressible three-dimensional scaled magnetohydrodynamics (SMHD) equations in the anisotropic horizontal viscosity and magnetic field regime. For the case of $H^1$-initial data case, we prove that global Leray-Hopf weak solutions of the three-dimensional SMHD equation strongly converge to the global strong solutions of the PEM. In the $H^2$-initial data case, the strong solution of the SMHD can be extended to be a global one for small $\v$. As a consequence, we observe that the global strong solutions of the SMHD strong converge to the global strong solutions of the PEM. As a byproduct, the convergence rate is of the same order as the aspect ratio parameter.
