The primitive equations with magnetic field approximation of the 3D MHD equations

Lili Du; Dan Li

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arXiv · 2208.01985

The primitive equations with magnetic field approximation of the 3D MHD equations

Lili Du·Dan Li

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Abstract

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.

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Lili Du, Dan Li. 2022-08-03. The primitive equations with magnetic field approximation of the 3D MHD equations. https://arxiv.org/abs/2208.01985

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