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

TL;DR

This paper rigorously justifies deriving the primitive equations with magnetic field as a limit of the 3D scaled magnetohydrodynamics equations, establishing convergence of solutions in different initial data regimes.

Contribution

It provides a rigorous mathematical derivation of the primitive equations with magnetic field from 3D MHD equations as the aspect ratio tends to zero, including convergence rates.

Findings

Global weak solutions of SMHD converge to strong solutions of PEM for H^1 initial data.

Strong solutions of SMHD extend globally for small aspect ratio in H^2 case.

Convergence rate matches the order of the aspect ratio parameter.

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.