The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain

Abstract

Abstract In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter ε ∈ ( 0 , ∞ ) , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are μ = O ( 1 ) and ν = O ( ε α ) , and the orders of magnetic diffusion coefficients κ and σ are κ = O ( 1 ) and σ = O ( ε α ) , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For H 1 -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For H 1 -initial data with additional regularity ( ∂ z A ~ 0 , ∂ z B ~ 0 ) ∈ L p ( Ω ) ( 2 < p < ∞ ) , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal. 272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For H 2 -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order O ( ε γ / 2 ) , where γ = min { 2 , α − 2 } with α ∈ ( 2 , ∞ ) . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order O ( ε ) .