Mixed 3D-2D asymptotics for the weak and strong solutions of the rotating magnetohydrodynamic system

2025/07/09 by Frédéric Charve, Charve, Frédéric, Van-Sang Ngo +1
Earth and Planetary Sciences · Mathematics · #Analysis of PDEs (math.AP) #Aquatic and Environmental Studies #Differential Equations and Numerical Methods #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2507.06678

openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we consider the 3D-rotating magnetohydrodynamic (MHD) system when the initial velocity and magnetic field both feature some 2D-part (i.-e. depending only on the horizontal space variables). We prove for weak and strong solutions, that the limit system, when the Rossby number goes to zero (i.-e. for strong rotation), is a 2D-MHD system with three components. Moreover we are able to provide explicit global-in-time convergence rates thanks to adapted Strichartz estimates and with the help of an additional 3D magnetic field transported by the 2D limit velocity.

Citations

• New asymptotics for strong solutions of the strongly stratified Boussinesq system without rotation and for large ill-prepared initial data
• Hidden asymptotics for the weak solutions of the strongly stratified Boussinesq system without rotation
• Sharper dispersive estimates and asymptotics for a Boussinesq-type system with larger ill-prepared initial data
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• A priori estimates for the 3D quasi-geostrophic system
• Higher order commutator estimates and local existence for the non-resistive MHD equations and related models
• The 2D Incompressible Magnetohydrodynamics Equations with only Magnetic Diffusion
• Two regularity criteria for the 3D MHD equations
• Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion
• Global existence for the magnetohydrodynamic system in critical spaces

Related

• Asymptotic Analysis of MHD Systems
• Large time behavior of the solutions to 3D incompressible MHD system with horizontal dissipation or horizontal magnetic diffusion
• Sharp decay estimates and asymptotic stability for incompressible MHD equations without viscosity or magnetic diffusion
• On the fast rotation asymptotics of a non-homogeneous incompressible MHD\n system
• On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state