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

TL;DR

This paper studies the behavior of 3D rotating magnetohydrodynamic systems with initial conditions partly 2D, showing convergence to a 2D-MHD system as rotation intensifies, and providing explicit convergence rates.

Contribution

It establishes the limit system for weak and strong solutions of the 3D-rotating MHD system as the Rossby number approaches zero, with explicit convergence rates.

Findings

Weak and strong solutions converge to a 2D-MHD system with three components.

Explicit global-in-time convergence rates are derived.

The limit involves a 2D velocity field transporting a 3D magnetic field.

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.