Existence of solutions for a model of the Earth's magnetic field

TL;DR

This paper proves the existence of weak solutions for a comprehensive mathematical model of Earth's core dynamics, integrating magneto-hydrodynamics and solid physics with boundary conditions.

Contribution

It introduces a novel functional framework to handle fluid-structure interaction and magnetic transmission in Earth's core modeling, establishing existence results.

Findings

Existence of Leray-Hopf type weak solutions proven.

Functional framework for magnetic field and fluid interaction developed.

Handling of boundary conditions for inner core and exterior insulator achieved.

Abstract

We study a physically realistic, whole-core mathematical model of the dynamics in the Earth's core and we prove existence of Leray-Hopf type weak solutions to the model. Our model combines Magneto-Hydrodynamic equations in the liquid outer core with solid physics for the electrically conducting inner core, and treats everything exterior to the core as a perfect insulator governed by Maxwell's equations. We prove existence of weak solutions using Galerkin approximations. In order to control the nonlinearities, we must define an appropriate function space for the magnetic field and prove a Biot-Savart type result. The main new difficulty here is properly setting up the functional framework to simultaneously deal with the fluid structure interaction with the inner core and the magnetic transmission problem, with both the perfectly conducting inner core and the perfectly insulating mantle/exterior.