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

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.