Well-posedness of some initial-boundary-value problems for dynamo-generated poloidal magnetic fields

TL;DR

This paper analyzes the mathematical well-posedness of boundary value problems related to dynamo-generated magnetic fields in magnetohydrodynamics, establishing existence, completeness, and smooth solutions for relevant equations.

Contribution

It proves existence and smoothness of solutions for a class of boundary value problems modeling magnetic fields in dynamo theory, including eigenfunction completeness and solution construction.

Findings

Existence of smooth solutions for the boundary value problem.

Completeness of poloidal free decay modes in b2b7b3.

Construction of solutions via Galerkin approximation.

Abstract

Given a bounded domain $G \subset \R^d$, $d\geq 3$, we study smooth solutions of a linear parabolic equation with non-constant coefficients in $G$, which at the boundary have to $C^1$-match with some harmonic function in $\R^d \setminus \ov{G}$ vanishing at spatial infinity. This problem arises in the framework of magnetohydrodynamics if certain dynamo-generated magnetic fields are considered: For example, in the case of axisymmetry or for non-radial flow fields, the poloidal scalar of the magnetic field solves the above problem. We first investigate the Poisson problem in $G$ with the above described boundary condition as well as the associated eigenvalue problem and prove the existence of smooth solutions. As a by-product we obtain the completeness of the well-known poloidal "free decay modes" in $\R^3$ if $G$ is a ball. Smooth solutions of the evolution problem are then obtained by Galerkin approximation based on these eigenfunctions.