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

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.