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

Published online by Cambridge University Press:  04 November 2009

Ralf Kaiser  and
Hannes Uecker
Ralf Kaiser
Affiliation:
Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, 95440 Bayreuth, Germany, ([email protected])
Hannes Uecker
Affiliation:
Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany, ([email protected])
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Abstract

Given a bounded domain G ⊂ ℝd, d ≥ 3, we study smooth solutions of a linear parabolic equation with non-constant coefficients in G, which at the boundary have to C1-match with some harmonic function in ℝd \ Ḡ 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 boundary condition described above 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 ℝ3 if G is a ball. Smooth solutions of the evolution problem are then obtained by Galerkin approximation based on these eigenfunctions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 139 , Issue 6 , December 2009 , pp. 1209 - 1235
Copyright
Copyright © Royal Society of Edinburgh 2009

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