Nonbneor Anulys~s. Theory, Methods & Applications, Vol. 26, No. 4. pp. 659-689, 1996 Copyright 0 1995 Elsevier Science Ltd Prmted m Great Britain. All rights reserved 0362-546X/% $15.00+ .oO 0362-546X(94)00308-4 VARIATIONAL METHODS FOR STATIONARY MHD FLOW UNDER NATURAL INTERFACE CONDITIONS-f- A. J. MEIR and PAUL G. SCHMIDT Department of Mathematics, Auburn University, AL 36849, U.S.A. (Received 8 February 1994; received for publication 21 October 1994) Key words and phrases: Magnetohydrodynamics, Navier-Stokes equations, Maxwell’s equations, variational methods, boundary integral methods. 0. INTRODUCTION Magnetohydrodynamics (or MHD) is the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. It is of importance in connection with many engineering problems, such as sustained plasma confinement for controlled thermonuclear fusion, liquid- metal cooling of nuclear reactors, and electromagnetic casting of metals (see, e.g. [l-3]). It also finds applications in geophysics and astronomy, where prominent examples are the so-called dynamo problem, that is, the question of the origin of the Earth’s magnetic field in its liquid metal core, and the equilibrium and stability of magnetic stars in their own gravitational fields (see, e.g. [4-61). Assuming the fluid to be viscous, resistive and incompressible (as we will do throughout this paper), MHD flow is governed by the Navier-Stokes equations (for the fluid velocity) and Maxwell’s equations (for the magnetic field); the equations are nonlinearly coupled via Ohm’s law and the Lorentz force (cf. [7, Chapter 21). The magnetic field will in general transcend the fluid region and, ideally, extend to all of space, satisfying different equations in the fluid and in the solid material or vacuum outside, and certain matching conditions at the interface. It is mainly this phenomenon, the electromagnetic principle of action at Q distance, which distinguishes MHD from ordinary hydrodynamics: typically, attention cannot be confined to the conducting fluid itself; its interaction with the outside world must be analyzed too. This adds considerably to the intricacies of the problem and may explain why so much theoretical work has been devoted to situations where, due to special circumstances, external conditions are either pre-imposed or irrelevant. Of course it is then necessary to specify boundary conditions for the magnetic field (or related quantities) at the surface of the fluid region. In doing so, one either implicitly assumes to have complete control over the exterior field so that artificial boundary conditions can be enforced no matter what the feedback from within the fluid, or one resorts to very restrictive assumptions on the nature of the interface, assumptions that give rise to natural boundary conditions. To quote from A Textbook of Magnetohydrodynamics by Shercliff, “it is for this reason that a perfectly conducting solid boundary is so much favoured by theoreticians because it shields the fluid from the external conditions and eliminates consideration of all but the fluid region. This selection usually consti- tutes escapism, however, justifiable only as a simplifying first approximation” [7, p. 1131. Shercliff was probably referring to theoretical physicists, but his remark applies to mathematicians as well. Indeed, most prior mathematical work on the MHD equations has t Partially based upon work supported by the National Science Foundation under Grant No. DMS-9404440. 6.59