The essential spectrum of a linear magnetohydrodynamic model containing a vcuum region

Abstract

The essential spectrum of the force operator of the ideal linear magnetohydrodynamics has been extensively studied in the mathematical literature. However, all the rigorous from mathematical point of view works on this topic concern the magnetohydrodynamic (MHD) model of a plasma confined in a bounded domain O{sub p} {contained_in} R{sup 3} with perfectly conducting boundary. On the other hand, there exists another MHD model which is more realistic from physical point of view. According to this model, the plasma region O{sub p} is surrounded by a vacuum region O{sub v} whose boundary consists of two disjoint surfaces S{sub p} and S{sub v}. The surface S{sub p} coincides with the plasma-vacuum Interface, while the outer surface S{sub v} is perfectly conducting. The interaction between the plasma filling O{sub p} and the exterior magnetic field is described by the MHD equations, while the dynamics of the electromagnetic field in the vacuum region O{sub v} is governed by the Maxwell equations. Usually the domains O{sub p} and O{sub v} are assumed to be axisymmetric. In other words, they are obtained by the rotation of two plane domains {Omega}{sub p} and {Omega}{sub v} ({Omega}{sub p} being surrounded by {Omega}{sub v}), around air axis situatedmore » at a positive distance from the closure of {Omega}{sub p} {union} {Omega}{sub v}. In the case where the ratio of the small and tire large characteristic radii of the toroidal domain O{sub v} {union} O{sub p} is sufficiently small, one may consider O{sub p} arid, respectively, O{sub v} as the cylindrical manifolds {Omega}{sub p} x S{sup 1} and, respectively, {Omega}{sub v} x S{sup 1} where S{sup 1} := R/2{pi}Z. This is the exact meaning of the notations O{sub p} and O{sub v} adopted in the present paper. We denote by {Lambda}p and {Lambda}{sub v} respectively the boundaries of {Omega}{sub p} and {Omega}{sub v}. Thus we have S{sub p} = {Lambda}{sub p} x S{sup 1} and S{sub v} = {Lambda}{sub v} x S{sup 1}.« less