On a Class of Degenerate Abstract Parabolic Problems and Applications to Some Eddy Current Models

TL;DR

This paper develops an abstract framework for degenerate parabolic equations that can change type and applies it to a degenerate eddy current model, bridging hyperbolic and elliptic behaviors in electromagnetic problems.

Contribution

It introduces a minimal-assumption functional analytic approach for degenerate parabolic equations and applies it to justify a degenerate eddy current model as a limit of Maxwell's equations.

Findings

Framework accommodates equations changing from parabolic to elliptic.

Degenerate eddy current model derived as a limit of hyperbolic Maxwell models.

Minimal boundary and interface regularity assumptions used.

Abstract

We present an abstract framework for parabolic type equations which possibly degenerate on certain spatial regions. The degeneracies are such that the equations under investigation may admit a type change ranging from parabolic to elliptic type problems. The approach is an adaptation of the concept of so-called evolutionary equations in Hilbert spaces and is eventually applied to a degenerate eddy current type model. The functional analytic setting requires quite minimal assumptions on the boundary and interface regularity. The degenerate eddy current model is justified as a limit model of non-degenerate hyperbolic models of Maxwell's equations.