Existence results for the A & φ magnetodynamic formulation of the Maxwell system

Abstract

The A/ϕ magnetodynamic Maxwell system given in its potential and space/time formulation is a popular model considered in the engineering community. We establish existence of strong solutions with the help of the theory of Showalter on degenerated parabolic problems; using energy estimates, existence of weak solutions are also deduced.

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

“…It is the aim of this section to show that restricting our attention to the orthogonal complement of N (η) ∩ N (C) as well as assuming an estimate of the type (10) for U attaining values in…”

“…More specifically, our investigation is inspired by a series of papers by S. Nicaise et al, [11,9,10]. We will employ the theory of evolutionary equations as laid out in Section 1, see [18,16], to analyse the structure of the degenerate eddy current problem.…”

Space-Time Residual-Based A Posteriori Estimators for the A−ϕ Magnetodynamic Formulation of the Maxwell System

“…Using the theory of Showalter on degenerated parabolic problem [29, Theorem V4.B], and appropriated energy estimates, the following existence result for problem (14) is proved in Theorem 2.1 of [23].…”