Preprint 2022-03
Jens A. Griepentrog and Joachim Naumann
Well-Posedness Of The Maxwell Equations With Nonlinear Ohm Law.
Abstract: This paper is concerned with weak solutions (e, h) ∈ L2 × L2 of the Maxwell
equations with nonlinear Ohm law and under perfect conductor boundary conditions.
These solutions are defined in terms of integral identities with appropriate test functions.
The main result of our paper is an energy equality that holds for any weak solution (e, h).
The proof of this result makes essential use of the existence of time-continuous repre-
sentatives in the equivalence classes (e, h). As a consequence of the energy equality, we
prove the well-posedness of the L2-setting of the Maxwell equations with regard to the
initial-boundary conditions under consideration. In addition, we establish the existence of
a weak solution via the Faedo-Galerkin method. An appendix is devoted to the proof of
a Carath ́eodory solution to an initial-value problem for an ordinary differential equation.
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2022-03
36 pp.