Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

TL;DR

This paper establishes the local wellposedness of quasilinear Maxwell equations with perfectly conducting boundary conditions in high regularity Sobolev spaces, addressing gaps in existing results for these physically relevant boundary conditions.

Contribution

It develops a comprehensive wellposedness theory for Maxwell equations with conducting boundaries in $H^m$, including existence, uniqueness, continuous dependence, and finite speed of propagation.

Findings

Proves local existence and uniqueness in $H^m$ for $m \\geq 3$.

Characterizes finite time blowup via Lipschitz norm.

Shows solutions depend continuously on initial data.

Abstract

In this article we develop the local wellposedness theory for quasilinear Maxwell equations in $H^m$ for all $m \geq 3$ on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in $H^3$ is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in $H^m$ with $m \geq 3$ of the corresponding initial boundary value problem if the material laws and the data are accordingly regular and compatible. We further characterize finite time blowup in terms of the Lipschitz norm and we show that the solutions depend continuously on their data. Finally, we establish the finite propagation speed of the solutions.