Communications in Mathematical Sciences

Volume 20 (2022)

Number 8

Local wellposedness of quasilinear Maxwell equations with conservative interface conditions

Pages: 2265 – 2313

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n8.a6

Authors

Roland Schnaubelt (Department of Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany)

Martin Spitz (Department of Mathematics, University of Bielefeld, Germany)

Abstract

We establish a comprehensive local wellposedness theory for the quasilinear Maxwell system with interfaces in the space of piecewise $H^m$-functions for $m \geq 3$. The system is equipped with instantaneous and piecewise regular material laws and perfectly conducting interfaces and boundaries. We also provide a blow-up criterion in the Lipschitz norm and prove the continuous dependence on the data. The proof relies on precise a priori estimates and the regularity theory for the corresponding linear problem also shown here.

Keywords

nonlinear Maxwell system, perfectly conducting boundary/interface conditions, local wellposedness, blow-up criterion, continuous dependence, piecewise regular

2010 Mathematics Subject Classification

35L50, 35L60, 35Q61

Received 21 January 2021

Received revised 22 February 2022

Accepted 21 March 2022

Published 29 November 2022