Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions

TL;DR

This paper establishes local well-posedness for quasilinear Maxwell equations with absorbing boundary conditions, including existence, uniqueness, and continuous dependence on data, using detailed a priori estimates.

Contribution

It develops a comprehensive well-posedness theory for Maxwell equations with nonlinear boundary conditions in Sobolev spaces, including new a priori estimates and regularity results.

Findings

Proved local existence and uniqueness of solutions.

Established blow-up criteria in the Lipschitz norm.

Demonstrated continuous dependence on initial and boundary data.

Abstract

In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in $\mathcal{H}^m$ for $m \geq 3$. The Maxwell equations are equipped with instantaneous nonlinear material laws leading to a quasilinear symmetric hyperbolic first order system. We consider both linear and nonlinear absorbing boundary conditions. We show existence and uniqueness of a local solution, provide a blow-up criterion in the Lipschitz norm, and prove the continuous dependence on the data. In the case of nonlinear boundary conditions we need a smallness assumption on the tangential trace of the solution. The proof is based on detailed apriori estimates and the regularity theory for the corresponding linear problem which we also develop here.