Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Abstract

In this article we develop the local wellposedness theory for quasilinear Maxwell equations in Hm for all m≥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 H3 is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in Hm with m≥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.