Local wellposedness of quasilinear Maxwell equations on domains
A talk in the Oberseminar Analysis series by
Martin Spitz
| Abstract: | In this talk we study the macroscopic Maxwell equations on
domains with perfectly conducting and absorbing boundary conditions as
well as the Maxwell interface problem. Equipped with instantaneous
material laws the Maxwell equations lead to a first order quasilinear
system.
We develop a local wellposedness theory in $H^m$ for all $m \geq 3$.
First we construct a unique solution of the nonlinear system assuming
that the initial value, the inhomogeneity, and the boundary value
satisfy certain compatibility conditions. We then characterize the case
of a finite maximal existence time by a blow-up condition in the
Lipschitz norm and show the continuous dependence on the data.
Our construction relies on the wellposedness theory for linear
nonautonomous Maxwell equations, which we derive using energy-type
$H^m$-estimates and several regularization techniques. Since the Maxwell
system has characteristic boundary, we also have to exploit the
structure of the Maxwell equations to avoid a loss of regularity. The
talk is based on joint work with Roland Schnaubelt. |