On a singular limit problem for nonlinear Maxwell's equations

Hong-Ming Yin

\newcommand{\E}{ {\bf E}} \newcommand{\F}{ {\bf F}} \newcommand{\g}{\nabla \times } \newcommand{\vh}{ {\bf H}} In this paper we study the following nonlinear Maxwell's equations \\ $\varepsilon \E_{t}+\sigma(x,|\E|)\E= \g \vh +\F,\, \vh_{t}+\g \E=0$, where $\sigma(x,s)$ is a monotone graph of $s$. It is shown that the system has a unique weak solution. Moreover, the limit of the solution as $\varepsilon\rightarrow 0$ converges to the solution of quasi-stationary Maxwell's equations.