On a singular limit problem for nonlinear Maxwell's equations
Hong-Ming Yin
In this paper we study the following nonlinear Maxwell's equations $$\varepsilon {\mathbf E}_{t}+\sigma(x,|{\mathbf E}|){\mathbf E}= {\nabla \times\,} {\mathbf H} +{\mathbf F},\ \ \ \ {\mathbf H}_{t}+{\nabla \times\,} {\mathbf E}=0,$$ 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.