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Non-linear Maxwell equations

We consider the following well-posed initial-boundary value problem:
    $\displaystyle D_t- {\rm rot} H=-j\quad \mbox{in} \quad G\times (0,T),$  
    $\displaystyle B_t+{\rm rot} E=0\quad \mbox{in} \quad G\times (0,T)$  


Here denote $G$ a bounded domain in $\R^3$. The electric and magnetic induction are given in such a way that

\begin{eqnarray*}B\vert _{t=0}=0,\quad D\vert _{t=0}=D_0\end{eqnarray*}
Introducing the charge density $q=q(x,t)$ by the relation $ q={\rm div}\; D$ we obtain
\begin{eqnarray*}q_t+{\rm div} j=0\end{eqnarray*}
Furthermore we have
\begin{eqnarray*}{\rm div} D(t)=-\int\limits _0^t{\rm div} jd\tau + {\rm div} D_0\end{eqnarray*}
In case of no current in the space we find ${\rm div} D_t=0$ and so ${\rm div} D_0=0$ From the above equations ([*]) we get
\begin{eqnarray*}&&{\rm div} D_t=-{\rm div } j\\&&{\rm div} B=0.\end{eqnarray*}
Assume the material relations $D=D(E),\;B=B(H)\;j=j(E,H)\;$ and $D(0)=B(0)=0$. We consider the linearized system
    $\displaystyle (\varepsilon E)_t-{\rm rot} H=-j$  
    $\displaystyle (\mu H)_t+{\rm rot} E=0$  


Here $\varepsilon $ and $\mu$ are matrices which are assumed to be symmetric. Compatibility equations follows easily:

\begin{eqnarray*}&&{\rm div}(\varepsilon E)={\rm div}(\varepsilon E_0)-\int\limits _0^t{\rm div} j d\tau=: k\\&&{\rm div}(\mu H)=0\end{eqnarray*}
where we used that ${\rm div}(\mu(0)H_0)=0$. System ([*]) is versuitable for a proof of existence and uniqueness if $\varepsilon =\varepsilon (t,x)$ and $\mu=\mu(t,x)$ but not in the nonlinear case. Therefore consideration of the system
\begin{eqnarray*}&&\varepsilon E_t-{\rm rot} H=-j\\&&\mu H_t+{\rm rot} E=0\end{eqnarray*}
Finally the problem will be reduced to the symmetric hyperbolic system
\begin{eqnarray*}Lu:=A_0u_t+\sum A_iu_{x_i}=F\end{eqnarray*}
where $u=(u_1,...,u_6)=(E,H),\quad F(-j,0)$ and $A_i$ are $ 6\times 6$ symmetric matrices such that
\begin{eqnarray*}A_0=\left(\begin{array}{cc}\varepsilon &0\\0&\mu\\\end{ar......s=\left(\begin{array}{cc}0&I_s\\I_s^T&0\\\end{array}\right)\end{eqnarray*}
where $I_s$ is the fully antisymmetric Ricci tensor.
 

A late reference: R. H. Picard, W.M. Zajaczkowski (1995) Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, MMAS 18, 169-199.
 


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Nächste Seite:Picard' s extended MaxwellAufwärts:Maxwell equations (ME)Vorherige Seite:ME and Dirac equation
collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg