THE SEMISTATIC LIMIT FOR MAXWELL'S EQUATIONS IN AN EXTERIOR DOMAIN

an exterior domain and the electrical conductivity vanishes on a subset 0 = n G. It is assumed that y (t; x; y) (x)jyj2 for all x 2 G; y 2 IR3 (1.14) 1 (G), > 0. may have mild degeneracies on G, i.e. it is not assumed with some 2 L that is uniformly positive. More precisely, ?1 2 Lr0 =(2?r0 ) (G) with some 6=5 < r (1.15) 0 2: Note that 1.8-1.13 can be considered as a problem of mixed type. In the case that is constant on and is linear and independent of x 2 G it is elliptic on 0 (with respect to the space variable x for xed time) and parabolic on (0; 1) G. It is shown in this paper that 1.8-1.13 admits for given h0 , j0 and Qk a unique global weak solution (E; h) with ? ? E 2 Wloc1( 0; 1); Lr0 ( \ BR0 )) \ Wloc1( 0; 1); L6(IR3 n BR0 )) \ L2 ( 0; 1); L2 (G)) loc 0 ((0; 1) 2 ( )); D ) and h 2 C ( 0; 1); L where ; r0 2 (6=5; 2] as in 1.14, 1.15 and L2 (G) is the weighted L2 - space and R0 > 0, such that G (IR3 n ) BR0 = fjxj < R0 g. Moreover, it is shown that "!0 h" ?! h in L1((0; T ); L2( )) weak ? for all T 2 (0; 1) "!0 and E" ?! E in D0 ((0; 1) ) : In the case that is independent of t and (E0; h0), j0 are su ciently regular it is shown that 1.8-1.13 admits global strong solution (E; h) with E 2 L2 ( 0; 1); Lr0 ( \ BR0 )) \ L2 ( 0; 1); L6(IR3 n BR0 )) \ L2 ( 0; 1); L2 (G)) loc loc loc 1; and h 2 Wloc1( 0; 1); L2( )). Moreover, one has in this case "!0 h" ?! h in W 1;1((0; T ); L2( )) weak ? "!0 curl h" ?! curl h in L2 ((0; T ); L2( )) weakly "!0 and E" ?! E in L2 ((0; T ); Lr0 ( \ BR0 ) \ L6 (IR3 n BR0 )) weakly : for all T 2 (0; 1). In the case that also @ is su ciently regular this implies also "!0 h" ?! h in C ( 0; T ]; L2( \ BR)) strongly for all R 2 (0; 1) and T 2 (0; 1). Finally, the asymptotic behavior of the solution to 1.8-1.13 for t ! 1 is investigated in the case j0 = 0 . It is shown that jjh(t)jjL2 decays exponentially provided that Z h0 gdx = 0 for all g 2 L2( ) with curl g = 0: This condition includes div ( h0) = 0 on and ~ h = 0 on @ weakly. n 3
E"(0) = E0and h"(0) = h0 :
Here IR3 is a domain with bounded complement, E"; h" denote the electric and magnetic eld respectively which depend on the time t 0 and the space-variable x 2 . On the perfectly conducting boundary @ the tangential-component of E must vanish, which is expressed by boundary condition 1.3. j0 is a prescribed external current, which is assumed to be divergence-free. The charge current (t; x; E") may depend nonlinearely on the electric eld. is a monotone function (with respect to E") with the property that (t; x; y) = 0 for all x 2
Abstract:
1 Introduction
This paper concerns the initial-boundary value problem for Maxwell's equations 6] in an exterior domain, that is the system curl h" = "@t E" + (t; x; E") + j0 (1.1) curl E" = ? @t h" supplemented by the initial-boundary-conditions ~ ^ E" = 0 on 0; 1) @ ; n (1.2) (1.3) (1.4)
0
(1.9) (1.10) (1.11) (1.12) (1.13)
supplemented by the initial-boundary-conditions ~ ^ E = 0 on 0; 1) @ ; n
Z
Ck
~ E(t; x)dS = Qk for all t 2 0; 1); k 2 f1; ::; N g: n
Z
Ck
~ E"(t; x)dS = Qk for all t 2 0; 1); k 2 f1; ::; N g: n
(1.7)
Here Ck ,k = 1; ::; N denote the connected components of IR3 n 0 . The physical meaning of 1.7 is that the total electric charge Qk on each Ck is invariant under the time evolution of E"; h" governed by 1.1-1.4. This is a consequence of the assumptions (t; x; y) = 0 for y 2 0 and div j0 = 0 on IR3. In many situations the displacement current "@tE" is small in comparison with the charge currents and is often neglected. Therefore, it is the aim of this paper to investigate the singular limit " ! 0. By letting " ! 0 one obtains fomally from 1.1-1.4 and 1.7 the quasi-stationary Maxwell-System curl h = (t; x; E) + j0 (1.8) curl E = ? @t h div E = 0 on 0; 1)
Z
Ck
~ E0(x)dS = Qk for all k 2 f1; ::; N g: n
(1.6)
If these compatibility conditions are ful lled then every solution of the system 1.1-1.4 satis es div E" = 0 on 0; 1) 0 and div h" = 0 on 0; 1) : and
and h(0) = h0 on
This problem has been investigated in 8] the case that the domain is bounded and is linear with respect to E and uniformly positive. See also 3] and 4], where a temperature dependent electrical conductivity is considered. However, in this paper is 2
THE SEMISTATIC LIMIT FOR MAXWELL'S EQUATIONS IN AN EXTERIOR DOMAIN
F.Jochmann Institut fur angewandte Mathematik Humboldt Universitat Berlin Unter den Linden 6 10099 Berlin
This paper provides a Lp-theory for the Maxwell-system in the semistatic limit case in an exterior domain. The problem under consideration is of mixed type, since the possibly nonlinear electric conductivity vanishes on a certain subset of the domain. The solution is obtained from a singular perturbation of the full Maxwell-system including the displacement current.
0 def
= nG 1
with some bounded set G . The dielectric susceptibility " > 0 is considered as a parameter in the above system. 2 L1(IR3) is the magnetic susceptibility, which is assumed to be uniformly positive. The initial-data must be compatible with Maxwell's equations, i.e. div E0 = 0 on 0 = n G; div h0 = 0 on (1.5) and