The semistatic limit for Maxwell’s equations in an exterior domain

F.Jochmann Institut für angewandte Mathematik Humboldt Universität Berlin Unter den Linden 6 10099 Berlin

                  jochmann@mathematik.hu-berlin.de

In this poster a L^{p}-theory for the Maxwell-system in the semistatic limit case in an exterior domain is described. 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, that is

curl h_{ε} = ε∂_{t}E_{ε} + S(E_{ε}) + j_{0} (1)

curl E_{ε} = −μ∂_{t}h_{ε} (2)

supplemented by the initial-boundary-conditions

n⃗ ∧ E_{ε} = 0 on [0, ∞) × ∂Ω, (3)

E_{ε}(0) = E_{0} and h_{ε}(0) = h_{0}. (4)

Here Ω ⊂ ℝ^{3} is a domain with bounded complement, E_{ε}, h_{ε} denote the electric and magnetic field respectively which depend on the time t ≥ 0 and the space-variable x ∈ Ω. On the perfectly conducting boundary ∂Ω the tangential-component of E must vanish, which is expressed by boundary condition 3. j_{0} is a prescribed external current, which is assumed to be divergence-free. The charge current S(E_{ε}) may depend nonlinearely and nonlocally on the electric field, such that

S(E)(t, x) = 0 if x ∈ Ω_{0} = Ω \ \overline{G}, (5)

with some bounded set G ⊂ Ω.

S : Lr̃ ; loc([0, ∞), G) → L_{loc}^{r̃^{∗}}([0, ∞), L^{2}(Ω) ∩ G^{∗}) with some r̃ ∈ [1, ∞) (6)

is a monotone operator, which is causal in the sense that

S(E)(t, x) = S(Ẽ)(t, x) for all t ∈ [0, T] (7)

provided that E(t, x) = Ẽ(t, x) for all t ∈ [0, T]. In 6 it is assumed that L^{2}(G) ⊂ G ⊂ L^{r_{0}}(G), where r_{0} ∈ [6/5, 2] dependes on Ω_{0}. The assumptions on S are fulfilled in the case that

S(E)(t, x) = σ(x, |E(t, x)|)E(t, x) (8)

with a field-dependent electric conductivity σ(x, u) = σ_{0}(x)(1 + u)^{r̃−2}. Here r̃ ∈ [r_{0}, 2] and σ_{0} ∈ L^{∞}(G) with σ_{0} > 0, which may have mild degeneracies on G with respect to x, i.e. it is not assumed that σ_{0} is uniformly positive. More precisely, σ_{0}^{−1} ∈ L^{r_{0}/(r̃−r_{0})}(G), where 6/5 < r_{0} ≤ 2 as above. In particular, the linear case r̃ = 2, i.e. σ(x, u) = σ_{0}(x) is possible.

In 2 μ ∈ L^{∞}(ℝ^{3}) is the magnetic susceptibility, which is assumed to be uniformly positive. The dielectric susceptibility ε > 0 is considered as a parameter in the above system. In many situations the displacement current ε∂_{t}E_{ε} 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 ε → 0. By letting ε → 0 one obtains fomally from 1-4 the quasi-stationary Maxwell-System

curl h = S(E) + j_{0} (9)

curl E = −μ∂_{t}h (10)

div E = 0 on [0, ∞) × Ω_{0} (11)

supplemented by the initial-boundary-conditions

n⃗ ∧ E = 0 on [0, ∞) × ∂Ω (12)

and h(0) = h_{0} on Ω (13)

∫_{∂C_{k}} n⃗E(t, x)dS = Q_{k} for all t ∈ [0, ∞), k ∈ {1, .., N}. (14)

Here C_{k} ,k = 1, .., N denote the connected components of C =^{def} ℝ^{3} \ \overline{Ω_{0}}, the complement of the ’vacuum-region’ Ω_{0}. The physical meaning of 14 is that the total electric charge Q_{k} on each C_{k} is prescribed. This problem has been investigated in [4] in the case that the domain Ω is bounded and σ is linear with respect to E and uniformly positive. See also [1] and [2], where a temperature dependent electrical conductivity is considered. However, in this paper Ω is an exterior domain and the electrical conductivity vanishes on a subset Ω_{0} = Ω \ \overline{G}. Note that 9-14 can be considered as a problem of mixed type. In the case that μ is constant on Ω and σ is linear and independent of x ∈ G it is elliptic on Ω_{0} (with respect to the space variable x for fixed time) and parabolic on (0, ∞) × G at least in the case 8. It is shown in this paper that 9-14 admits for given h_{0}, j_{0} and Q_{k} a unique global weak solution (E, h) with

E ∈ W_{loc}^{−1,∞} ([0, ∞), L^{r_{0}}(Ω ∩ B_{R_{0}}) ∩ L^{6}(ℝ^{3} \ B_{R_{0}})) ∩ Lr̃ ; loc([0, ∞), G)

⊂ D′((0, ∞) × Ω) and h ∈ C([0, ∞), L^{2}(Ω)),

where G, r_{0} ∈ (6/5, 2] and r̃ as in 6 and R_{0} > 0, such that C ⊂ B_{R_{0}} = {|x| < R_{0}}. Moreover, it is shown that

h_{ε} ⟶^{ε→0} h in L^{∞}((0, T), L^{2}(Ω)) weak − ∗ for all T ∈ (0, ∞)
and E_{ε} ⟶^{ε→0} E in D′ ((0, ∞) × Ω) .

Finally, the asymptotic behavior of the solution to 9-14 for t → ∞ is investigated in the case j_{0} = 0 under a suitable coerciveness-condition on S. It is shown under certain assumptions on the initial-state h_{0} that ||h(t)||_{L^{2}} decays exponentially.

[1] Hong-Ming Yin, Global Solutions of Maxwell’s Equations in an Electromagnetic Field with a Temperaturedependent Electrical Conductivity , Europ. J. Appl. Math. 5 (1994), 57 - 64.

[2] Hong-Ming Yin, Regularity of Solutions to Maxwell’s System in Quasi-stationary Electromagnetic Fields and Applications , Comm. Partial Differential Equations 22 (1997), 1029 - 1053.

[3] F. Jochmann, A Compactness Result for Vector Fields with divergence and curl in L^{q}(Ω) Involving Mixed Boundary Conditions, Appl. Anal. vol. 66 (1997), pp. 189-203.

[4] A. Milani, R. Picard, Weak Solution Theory for Maxwell’s Equations in the Semistatic Limit, J. Math. Anal. Appl. 191 (1995), 77 - 100.