On a Singular Limit Problem for Nonlinear Maxwell’s Equations

Hong-Ming Yin Department of Mathematics, University of Notre Dame Notre Dame, IN 46556.

Abstract: In this paper we study the following nonlinear Maxwell’s equations εE_{t} + σ(x, |E|)E = ∇ × H + F, H_{t} + ∇ × E = 0, where σ(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 ε → 0 converges to the solution of quasi-stationary Maxwell’s equations.

AMS(MOS) Subject Classifications: 35K20, 35Q20.

Key Words and Phases: Nonlinear Maxwell’s Equations, Singular limit.

1. Introduction

Let Ω be a bounded domain in R^{3} and Q_{T} = Ω × (0, T] for any fixed T > 0. Let E and H be the electric and magnetic fields, respectively, in Ω (here and thereafter a bold letter represents a vector in R^{3}). Let σ be the electric conductivity in the field, which is assumed to be a function of x and |E|. Consider the following Maxwell’s equations (see Landau-Lifschitz [13]):

εE_{t} + σ(x, |E|)E = ∇ × H + F,  (x, t) ∈ Q_{T} ,  (1.1)
H_{t} + ∇ × E = 0,  (x, t) ∈ Q_{T} ,  (1.2)
∇ × E = 0,  (x, t) ∈ ∂Ω × (0, T],  (1.3)
E(x, 0) = E_{0}(x), H(x, 0) = H_{0}(x),  x ∈ Ω,  (1.4)

where ε is the dielectric parameter and other physical parameters are normalized.

In some applications ([4, 11]), the electric conductivity, σ, strongly depends on the electric field |E|, hence the electric current density. Particularly, the electric conductivity may act like a switch-like function in some electromagnetic fields. On the other hand, for many types of micron devices and other industrial problems (such as microwave heating [11, 8, 17]) etc.) the experiment shows that the displacement current, εE_{t}, is often negligible since it is small in comparison of the eddy current, J = σE. This motivates us to investigate the nonlinear problem (1.1)-(1.4) and the singular limit problem as ε → 0. It is shown that there exists a unique global solution to (1.1)-(1.4). Moreover, the limit of the solution converges to the solution of the quasi-stationary system (i.e., the system (1.1)-(1.4) with ε = 0 in (1.1)). This limit solution provides new existence result for the quasi-stationary system. Indeed, when ε = 0, the system (1.1) becomes

σ(x, |E|)E = ∇ × H + F,  (1.5)

Thus, one can solve Eq. (1.5) for |E| in terms of |∇ × H| and known data,

|E| = g(x, |∇ × H|),

where g(x, s) is the inverse function of σ(x, s)s.

It follows from (1.2) that H satisfies

H_{t} + ∇ × [ρ(x, |∇ × H|)∇ × H] = 0,  (1.6)

where

ρ(x, |∇ × H||) = \frac{1}{σ(x, |E|)} = \frac{1}{σ(x, g(x, |∇ × H|))}

represents the electric resistivity in the field.

The research on Maxwell’s equations is of great interesting because of the important applications in plasma physics, semiconductor-superconductor modeling and other industrial problems ([8, 9, 13, 17] etc.). The study on the system (1.1)-(1.4) as well as the quasi-stationary form (1.6) received considerable attention recently. In [12], the authors established the well-posedness for a quasi-stationary system, where a constitutive relation between the magnetic field H and the magnetic induction B is assumed to be nonlinear. In [15], the author studied the regularity of weak solution to a linear system of (1.6) with minimal requirement on coefficients. There is a special interest when

ρ(|∇ × H|) = |∇ × H|^{p−2}, p > 2.

On one hand, if H is restricted in one direction (scalar field) then the evolution system (1.6) becomes the p-Laplacian which has been studied extensively (see [5] and the references therein). On the other hand, in a recent work [16] (also see [1, 3] for the scalar case), it is shown that the limit of the solution to (1.6) as p → ∞ is the unique solution to Bean’s critical-state model in the superconductivity theory ([2]). Thus, for large p the system (1.6) provides a good approximation to Bean’s model. More recently, the author of [10] studied the similar problem to this paper in a domain with a bounded complement in R^{3}. The conditions on σ in [10] is quite different from ours here. Like many nonlinear problems, the major difficulty is how to pass the weak limit of an approximate solution for a nonlinear function σ(x, s). This is done by employing a monotonicity argument ([6]). The monotonicity of σ(x, s) in s is essential in the proof.

In §2, we use the finite element method to establish the well-posedness of the system (1.1)-(1.4) for fixed ε > 0. In §3, we show that the singular limit of the solution to (1.1)-(1.4) has a unique limit. Moreover, the limit solution solves the quasistationary Maxwell’s equations. Some examples are also discussed in this section.

2. Existence and Uniqueness for fixed ε > 0

Introduce some standard spaces (see [4, 7]).

H(div, Ω) = {U ∈ L^{2}(Ω)^{3}, divU ∈ L^{2}(Ω)};
H(curl, Ω) = {U ∈ L^{2}(Ω)^{3}, curlU ∈ L^{2}(Ω)^{3}};
H(div0, Ω) = {U ∈ H(div, Ω) : divU = 0 in Ω},
H_{0}(curl, Ω) = {U ∈ H(curl, Ω) : N × U = 0 on ∂Ω},

where N is the exterior unit normal on ∂Ω.

Note that the trace of a function in H(curl, Ω) is well defined (see [4] for example). We shall assume the following conditions on σ(x, s) and data E_{0}(x), H_{0}(x) and

F(x, t). H(2.1): Let σ(x, s) be measurable in Ω × [0, ∞) and monotone increasing in s. Moreover,

∫_{0}^{s^{2}} σ(x, \sqrt{s})dx ≥ a_{0}s^{p+2} − a_{1}, if s is sufficiently large,
0 ≤ σ(x, s) ≤ b_{0}(1 + s^{p}), s ∈ [0, ∞), for some p ≥ 0,

where the constants a_{0} > 0, a_{1} ≥ 0 and b_{0} ≥ 0.

H(2.2): Assume that H_{0} ∈ H(curl, Ω) ⋂ H(div0, Ω), F ∈ H^{1}(0, T; H^{1}(Ω)).

Definition 2.1: A pair of vector fields (E(x, t), H(x, t)) is said to be a weak solution of the problem (1.1)-(1.4), if

E ∈ L^{2}(0, T; H_{0}(curl, Ω)) ⋂ L^{p+2}(0, T; Ω), H ∈ L^{2}(0, T; H(div0, Ω)) ⋂ H(0, T, L^{2}(Ω)) which satisfy the following integral identities:

∫ ∫_{Q_{T}} [−εE · Φ_{t} + σ(x, |E|)E · Φ] dxdt
= ∫ ∫_{Q_{T}} H · (∇ × Φ)dxdt + ε ∫_{Ω} E_{0} · Φ(x, 0)dx,  (2.1)
∫ ∫_{Q_{T}} [−H · Ψ_{t} + E · (∇ × Ψ)] dxdt = ∫_{Ω} [H_{0}(x) · Ψ(x, 0)] dx  (2.2)

for all test functions Φ ∈ H^{1}(0, T; H_{0}(curl, Ω)), Ψ ∈ H^{1}(0, T; H(curl, Ω)) with Φ(x, T) = Ψ(x, T) = 0.

First of all, we derive some energy estimates. A special attention is paid on how various constants depend on ε since we will study the singular limit problem in section 3.

Lemma 2.1: Under the assumptions H(2.1)-H(2.2) there exist constants C_{1}, C_{2} and C_{3} such that

sup_{[0,T]} ∫_{Ω} [ε|E|^{2} + |H|^{2} + |E|^{p+2}] dx + ∫_{0}^{T} ∫_{Ω} [ε|E_{t}|^{2} + |H_{t}|^{2}] dxdt

≤ C_{1} ∫_{Ω} [|E_{0}|^{2} + |H_{0}|^{2} + |∇ × H_{0}|^{2}] dx + C_{2} ∫ ∫_{Q_{T}} [|F|^{2} + |F_{t}|^{2}]dxdt + C_{3},

where C_{1}, C_{2} and C_{3} depend only on known data. Proof: Note that for any vector fields A, B ∈ H(curl, Ω) with either A or B in H_{0}(curl, Ω), the following identity holds:

∫_{Ω} A · (∇ × B)dx = ∫_{Ω} B · (∇ × A)dx.

Taking inner product to the system (1.1) and (1.2) by E and H, respectively, we add up the resulting equations to obtain

sup_{[0,T]} ∫_{Ω} [ε|E|^{2} + |H|^{2}] dx + ∫ ∫_{Q_{T}} σ(x, |E|)|E|^{2}dxdt

≤ C ∫_{Ω} [|E_{0}|^{2} + H_{0}|^{2}] dx + ∫ ∫_{Q_{T}} [|E · F|]dxdt  (2.3)

where the constant C depends only on known data, but not on ε.

We first assume that σ(x, s) is differentiable with respect to s. Then we formally differentiate Eq.(1.1) and Eq.(1.2) with respect to t to obtain

εE_{tt} + σ(x, |E|)E_{t} + σ_{s}(x, |E|)(|E|)_{t}E = ∇ × H_{t} + F_{t},
H_{tt} + ∇ × E_{t} = 0.

It is clear that

∫ ∫_{Q_{T}} (∇ × E_{t}) · H_{t}dxdt = ∫ ∫_{Q_{T}} (∇ × H_{t}) · E_{t}dxdt.

We take the inner product by E_{t} for the first equation and by H_{t} for the second equation and add up the resulting equations to obtain:

sup_{[0,T]} ∫_{Ω} [ε|E_{t}|^{2} + |H_{t}|^{2}] dxdt + ∫ ∫_{Q_{T}} [σ|E_{t}|^{2} + σ_{s}(|E|)_{t}E · E_{t}] dxdt

≤ C,

where C depends only on known data.

Note that σ_{s} ≥ 0, we see that

∫ ∫_{Q_{T}} σ_{s}(|E|)_{t}E · E_{t}dxdt

= ∫ ∫_{Q_{T}} σ_{s}(|E|)_{t} \frac{d}{dt} \frac{|E|^{2}}{2} dxdt

= ∫ ∫_{Q_{T}} σ_{s}|E|[|E|_{t}]^{2}dxdt ≥ 0.

It follows that

sup_{[0,T]} ∫_{Ω} [ε|E_{t}|^{2} + |H_{t}|^{2}] dxdt + ∫ ∫_{Q_{T}} [σ|E_{t}|^{2}dxdt ≤ C.

Since the above estimate does not depend on the differentiability of σ with respect to s, therefore the above estimate holds as long as σ is monotone increasing with respect to s.

Now we take the inner product by E_{t} to (1.1) and by H_{t} to (1.2) and then add up the resulting equations to obtain

∫ ∫_{Q_{T}} [ε|E_{t}|^{2} + |H_{t}|^{2}] dxdt + \frac{1}{2} ∫_{Ω} ∫_{0}^{|E(x,T)|^{2}} σ(x, \sqrt{s})dsdx

≤ ∫ ∫_{Q_{T}} E_{t} · Fdxdt + C ∫_{Ω} ∫_{0}^{|E_{0}(x)|^{2}} σ(x, \sqrt{s})dsdx + ∫_{Ω} |∇ × H_{0}|^{2}dx + C.

Now

∫ ∫_{Q_{T}} E_{t} · Fdxdt

= ∫_{Ω} [E(x, T) · F(x, T) − E_{0}(x) · F(x, 0)] dx − ∫ ∫_{Q_{T}} E · F_{t}dxdt

≤ ∫_{Ω} [\frac{a_{0}}{4} E(x, T)|^{p+2} + \frac{16}{a_{0}}|F_{t}|^{\frac{p+2}{p+1}} ] dxdt.

On the other hand, by the assumption H(2.1) we may assume that the growth condition of σ(x, s) on s holds for all s ≥ M_{0}, i.e.,

∫_{0}^{s^{2}} σ(x, \sqrt{s})dx ≥ a_{0}s^{p+2} − a_{1}, if s ≥ M_{0},

where M_{0} is a fixed constant.

It follows that

∫_{Ω} ∫_{0}^{|E(x,T)|^{2}} σ(x, s)dsdx
≥ a_{0} ∫_{Ω ⋂{x:|E(x,T)|≥M_{0}}} |E(x, T)|^{p+2}dx − C.  (2.4)

Combining (2.3)-(2.4) yields

sup_{0≤t≤T} ∫_{Ω}[ε|E|^{2} + |H|^{2}]dx + sup_{0≤t≤T} ∫_{Ω} |E(x, t)|^{p+2}dx + ∫_{0}^{T} ∫_{Ω}[ε|E_{t}|^{2} + |H_{t}|^{2}]dxdt

≤ C ∫ ∫_{Q_{T}} [|F|^{2} + |F_{t}|^{2}]dxdt + ∫_{Ω}[|E_{0}|^{2} + |H_{0}|^{2} + |∇ × H_{0}|^{2}]dx + C.

Q.E.D.

Theorem 2.2: Under the assumptions H(2.1)-H(2.2) the problem (1.1)-(1.4) has a unique weak solution. Moreover,

curlE ∈ L^{2}(Q_{T}), E_{t} ∈ L^{2}(Q_{T})

and

H_{t} ∈ L^{2}(Q_{T}), ∇ × H ∈ L^{\frac{p+2}{p+1}} (Q_{T}).

Proof: The proof is based on the finite element method (see [14] for parabolic equations). The monotonicity of σ(x, s) on s plays an important role. We shall first deal with the case where σ(x, s) is continuous on s. For convenience, we rewrite the system (1.1)-(1.4) to the following form:

εW_{tt} + σ(x, |W_{t}|)W_{t} = ∇ × [H_{0} − ∇ × W] + F,  (x, t) ∈ Q_{T} , (2.5)
N × (W_{t}) = 0,  x ∈ ∂Ω, 0 ≤ t ≤ T,  (2.6)
W(x, 0) = 0, W_{t}(x, 0) = E_{0}(x),  x ∈ Ω,  (2.7)

where W is defined as follows:

W(x, t) = ∫_{0}^{t} E(x, τ)dτ.

It is clear that if W is a solution of the system (2.5)-(2.7) then a pair of functions defined by

E(x, t) = W_{t}(x, t), H(x, t) = H_{0}(x) − ∇ × W(x, t)

will be a weak solution of (1.1)-(1.4). Let {e_{k}} = {e_{k}^{(1)}, e_{k}^{(2)}, e_{k}^{(3)}} be a smooth basis of H_{0}(curl, Ω) and orthonormal in L^{2}(Ω)^{3}, i.e.

< e_{i}, e_{j} >= δ_{ij},

where δ_{ij} = 1 if i = j and δ_{ij} = 0 if i ≠ j.

Now we expand the known data as follows:

H_{0}(x) = ∑_{k=1}^{∞} diag[a_{k} ∘ e_{k}],

E_{0}(x) = ∑_{k=1}^{∞} diag[b_{k} ∘ e_{k}],

F(x, t) = ∑_{k=1}^{∞} diag[g_{k}(t) ∘ e_{k}],

where a_{k}, b_{k} and g_{k} are 3 × 1 matrices, the symbol ∘ is the matrix product and diag[] represents the diagonal vector of a matrix.

Let

W_{n}(x, t) = ∑_{k=1}^{n} diag[c_{n}^{(k)} ∘ e_{k}],

where c_{n}^{(k)}(t) is a 3×1 vector which is determined by the following ordinary differential system:

ε\frac{d^{2}}{dt^{2}} c_{n}^{(k)} + σ(x, |W_{nt}|)\frac{d}{dt}c_{n}^{(k)} = A_{k}(W_{n}^{(k)}, e_{k}) + B_{k}(t),  (2.8)
c_{n}^{(k)})(0) = 0,  (2.9)
\frac{d}{dt}c_{n}^{(k)}(0) = b_{k},  (2.10)

where

W_{n}^{(k)} = diag[c_{n}^{(k)} ∘ e_{k}],
A_{k}(W_{n}^{(k)}, e_{k}) = ∫_{Ω} diag{[∇ × W_{n}^{(k)}] ∘ [∇ × e_{k}]}dx,
B_{k}(t) = ∫_{Ω} diag{(∇ × H_{0} + F) ∘ e_{k}}dx, k = 1, 2, ⋯ n.

Now we define the approximate solution (E_{n}, H_{n}) as follows:

E_{n}(x, t) = W_{nt}(x, t), H_{n}(x, t) = H_{0n} − ∇ × W_{n}(x, t),

where

H_{0n}(x) = ∑_{k=1}^{n} diag[a_{k} ∘ e_{k}].

Equivalently, then (E_{n}, H_{n}) satisfies the following system in the weak sense:

εE_{nt} + σ(x, |E_{n}|)E_{n} = ∇ × H_{n},  (x, t) ∈ Q_{T} ,  (2.11)
H_{nt} + ∇ × E_{n} = 0,  (x, t) ∈ Q_{T} .  (2.12)

Similar to Lemma 2.1, one can easily derive the following energy estimates :

sup_{[0,T]} ∫_{Ω} [ε|E_{n}|^{2} + |H_{n}|^{2} + |E_{n}|^{p+2}] dx + ∫_{0}^{T} ∫_{Ω} [ε|E_{nt}|^{2} + |H_{nt}|^{2}] dxdt

≤ C_{1} ∫_{Ω} [|E_{0n}|^{2} + |H_{0n}|^{2} + |∇ × H_{0n}|^{2}] dx + C_{2} ∫ ∫_{Q_{T}} [|F_{n}|^{2} + |F_{nt}|^{2}]dxdt + C_{3},

where C_{1}, C_{2} and C_{3} are independent of n and ε.

By the weak compactness property, we can extract a subsequence (still denoted by (E_{n}, H_{n})) such that

E_{n} → E, E_{nt} → E_{t}, H_{nt} → H_{t}, weakly in L^{2}(Q_{T}),
H_{n} → H, weakly in L^{2}(0, T; W^{1, \frac{p+2}{p+1}} (Ω)),
H_{n} → H, a.e. in Q_{T} .

Moreover,

E_{n} → E weakly in L^{p+2}(Q_{T}).

Next we claim that the sequence E_{n} converges to E strongly in L^{2}(Q_{T})^{3}. To prove the claim we only need to show that {E_{n}} is a Cauchy sequence in L^{2}(Q_{T})^{3}. Let

E_{n}^{∗}(x, t) = E_{n}(x, t) − E_{m}(x, t), H_{n}^{∗}(x, t) = H_{n}(x, t) − H_{m}(x, t).

By energy estimates, we see

sup_{0≤t≤T} ∫_{Ω}[|E_{n}^{∗}|^{2} + |H_{n}^{∗}|^{2}]dx +

∫ ∫_{Q_{T}} {[σ(x, |E_{n}|)E_{n} − σ(x, |E_{m}|)E_{m}] · [E_{n} − E_{m}]} dxdt

≤ C ∫_{Ω}[|E_{0n} − E_{0m}|^{2} + |H_{0n} − H_{0m}|^{2}]dx + C ∫ ∫_{Q_{T}} [|F_{n} − F_{m}|^{2}]dxdt,

where C is a constant independent of n and m.

Note that σ(x, s) is monotonic increasing in s, then

[σ(x, |E_{n}|)E_{n} − σ(x, |E_{m}|)E_{m}] · [E_{n} − E_{m}]

≥ \frac{σ(x, |E_{n}|) − σ(x, |E_{m}|)][|E_{n}|^{2} − |E_{m}|^{2}]}{2} ≥ 0.

It follows that

sup_{0≤t≤T} ∫_{Ω}[|E_{n}^{∗}|^{2} + |H_{n}^{∗}|^{2}dx

≤ C ∫_{Ω}[|E_{0n} − E_{0m}|^{2} + |H_{0n} − H_{0m}|^{2}]dx + C ∫ ∫_{Q_{T}} |F_{n} − F_{m}|^{2}dxdt.

This implies that E_{n}, H_{n} are Cauchy sequences since both E_{0n}, H_{0n} and F_{n} are Cauchy sequences in L^{2}(Q_{T})^{3}. Hence,

E_{n}, H_{n} → E, H strongly in L^{2}(Q_{T}).

After taking a subsequence if necessary, we see that

E_{n} → E, a.e. in Q_{T} .

To show the existence of a weak solution to (1.1)-(1.4), we only need to show

σ(x, |E_{n}|)E_{n} → σ(x, |E|)E in L^{1}(Q_{T}).

As σ(x, s) is continuous on s and E_{n} converges to E almost everywhere in Q_{T}, we know

σ(x, |E_{n}|)E_{n} → σ(x, |E|)E a.e. in Q_{T}.

We now show that σ(x, |E_{n}|)E_{n} is equip-integrable in Q_{T}. We adopt a technique used for scalar elliptic and parabolic equations. Let A be any measurable subset of Q_{T}. For any large m > 0,

∫ ∫_{A} σ(x, |E_{n})|E_{n}|dxdt
≤ ∫ ∫_{A ⋂{|E_{n}|≤m}} σ(x, |E_{n}|)|E_{n}|dxdt + ∫ ∫_{A ⋂{|E_{n}|≥m}} σ(x, |E_{n}|)|E_{n}|dxdt
≡ I_{1} + I_{2}.

The assumption on σ(x, s) yields

I_{1} ≤ C ∫ ∫_{A ⋂{|E_{n}|≤m}}[1 + |E_{n}|^{p}|E_{n}|]dxdt,

which can be arbitrarily small if |A| is small since E_{n} ∈ L^{p+2}(Q_{T}).

On the other hand,

I_{2} ≤ \frac{1}{m} ∫ ∫_{A ⋂\{|E_{n}|≥m\}} σ(x, |E_{n}|)|E_{n}|^{2}dxdt ≤ \frac{C}{m},

which is also small if m is sufficiently large.

This concludes that σ(x, |E_{n}|)E_{n} is equip-integrable in Q_{T}.

It follows by Vitali’s theorem that

σ(x, |E_{n}|)E_{n} → σ(x, |E|)E in L^{1}(Q_{T}).

Finally, we show that (E, H) is a weak solution of (1.1)-(1.4). By multiplying Eq.(2.11) and Eq.(2.12) by test functions Ψ and Φ, respectively, and then taking integration over Q_{T}, after some routine calculations and taking the limit, we see that (E, H) is a weak solution to the system (1.1)-(1.4).

Now we consider the case where σ(x, s) is discontinuous on s at some points. Without loss of generality, we may assume that σ(x, s) has a jump only at one point s = 1. In this case σ(x, s) is not uniquely defined at s = 1. We shall understand the value of σ(x, 1) in the following sense:

σ(x, 1) ∈ [σ(x, 1−), σ(x, 1+)],

where σ(x, 1±) represents the right or left limit as s → 1.

By the standard approximation, we can construct a smooth approximation sequence σ_{m}(x, s) such that

(i) σ_{m}(x, s) is monotonic increasing for all s ≥ 0,
(ii) σ_{m}(x, s) = σ(x, s), if |s − 1| ≥ \frac{1}{m}.

Let (E_{m}, H_{m}) be a solution of (1.1)-(1.4) in which σ(x, s) is replaced by σ_{m}(x, s). By the same energy estimate we see that there exists a measurable function β(x, t) ∈ L^{\frac{p+2}{p+1}} (Q_{T}) such that

σ_{m}(x, |E_{m}|) → β(x, t), weakly in L^{\frac{p+2}{p+1}} (Q_{T}).

Define

A_{m} = \{(x, t) : 1 − \frac{1}{m} ≤ |E| ≤ 1 + \frac{1}{m}\},

A = {(x, t) : |E(x, t)| = 1}.

Since σ(x, s) is continuous except at s = 1, we see

β(x, t) = σ(x, |E|)E, if (x, t) ∈ Q_{T} \A.

Now it is clear that

A = ⋂_{m=1}^{∞} A_{m}.

Recall that σ_{m}(x, s) = σ(x, s) if |s − 1| ≥ \frac{1}{m}. It follows that for all (x, t) ∈ A_{m}

σ(x, 1 − \frac{1}{m}) ≤ σ_{m}(x, |E|) ≤ σ(x, 1 + \frac{1}{m}).

Consequently, as m → 0,

σ(x, 1−) ≤ β(x, t) ≤ σ(x, 1+), (x, t) ∈ A.

Thus, (E, H) is a weak solution of (1.1)-(1.4).

Finally, we show the uniqueness. Suppose (E, H) and (E^{∗}, H^{∗}) are two solutions of (1.1)-(1.4). Let

Ê = E − E^{∗}, Ĥ = H − H^{∗}.

Similar to the calculation in deriving energy estimates, we find

sup_{0≤t≤T} ∫_{Ω}[|Ê|^{2} + |Ĥ|^{2}]dx + ∫ ∫_{Q_{T}} [σ(x, |E|)E − σ(x, |E^{∗}|)E^{∗}] · [E − E^{∗}] dxdt

≤ 0.

The monotonicity of σ(x, s) implies that the second term in the above inequality is nonnegative. It follows that

sup_{0≤t≤T} ∫_{Ω}[|Ê|^{2} + |Ĥ|^{2}]dx ≤ 0.

Therefore, the uniqueness follows immediately.

Q.E.D.

3. Singular Limit Problem

In this section we shall show that the solution of (1.1)-(1.4) has a limit as ε → 0, which solves Maxwell’s equations in quasi-stationary fields, i.e. the system (1.1)-(1.4) with ε = 0. A weak solution of the quasi-stationary system is defined as in Definition 2.1 with ε = 0.

From now on we denote by (E_{ε}, H_{ε}) the weak solution of the system (1.1)-(1.4).

Theorem 3.1: The limit of (E_{ε}, H_{ε}) as ε → 0 solves the quasi-stationary system (1.1)-(1.4) with ε = 0 in the weak sense. Moreover, the weak solution is unique if σ(x, s) > 0 for all (x, s) ∈ Ω × R^{+}. Proof: The crucial step in proving the convergence is to show

σ(|E_{ε}|)E_{ε} → σ(|E|)E, a.e. in Q_{T} as ε → 0.

The monotonicity of σ(x, s) in s plays a key rule.

First of all, from Lemma 2.1 and the weak compactness we see

E_{ε} → E, H_{εt} → H_{t}, weakly in L^{2}(Q_{T}),
∇ × H_{ε} → ∇ × H, weakly in L^{\frac{p+2}{p+1}} (Q_{T}),
σ(x, |E_{ε}|)E_{ε} → J(x, t), weakly in L^{\frac{p+2}{p+1}} (Q_{T}),

where J(x, t) ∈ L^{\frac{p+2}{p+1}} (Q_{T}). Moreover, as divH_{ε}(x, t) = 0, by the decomposition property of H^{1}(Ω) property, after extracting a subsequence if necessary we see that

H_{ε} → H, strongly in L^{2}(Q_{T})

and

H_{ε} → H, a.e. in Q_{T} .

Next we show

J(x, t) = σ(x, |E|)E, a.e.in Q_{T} .

We use a monotonicity argument. As a first step, we show

lim_{ε→0} ∫ ∫_{Q_{T}} σ(x, |E_{ε}|)|E_{ε}|^{2}dxdt = ∫ ∫_{Q_{T}} J · Edxdt.

Here we adopt an idea from [10]. Let λ(t) be a nonnegative smooth function and

λ′(t) ≤ 0, λ(0) = 1, λ(T) = 0.

Define an operator L in L^{p+2}(Q_{T})^{3} as follows:

L[E] = σ(x, |E|)E.

Since σ(x, s) is monotonic increasing in s, then the operator L is monotonic increasing, that is,

< L[E_{ε}] − L[E], E_{ε} − E >≥ 0.

It is clear that

< L[E_{ε}] − L[E], E_{ε} − E >
=< L[E_{ε}], E_{ε} > − < L[E_{ε}], E > − < L[E], E_{ε} > + < L[E], E >

It follows that

lim_{ε→0} inf < L[E_{ε}], E_{ε} >≥< J, E > .  (3.1)

On the other hand, from the system (1.1)-(1.2) we have

∫_{0}^{T} ∫_{Ω} σ(E_{ε})|E_{ε}|^{2}λ(t)dxdt

= − ∫_{0}^{T} ∫_{Ω} [ελ(t)E_{εt} · E_{ε} + λH_{εt} · H_{ε}] dxdt + ∫ ∫_{Q_{T}} F · E_{ε}dxdt

= − ∫_{0}^{T} ∫_{Ω} \{\frac{∂}{∂t} [\frac{1}{2}(ε|E_{ε}|^{2} + |H_{ε}|^{2})λ(t)] − λ′(t) [\frac{1}{2}(ε|E_{ε}|^{2} + |H_{ε}|^{2})]\} dxdt +

∫ ∫_{Q_{T}} F · E_{ε}dxdt

≤ ∫_{0}^{T} ∫_{Ω} \{λ′(t) [\frac{1}{2}|H_{ε}|^{2}]\} dxdt + \frac{1}{2} ∫_{Ω} [ε|E_{ε}(x, 0)|^{2} + |H_{ε}(x, 0)|^{2}] dx +

∫ ∫_{Q_{T}} F · E_{ε}dxdt.  (3.2)

Since λ′(t) ≤ 0, it follows that

lim_{ε→0} sup ∫_{0}^{T} ∫_{Ω} σ(|E_{ε}|)|E_{ε}|^{2}λ′(t)dxdt

≤ \frac{1}{2} ∫_{0}^{T} ∫_{Ω} λ(t)|H|^{2}dxdt + \frac{1}{2} ∫_{Ω} |H_{0}(x)|^{2}dx + ∫ ∫_{Q_{T}} E · Fdxdt.  (3.3)

Recall from Definition 2.1 that (E_{ε}, H_{ε}) satisfies the following integral equations:

∫_{0}^{T} ∫_{Ω} [−εE_{ε} · Φ_{t} + σ(x, |E_{ε}|)E_{ε} · Φ] dxdt
= ∫_{0}^{T} ∫_{Ω} H_{ε} · (∇ × Φ)dxdt + ∫ ∫_{Q_{T}} F · Φ_{ε}dxdt + ∫_{Ω} εE_{0} · Φ(x, 0)dx,
∫_{0}^{T} ∫_{Ω} [−H_{ε} · Ψ_{t} + E_{ε} · (∇ × Ψ)] dxdt = ∫_{Ω} [H_{0}(x) · Ψ(x, 0)] dx.

Note by Lemma 2.1 that

| ∫_{0}^{T} ∫_{Ω} εE_{εt} · Φdxdt|
≤ (∫_{0}^{T} ∫_{Ω} ε|E_{εt}|^{2}dxdt)^{1/2} (∫_{0}^{T} ∫_{Ω} ε|Φ|^{2}dxdt)^{1/2}
→ 0 as ε → 0.

We take the limit as ε → 0 to obtain

∫_{0}^{T} ∫_{Ω} [J · Φ] dxdt = ∫_{0}^{T} ∫_{Ω} H · (∇ × Φ)dxdt + ∫ ∫_{Q_{T}} F · Φdxdt (3.4)
∫_{0}^{T} ∫_{Ω} [−H · Ψ_{t} + E · (∇ × Ψ)] dxdt = ∫_{Ω} [H_{0}(x) · Ψ(x, 0)] dx.  (3.5)

Now by choosing Φ = λ(t)E and Ψ = λ(t)H (note that the condition at t = T is satisfied since λ(T) = 0), we obtain

∫_{0}^{T} ∫_{Ω}[λ(t)J · E]dxdt = \frac{1}{2} ∫_{0}^{T} ∫_{Ω}[λ′(t)|H|^{2}]dxdt + \frac{1}{2} ∫_{Ω} |H_{0}(x)|^{2}]dx + ∫ ∫_{Q_{T}} E · Fdxdt.

It follows by (3.3) that

lim_{ε→0} sup < λ(t)L[E_{ε}], E_{ε} >≤< λ(t)J, E > .  (3.6)

Combination of (3.1) and (3.6) yields

lim_{ε→0} < λ(t)L[E_{ε}], E_{ε} >=< λ(t)J, E > .  (3.7)

Consequently, by choosing λ(t) properly we have

lim_{ε→0} < L[E_{ε}], E_{ε} >=< J, E > .  (3.8)

For any vector field W ∈ L^{2}(Q_{T}) ⋂ L^{p+1}(Q_{T}), the monotonicity of σ(x, s) in s implies

∫ ∫_{Q_{T}} [σ(x, |E_{ε}|)E_{ε} − σ(x, |W|)W] · [E_{n} − W]dxdt ≥ 0.  (3.9)

We rewrite the above inequality to the following form:

∫ ∫_{Q_{T}} {σ(|E_{ε}|)|E_{ε}|^{2} − σ(|E_{ε}|)E_{ε} · W} dxdt
≥ ∫ ∫_{Q_{T}} [σ(x, |W|)W] · [E_{ε} − W]dxdt.  (3.10)

We take the limit as ε → 0 and use (3.8) for the first term in (3.10) to obtain

∫ ∫_{Q_{T}} {J · E − J · W} dxdt
≥ ∫ ∫_{Q_{T}} [σ(x, |W|)W] · [E − W]dxdt.

Equivalently,

∫ ∫_{Q_{T}} {[J − σ(x, |W|)W] · [E − W]} dxdt ≥ 0.  (3.11)

Set W = E + δY, where δ > 0 is small parameter and Y ∈ L^{2}(Q_{T}) ⋂ L^{p+1}(Q_{T}) is arbitrary.

With the above choice of W in the equality (3.10), we obtain

∫ ∫_{Q_{T}} {Y · [J − σ(|E + δY|)[E + δY]} dxdt ≥ 0.

When σ(x, s) is continuous in s, then we let δ → 0 to obtain

J(x, t) = σ(x, |E|)E,

since Y(x, t) is arbitrary in L^{2}(Q_{T}) ⋂ L^{p+1}(Q_{T}).

When σ(x, s) has a jump at a point, say, s = 1. Then as in §2 we understand that the value of σ(x, s) at s = 1 is

σ(x, 1−) ≤ σ(x, 1) ≤ σ(x, 1+).

By using the same procedure as in §2, we can derive the above inequality.

Finally, by taking limit for (2.1)-(2.2) we see that (E, H) is a weak solution of the quasi-stationary system.

To prove the uniqueness, we assume that (E, H) and (E^{∗}, H^{∗}) are two weak solutions to the quasi-stationary system. Let Ê = E − E^{∗} and Ĥ = H − H^{∗}. Then the energy estimate implies

sup_{0≤t≤T} ∫_{Ω} |Ĥ|^{2}dx + ∫ ∫_{Q_{T}} [σ(x, |E|)E − σ(x, |E^{∗}|)E^{∗}] · [E − E^{∗}]dxdt ≤ 0.

The monotonicity of σ(x, s) in s implies the second term in the above inequality is nonnegative. It follows that

Ĥ = 0, a.e.in Q_{T} .

From the definition of weak solution, we have Ê = 0 as long as σ > 0.

Q.E.D.

To conclude this section, we consider two special classes of electric conductivity σ(x, s) = s^{p} and

σ(x, s) = \{ \begin{matrix}a, if |s| ≤ 1; \\ b, if |s| > 1, \end{matrix}

where 0 < a < b.

For the first case with F = 0, it is easy to see that

E = |∇ × H|^{− \frac{p}{p+1}} ∇ × H.

It follows from Theorem 3.1 that there exists a unique weak solution to the following evolution system

H_{t} + ∇ × [|∇ × H|^{− \frac{p}{p+1}} ∇ × H] = G, (x, t) ∈ Q_{T} ,

subject to the initial-boundary conditions:

N × (∇ × H) = 0, (x, t) ∈ ∂Ω × [0, T],
H(x, 0) = H_{0}(x), x ∈ Ω,

where G is a known exterior magnetic field. This existence result is not covered in [16]. More regularity of the weak solution can be established as in [16]. We shall not repeat it here.

For the second case, we see that

E = \frac{1}{a}∇ × H + \frac{1}{a}F,

in the region Q_{T}^{−} = {(x, t) : |E| < 1} and

E = \frac{1}{b}∇ × H + \frac{1}{b}F,

in the region Q_{T}^{+} = {(x, t) : |E| > 1}. Note that

divH(x, t) = divH_{0}(x) = 0,

we see that

∇ × ∇ × H = −ΔH.

It follows that H satisfies the parabolic equation:

H_{t} − \frac{1}{a}ΔH = −\frac{1}{a}∇ × F, (x, t) ∈ Q_{T}^{−}

and

H_{t} − \frac{1}{b}ΔH = −\frac{1}{b}∇ × F, (x, t) ∈ Q_{T}^{+}.

The regularity theory of parabolic equations implies that H(x, t) is smooth in Q_{T}^{±}.

The interface between Q_{T}^{−} and Q_{T}^{+} is defined by

Γ = {(x, t) : |E| = 1},

which is a free boundary.

Remark 3.1: We may allow that σ(s) = 0 if |E| < 1 in the above example. In this case, one must consider the full system (1.1)-(1.2) in order to define a weak solution. However, in this case the uniqueness of the weak solution does not hold. Remark 3.2: It is not clear whether or not Γ is indeed a hypersurface in R^{3} ×(0, ∞]. It would be of great interesting to study the smoothness of the interface Γ and to find the free boundary conditions for H.

Acknowledgment: This work was done when author was visiting University of California and MSRI at Berkeley. The author would like to thank them for the hospitality. The author is also grateful to Professor L.C. Evans and Professor Guy David for many helpful discussions.

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