Energy decay for Maxwell's equations with Ohm's law on partially cubic domains

We prove a polynomial energy decay for the Maxwell's equations with Ohm's law on partially cubic domains with trapped rays.


Introduction
The problems dealing with Maxwell's equations with nonzero conductivity are not only theoretical interesting but also very important in many industrial applications (see e.g. [3], [7], [8]).
Let Ω be a bounded open connected region in R 3 , with a smooth boundary ∂Ω. We suppose that Ω is simply connected and ∂Ω has only one connected component. The domain Ω is occupied by an electromagnetic medium of constant electric permittivity ε o and constant magnetic permeability µ o . Let E and H denote the electric and magnetic fields respectively. The Maxwell's equations with Ohm's law are described by in Ω . (1.1) Here, (E o , H o ) are the initial data in the energy space L 2 (Ω) 6 and ν denotes the outward unit normal vector to ∂Ω. The conductivity is such that σ ∈ L ∞ (Ω) and σ ≥ 0. It is well-known that when the conductivity is identically null, then the above system is conservative and when σ is bounded from below by a positive constant, then an exponential energy decay rate holds for the Maxwell's equations with Ohm's law in the energy space. The situation becomes more delicate when we only assume that σ (x) ≥ constant > 0 ∀x ∈ ω for some non-empty connected open subset ω of Ω. Observe that the condition div (ε o E) = 0 in Ω × [0, +∞) does not appear because the free divergence is not preserved by the Maxwell's equations with Ohm's law. Here we know that the above system is dissipative and its energy tends to zero in large time. However, we would like to establish the energy decay rate as well. In the field of control theory, the exponential energy decay rate of a linear dissipative system is deduced from an observability estimate. Precisely, in order to get an exponential decay rate in the energy space we should have the following observability inequality or simply, in virtue of a semigroup property, for any initial data (E o , H o ) in the energy space L 2 (Ω) 6 . We can also look for establishing the above observability inequality for any initial data in the energy space intersecting suitable invariant subspaces but not with the condition div E o = 0 in Ω. Such estimate is established in [11] under the geometric control condition of Bardos, Lebeau and Rauch [2] for the scalar wave operator and when the conductivity has the property that σ (x) ≥ constant > 0 for all x ∈ ω and σ (x) = 0 for all x ∈ Ω \ω . From now, we consider a subset ω such that the geometric control condition for the scalar wave operator or other assumptions based on the multiplier method fail. In such geometry, we do not hope an exponential energy decay rate in the energy space. Our geometry (described precisely in Section 3) presents parallel trapped rays and can be compared to the one in [12] or in [4], [10] for the two dimensional case. It generalises the cube (see [8]) and therefore explicit and analytical results are harder to obtain. Our main result gives a polynomial energy decay with regular initial data. Our proof is based on a new kind of observation inequality (see (4.33) below) which can also be seen as an interpolation estimate. It relies with the construction of a particular solution for the operator i∂ s + h ∆ − ∂ 2 t inspired by the gaussian beam techniques. Also the dispersion property for the one dimensional Schrödinger operator will play a key role.
The plan of the paper is as follows. In the next section, we recall the known results about the Maxwell's equations with Ohm's law that will be used in the following. Section 3 contains the statement of our main result, while Section 4 is concerned with its proof. In Section 5, we present the interpolation estimate, while Section 6 includes its proof. Finally, two appendix are added dealing with inequalities involving Fourier analysis.
It is well-known that if (E o , H o ) ∈ V, there is a unique weak solution (E, H) ∈ C 0 ([0, +∞) , V). Further, if (E o , H o ) ∈ W, there is a unique strong solution (E, H) ∈ C 0 ([0, +∞) , W)∩C 1 ([0, +∞) , V). Let us define the functionals of energy We can easily check that the energy E is a continuous positive non-increasing real function on [0, +∞) and further for any initial data ( (2.1.5) and for any t 2 > t 1 ≥ 0,

Orthogonal decomposition
Both E and µ o H can be described, by means of the scalar and vector potentials p and A with the Coulomb gauge, in an unique way as follows.
and we have the following relations The proof is essentially given in [11, page 121] from a Hodge decomposition and is omitted here. Now, the vector field A has the nice property of free divergence and satisfies a second order vector wave equation with homogeneous boundary condition A × ν = div A = 0 and with a second member in C 1 [0, +∞) , L 2 (Ω) 3 bounded by 2µ o σE L 2 (Ω) 3 . For the sake of simplicity, we assume from now that ε o µ o = 1.
We recall that the range of the curl, curl [6, page 257] or [5, page 54]) and The space W ∩ S σ is stable for the system of Maxwell's equations with Ohm's law, which can be seen by multiplying by Then, we can add the following well-posedness result. If It has been proved (see [11, page 124]) that if ω − is a non-empty connected open set then lim t→+∞ E (t) = 0 for any initial data (E o , H o ) ∈ V ∩ S σ . Further, the following result (see [11, page 124]) plays a key role.

Proposition 2.2 -.
If ω − is a non-empty connected open set, then there exists c > 0 such that for all initial data (E o , H o ) ∈ W ∩ S σ of the system (1.1) of Maxwell's equations with Ohm's law, we have Indeed, the proof given in [11, page 127] can be divided into two steps. In the first step, we begin to establish the existence of c > 0 such that ∇p Here, we used a standard compactness-uniqueness argument for H, (2.2.5) of Proposition 2.1 for ∂ t A, and for ∇p from the fact that σ (x) ≥ constant > 0 for all x ∈ ω + and (2.1.5). Till now, we did not need that ω − is a connected set. The second step (see [11, page 128]) did consist to prove that ∇p Finally, we concluded by virtue of (2.2.3) of Proposition 2.1. This last estimate becomes easier to obtain under the assumption ∂ω + ∩ ∂Ω = ∅ and without adding the hypothesis saying that ω − is a connected set. Indeed, since (E o , H o ) ∈ W ∩ S σ and −∆p = divE, p ∈ H 1 0 (Ω) solves the following elliptic system Thus, by the elliptic regularity, the trace theorem and the Poincaré inequality, we have the following estimate for suitable constants c 1 , c 2 , c 3 > 0. The exponential energy decay rate for the Maxwell's equations with Ohm's law in the energy space is as follows.
Proposition 2.4 -. Let ϑ be a subset of Ω such that any generalized ray of the scalar wave operator ∂ 2 t − ∆ meets ϑ. Suppose that ϑ ∩ Ω ⊂ ω + . Further if ∂ω + ∩ ∂Ω = ∅ or ω − is a non-empty connected open set, then there exist c > 0 and β > 0 such that for all initial data (E o , H o ) ∈ V ∩ S σ of the system (1.1) of Maxwell's equations with Ohm's law, we have The proof of Proposition 2.4 is done in [11, page 129] when ω − is a non-empty connected open set. Here, we simply recall the key points of the proof. From the geometric control condition, the following estimate holds without using the fact that ω − is a non-empty connected open set.
Finally, we concluded by virtue of a semigroup property. The proof works as well when ∂ω + ∩ ∂Ω = ∅ thanks to Remark 2.3.

Geometric setting and main result
Let us introduce the geometry on which we work in this paper.
After these preparations, we are now able to state our main result.
for all x ∈ ω and σ (x) = 0 for all x ∈ Ω \ω , then there exist c > 0 and γ > 0 such that for any t ≥ 0 for every solution of the system (1.1) of Maxwell's equations with Ohm's law with initial data Remark 3.2 -. From now ω = Θ ∩ Ω and σ ∈ L ∞ (Ω) is such that σ (x) ≥ constant > 0 for all x ∈ ω and σ (x) = 0 for all x ∈ Ω \ω . Notice that ω and Ω \ω are two non-empty connected open sets with Lipschitz boundaries. Therefore by Proposition 2.2, there exists c > 0 such that for all initial data (E o , H o ) ∈ W ∩ S σ of the system (1.1) of Maxwell's equations with Ohm's law, E (t) ≤ c E 1 (t) for any t ≥ 0.

Proof of the main result
Let us consider the solution U of the following system It is well-known that the above system is well-posed with a unique solution U such that (U (·, t) , ∂ t U (·, t)) and ∂ t U (·, t) , ∂ 2 t U (·, t) belong to X for any t ∈ R. Let us define the following two conservations of energies.
Further, for such solution U , the following two inequalities hold by standard compactness-uniqueness argument and classical embedding (see [1] and [5, page 50]).
for some c > 0 and any t ∈ R.
We start by choosing ω o ⊂ ω o such that ω ∪ ω o is a non-empty connected open set and such that the boundaries ∂ (ω ∪ ω o ) and ∂ Ω ω ∪ ω o are Lipschitz. Notice that ∂ (ω ∪ ω o ) ∩ ∂Ω = ∅ and there exists ϑ a subset of Ω such that ϑ ∩ Ω ⊂ (ω ∪ ω o ) and such that any generalized ray of the scalar wave operator ∂ 2 t − ∆ meets ϑ.
Let ζ, T h ≥ 0. Let E, H denote the electromagnetic field of the following Maxwell's equations with Ohm's law in Ω . (4.9) The conductivity σ + 1 | ωo is such that σ + 1 | ωo ≥ constant > 0 in ω ∪ ω o and σ + 1 | ωo = 0 in Ω ω ∪ ω o . Also notice that (E, H) ∈ W ∩ S σ ⊂ V ∩ S σ+1 | ωo . Therefore by Proposition 2.4, there exist c, β > 0 (independent of ζ, T h ) such that for any t ≥ 0 we have in Ω , (4.11) and by a standard energy method and the fact that ω o ⊂ ω o , we get that for any t ≥ 0 Now we are able to bound the quantity |E (x, s)| 2 dxds as follows. By using (4.10) and (4.12), we deduce that (4.13) which implies by taking t large enough, the existence of constants C, T c > 1 such that Recall the existence of the vector potential A from Proposition 2.1 and let U be the solution of in Ω , then by a standard energy method, for any Now we are able to bound the quantity we deduce by (4.14) and (4.17) that (4.18) Here and hereafter, C will be used to denote a generic constant, not necessarily the same in any two places.
(4.22) and finally, combining (4.18) and (4.22), we get We have proved that there exist h o , c, γ > 0 such that for any ζ ≥ 0 and h ∈ (0, h o ], the solution (E, H) of (1.1) satisfies By formula (2.1.6) and since G ∂ t U, ζ + mc 1 h γ = G (∂ t U, ζ) for any m, this last inequality becomes where M is the m-accretive operator in V with domain D (M) = W, defined as follows.
Therefore, combining (4.26) and (4.27), we get the existence of constants c, γ > 0 such that for any Denote (T (t)) t≥0 the unique semigroup of contractions generated by −M. First, suppose that (E o , H o ) ∈ D M 3 and let us define the functional of energy which satisfies . Further, by uniqueness of the orthogonal decomposition in (2.1.1) of Proposition 2.1, (4.29) implies that for any ( Since by Proposition 2.2, E (ζ) ≤ cE 1 (ζ) and in a similar way E 1 (ζ) ≤ cE 2 (ζ) for some c > 0, taking account of the first line of (2.2.1) and (2.2.4), (4.32) becomes . (4.34) Since H ≤ 1, the inequality (4.33) holds for any h > 0. Taking h = c 0 H (ζ) with some suitable small constant c 0 , we get the existence of constants c, γ > 0 such that for any ζ ≥ 0, The function H is a continuous positive decreasing real function on [0, +∞), bounded by one and satisfying from (2.1.6), (2.1.7), (4.31) and (4.35), From [12, p.122, Lemma B], we deduce that there exist C, γ > 0 such that for any t > 0 Since M is an m-accretive operator in V with dense domain, one can restrict it to D M 2 in a way that its restriction operator is m-accretive. Thus the following two properties holds. Consequently, Since M is an m-accretive operator in V with dense domain, one can restrict it to D (M) in a way that its restriction operator is m-accretive. Thus the following two properties holds.

Proof of Proposition 4.1
Recall that the definition of ω o and the solution U were given in Section 4.
Notice that the hypothesis saying that denotes the ball of center x o and radius r. The proof of Proposition 4.1 comes from the following result.
We shall leave the proof of Proposition 5.1 till later (see Section 6). Now we turn to prove Proposition 4.1.
We begin by covering ω o with a finite collection of and further, By a translation in time, Proposition 4.1 follows.
By integrations by parts on the time variable, we can check that Let us introduce for any θ ∈ {1, 2}, By the Fourier inversion formula, for any λ ≥ 1. Here we recall that when F and F belong to L 1 R 4 3 . On the other hand, from (A1) of Appendix A, we have that It remains to study the following two quantities We claim that with T given by (5.2) which implies Proposition 5.1 using (6.1), (6.5) and Cauchy-Schwarz inequality.
The proof of our claim is divided into nine subsections. In the next subsection, we introduce suitable sequences of Fourier integral operators. First, we add a new variable s ∈ [0, L]. Next, we construct a particular solution of the equation (6.1.10) below for (x, t, s) ∈ R 4 × [0, L] with good properties on Γ 1 ∪ Γ 2 .

Fourier integral operators
First, let us introduce for any (x, t, s) ∈ R 4 × [0, L] and n ∈ Z, Next, let us introduce for any (x, t, s) ∈ R 4 × [0, L], where (P, Q) ∈ N 2 is the first couple of integer numbers satisfying    P ≥ 1 4ρ (|ξ o3 | + 2) (L 2 + 1) + 2 (|ξ o3 | + 1) L , We check after a lengthy but straightforward calculation that for any ( and that for any ( Let f j = f j (x, t) ∈ L ∞ R; L 2 R 3 be such that ϕf j ∈ L 1 R 4 for any j ∈ {1, 2, 3}. Let us introduce On another hand, let U be the solution of (4.1). Denote because div U = 0 and U × ν = 0. Further, by (4.3) and (4.6), (6.1.13) The different terms of the last equality will be estimated separately. The quantity I 1 will allow us to recover (6.6) (resp. (6.7)) when θ = 2 and ϕF = ϕ 2 U (resp. when θ = 1 and ϕF = ϕ 1 ∂ 2 t U ). The dispersion property for the one dimensional Schrödinger operator will be used for making I 2 small for large L. We treat I 3 (resp. I 4 ) by applying the formula (6.1.7) (resp. (6.1.8)). The quantity I 5 and I 7 will correspond to a term localized in ω. Finally, an appropriate choice of T will bound I 6 and give the desired inequality (6.9.2) below.
(6.2.3) Remark that for any (x o1 , x o2 , x o3 ) ∈ ω o and (x 1 , x 2 , x 3 ) ∈ Ω, the following two cases appears. If ξo3−1 |τ |<λ ϕf (ξ, τ ) dξdτ (6.2.5) where in the last line we have used the fact that the solution U has the following property, from Cauchy-Schwarz inequality and (6.1.12), Here and hereafter, c will be used to denote a generic constant, not necessarily the same in any two places. On the other hand, we cut the integral on time into two parts to obtain (6.2.7) and, by using (6.1.12) and Cauchy-Schwarz inequality, we have (6.2.8) We conclude from (6.2.5), (6.2.7) and (6.2.8) that (6.2.10) This completes the proof.