Long time behaviour of the solution of Maxwells equations in dissipative generalized Lorentz materials I A frequency dependent Lyapunov function approach

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Long time behaviour of the solution of Maxwell’s equations in

dissipative generalized Lorentz materials (I) A frequency

dependent Lyapunov function approach

Maxence Cassiera, Patrick Jolyband Luis Alejandro Rosas Mart´ınezb

aAix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France

bENSTA / POEMS1, 32 Boulevard Victor, 75015 Paris, France

(maxence.cassier@fresnel.fr, patrick.joly@inria.fr, alejandro.rosas@ensta-paris.fr)

October 19, 2022

Abstract

It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled

by generalized Drude-Lorentz models. The present paper is the ﬁrst of two articles dedicated to

dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media

in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy

problem. By using an approach based on frequency dependent Lyapounov estimates, we show that

this decay is polynomial in time. These results extend to an unbounded structure the ones obtained

for bounded media in [18] via a quite diﬀerent method based on the notion of cumulated past history

and semi-group theory. A great advantage of the approach developed here is to be less abstract and

directly connected to the physics of the system via energy balances.

Keywords: Maxwell’s equations, passive electromagnetic media, dissipative generalized Lorentz models,

long time electromagnetic energy decay rate, frequency-dependent Lyapunov estimates.

1 Introduction and motivation

The study of the long time behaviour of solutions of dispersive and dissipative models for linear wave

propagation has already been extensively studied in the literature, primarily for applications in visco-

elasticity and more recently in electromagnetism. The subject has recently known a regain of interest

related to metamaterials. We can refer for instance, in electromagnetism, to the article [5] in which we

presented a systematic construction of mathematical models compatible with physically motivated prin-

ciple such as causality and passivity (see also [14,6,23]). The common point to all these models lies in

that the constitutive laws include memory eﬀects corresponding to time convolution nonlocal operators

that induce dispersion (the velocity of waves is frequency dependent) and dissipation (the energy decay

of the solution) that are in often intimately related.

For such models one of the most natural question is the study of the long time behaviour of the corre-

sponding Cauchy problem: prove that the energy of the solution tends to 0 when ttends to +∞and study

the rate of decay. This is of course closely related to the theory of control and stabilization of dynamical

1POEMS (Propagation d’Ondes: Etude Math´ematique et Simulation) is a mixed research team (UMR 7231) between

CNRS (Centre National de la Recherche Scientiﬁque), ENSTA Paris (Ecole Nationale Sup´erieure de Techniques Avanc´ees)

and INRIA (Institut National de Recherche en Informatique et en Automatique).

arXiv:2210.09360v1 [math.AP] 17 Oct 2022

systems where one commonly distinguishes the notion of exponential stability (which corresponds to an

exponential decay of the energy) and polynomial stability (the energy decays as the inverse of a positive

power of t).

1.1 Maxwell’s equations in dispersive media

1.1.1 General features

Maxwell’s equations relate the electric and magnetic inductions D(x, t) and B(x, t) (x∈R3and t > 0

are respectively the space and variables) to the the electric and magnetic ﬁelds E(x, t) and H(x, t):







∂tD−∇×H= 0,

∂tB+∇ × E= 0.

(1.1)

On the other hand, one deﬁnes the electric polarization and magnetization by







D=ε0E+Ptot,Ptot : electric polarization,

B=µ0E+Mtot,Mtot : magnetization.

(1.2)

where ε0>0 and µ0>0 are the vacuum permittivity and permeability. The above equations are

completed by the following non local constitutive laws (we consider the case of a homogeneous medium)











Ptot(·, t) = ε0Zt

χe(t−s)E(·, s) ds,

Mtot(·, t) = µ0Zt

χm(t−s)H(·, s) ds,

(1.3)

where χeand χmare the electrical and magnetic susceptibilities of the material (convolutions products

being understood in the distributional sence, see for e.g. [7,24], for (χe, χm) not in L1).

In the Fourier-Laplace domain

E(·, t)−→ ˆ

E(·, ω) = Z+∞

E(·, t)eiωt dt, Im ω > 0,

(1.2) and (1.3) reduce to 





D(·, ω) = ε(ω)ˆ

E(·, ω),

B(·, ω) = µ(ω)ˆ

H(·, ω),

(1.4)

where the complex permittivity ε(ω) and the complex permeability µ(ω) are given in terms of the Fourier-

Laplace transform of the susceptibility functions:

ε(ω) = ε01 + ˆχe(ω)and µ(ω) = µ01 + ˆχm(ω).(1.5)

where ε(ω)→ε0and µ(ω)→µ0when ω→ ∞ in C+:= {ω∈C/Im ω > 0. In other words, the

material behaves as the vacuum at high frequencies. In the frequency domain, passivity, causality and

the high frequency behaviour are traduced by the fact that (see [1,5,6,23,24] for more details)

ω7→ ω ε(ω) and ω7→ ω µ(ω) are Herglotz functions, (1.6)

that is to say analytic functions from C+into its closure C+. Furthermore as the susceptibilities χeand

χmare real-valued functions in the time domain, the permittivity and permeability satisfy ε(ω) = ε(−ω)

and µ(ω) = µ(−ω),∀ω∈C+.

Remark 1.1. [About the notion of passivity] The condition (1.6) is the condition which is most often

used to deﬁne passive materials: we called it mathematical passivity in [5]. In the same article, we deﬁne

the related notion physical passivity which is associated to the Cauchy problem associated to (1.1., 1.2,

1.3), seen as an evolution problem with respect to the electromagnetic ﬁeld (E,H). In other words, we

look at the free evolution of the system i.e in the absence of external sources. More precisely a material

is physical passive if and only if the electromagnetic energy

E(t)≡ E(E,H, t) := 1

2ε0ZR3|E(x,t)|2dx+µ0ZR3|H(x,t)|2dx.(1.7)

can never exceeds its value at t= 0, that is to say

∀t≥0,E(t)≤ E(0).(1.8)

It is emphasized in [5] that the above property does not mean that the electromagnetic energy is a

decreasing function of time.

1.1.2 The generalized Lorentz media

In this paper, we shall concentrate of the most well-known subclass of models: the (dissipative) generalized

Lorentz media. Such model will be called local because of the relationship between Dand Eor Band H

can be written with ordinary diﬀerential equations. More precisely, these correspond to

Ptot =ε0

j=1

Ω2

e,j Pj,Mtot =µ0

`=1

Ω2

m,` M`,(1.9)

where each Pj(resp. each M`) is related to E(resp. H) by an ordinary diﬀerential equation







∂2

tPj+αe,j ∂tPj+ω2

e,j Pj=E,1≤j≤Ne,

∂2

tM`+αm,` ∂tM`+ω2

m,` M`=H,1≤`≤Nm,

(1.10)

completed by 0 initial conditions







Pj(·,0) = ∂tPj(·,0) = 0,1≤j≤Ne,

M`(·,0) = ∂tM`(·,0) = 0,1≤`≤Nm.

(1.11)

In the above equations, the (real) coeﬃcients Ωe,j ,Ωm,`, ωe,j , ωm,` are supposed to satisfy

Ωe,j >0, ωe,j ≥0,1≤j≤Ne,Ωm,` >0, ωm,` ≥0,1≤`≤Nm,(1.12)

while for stability/dissipation issues the coeﬃcients (αm,`, αm,`) must be positive

αe,j ≥0,1≤j≤Ne, αm,` ≥0 1 ≤`≤Nm.(1.13)

Moreover, the reader will easily check that one can assume without any loss of generality that the couples

(αe,j , ωe,j ) (resp. (αm,`, ωm,`)) are all distinct the ones from the others.

Note that (1.9,1.10) corresponds to

(a)ε(ω) = ε01−

j=1

Ω2

e,j

ω2+i αe,j ω−ω2

e,j ,(b)µ(ω) = µ01−

`=1

Ω2

m,`

ω2+i αm,` ω−ω2

m,` .(1.14)

Straightforward calculations show that (1.9,1.10) are equivalent to (1.3) with

χe=

j=1

Ω2

e,j χe,j , χm=

`=1

Ω2

m,l χm,` (1.15)

where the expression of each χν,j for ν=e, m and j∈ {1, . . . , Nν}is given by

(i)χν,j (t)=2δ−1

ν,j sinh δν,j t/2e−(αν,j t/2),if αν,j >2ων,j ,

(ii)χν,j (t)=2δ−1

ν,j sin δν,j t/2e−(αν,j t/2),if αν,j <2ων,j ,

(iii)χν,j (t) = t e−(αν,j t/2),if αν,j = 2 ων,j ,

(1.16)

where we have set δν,j =qα2

ν,j −4ω2

ν,j if αν,j ≥2ων,j and δν,j =q4ω2

ν,j −α2

ν,j if αν,j <2ων,j .

Note that each kernel χν,j is not monotonous with respect to time as soon as ων,j >0 or αν,j >0 and

tends to 0 when t→+∞if (and only if) αν,j >0 (see ﬁgure 1.1.2, ﬁrst two pictures). As a consequence

χνdoes not tend to 0 at inﬁnity as soon as one of the αν,j vanishes (see ﬁgure 1.1.2, third picture).

Figure 1: Kernels as functions of time. Left: χν,j for αν,j >2ων,j >0. Center: χν,j for 0 < αν,j <2ων,j .

Right: χνfor Nν= 2, αν,1= 0, αν,2>0.

1.2 A brief review of the literature

As said in introduction, there are already many existing results on the long time behaviour of the solution

of dissipative dispersive systems. In this paragraph, we discuss in some detail some of the most signiﬁcant

contributions that are in close connection with the present work.

In the article [13], the authors considered a very abstract evolution model that in particular includes

(1.3) with χm= 0 and a function χefor which ωˆχe(ω) is a Herglotz function. If one assumes that this

function satisﬁes the additional assumption that

for a. e. ω∈R, γe(ω) := lim

ζ∈C+→ωIm ζˆχe(ζ) exists, (1.17)

(which is satisﬁed by most of dispersive materials in physics and in particular by generalized Lorentz

materials) then the suﬃcient dissipation condition (6.4) of [13] is equivalent to

for a. e. ω∈R, γe(ω)>0, γ−1

e∈L1

loc(R).(1.18)

When applied to the generalized Lorentz model (see appendix A.1), namely when ε(ω) is given by (1.14)

the above condition corresponds to

∃1≤j≤Nesuch that ωe,j = 0 and αe,j >0.(1.19)

Under this condition, it is proven that the electromagnetic energy (see deﬁnition 1.7) tends to 0 for any

initial data (E0,H0) in L2(R3)3×L2(R3)3:

∀(E0,H0)∈L2(R3)3×L2(R3)3,lim

t→+∞E(t)=0.(1.20)

This result is proven in [5] (section 4.4) in a much more pedestrian way on a toy problem corresponding

the Drude model with Ne=Nm= 1, ωe,1=ωm,1= 0 and αe,1, αe,m >0.

In the above references, the question of the rate of convergence to 0 of the electromagnetic energy is not

discussed. This question is addressed in a series of work by S. Nicaise and her collaborator C. Pignotti

[17], [18] (which generalizes [17]) (see also [19] for local dissipation models). These works consider the

initial value problem in a bounded domain Ω ⊂R3with perfectly conducting boundary conditions for

which they prove polynomial stability in the sense mentioned below (see estimate (1.25)). The conditions

for polynomial stability in [18] are two fold:

(i) The ﬁrst condition is expressed in the time domain, more precisely in terms of regularity and decay

properties of the kernels χeand χm:

χ=χeor χmsatisﬁes χ∈C2(R+),lim

t→+∞χ0(t)=0,|χ00(t)| ≤ C e−δt (with C, δ > 0.) (1.21)

Note that (1.21) implies that it exists C1>0 such that |χ0(t)| ≤ C1δ−1e−δt and |χ(t)| ≤ C1for

t≥0 (see e.g. the appendix of [18] for the details). Hence, it follows (by integrations by parts of

the Fourier-Laplace integral) that

ωbχ(ω) = i b

χ0(ω)+iχ(0) and ωbχ(ω) = −c

χ00(ω) + χ0(0)ω−1+ i χ(0),∀ω∈C+(1.22)

where, as the Laplace-Fourier transform of a L1causal function, ω7→ b

χ0(ω) and ω7→ c

χ00(ω) are

analytic on C+, continuous on C+and decay to 0 when ω→ ∞ in C+. Thus, using (1.22), one

observes that (1.21) implies that the functions ω7→ ωˆχe(ω) and ω7→ ωˆχm(ω) are analytic on

C+and can be extended as continuous and bounded functions in the closed upper half-plane C+.

Furthermore, ˆχ= ˆχe,ˆχm, one has ωˆχ(ω) = iχ(0) −χ0(0) ω−1+o(ω−1) when ω→ ∞ in C+.

(ii) The second condition is expressed in the frequency domain for real frequencies. It also has two

parts. The ﬁrst one is a strict positivity condition

∀ω∈R∗, γe(ω) =: Im ωˆχe(ω)>0, γm(ω) := Im ωˆχm(ω)>0.(1.23)

which is completed the additional assumption

∃ω0, q, C > 0 such that |ω| ≥ ω0=⇒γe(ω)≥C|ω|−q, γm(ω)≥C ω−q,(1.24)

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LongtimebehaviourofthesolutionofMaxwell'sequationsindissipativegeneralizedLorentzmaterials(I)AfrequencydependentLyapunovfunctionapproachMaxenceCassiera,PatrickJolybandLuisAlejandroRosasMartnezbaAixMarseilleUniv,CNRS,CentraleMarseille,InstitutFresnel,Marseille,FrancebENSTA/POEMS1,32BoulevardVictor,...

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Long time behaviour of the solution of Maxwells equations in dissipative generalized Lorentz materials I A frequency dependent Lyapunov function approach

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