Asymptotic analysis of abstract two-scale wave propagation problems

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propagation problems

Sébastien Imperiale

To cite this version:

Sébastien Imperiale. Asymptotic analysis of abstract two-scale wave propagation problems. 2021.

�hal-03306856�

PROPAGATION PROBLEMS

S. IMPERIALE ^: Abstract.

This work addresses the mathematical analysis, by means of asymptotic analysis, of linear wave propagation problems involving two scales, represented by a single small parameter, and written in an abstract setting. This abstract setting is defined using linear operators in Hilbert spaces and also enters the framework of semi-group theory. In this setting, we show, under some assumptions on the structure of the wave propagation problems, weak and strong convergence of solutions with respect to the small parameter towards the solution of a well-defined limit problem.

Key word.Wave equation, Asymptotic analysis, Operator theory AMS subject classifications.35L05, 35B40, 47A50

1. Introduction. In the analysis of life science problems or industrial prob- lems by means of partial differential equations (PDE), a class of problem that is frequently encountered is the class of two-scales problems. These problems – after non-dimensionnalisation – involve a non-dimensionnalised small parameter – denoted δin this work. This parameter represents a ratio of two length scales or of two veloc- ity scales for instance. In the most interesting cases, whenδ is small, the behavior of the underlying PDE is not trivially understood and some kind of asymptotic analysis process is required. More precisely, one study the limit of the solution to the PDE whenδgoes to zero to obtain at the limit simpler PDEs in which the small parameter δis absent.

In applied mathematics such problems are studied for linear problems since decades, see for instance [3] in the context of periodic structures and homogenization, [8] for the study of the mechanical behavior of plates (i.e. thin domains) or [18,10] for the study of stokes equations in nearly incompressible fluids. Many generalization of these works has been published since then, see for instances [7,5,1].

Although in semi-group theory the definition of an abstract framework based on linear operator theory [15] is now very standard (see [21,2,9] or [19,4] in the context of control theory) we have find very few work in the literature that construct and analyze a similar abstract framework (see however [12]) that try to unify (part of) the previously mentioned problems.

This is precisely the objective of this work. Of course the mentioned class of problems is too large to be meaningfully represented in a single abstract framework, therefore we will restrict our analysis to wave propagation problems that possesses what is denoted later a two-scale structure. The abstract framework that is presented in this work encompasses many relevant applications in elastodynamics, electromag- netic or piezoelectric problems (see [13,14,6]). In the present paper we only focus on the description of the abstract framework and its rigorous analysis, while we leave its application to the mentioned electro-mechanical problems for a forthcoming work.

The paper is organized as follows.

˚Submitted to the editors July 29, 2021.

:sebastien.imperiale@inria.fr. Inria, Université Paris-Saclay, France. LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay, France.

1

‚ Section2 is devoted to the abstract definition of two-scale wave propagation problems.

‚ We first define and study what is denoted a sequence of two-scale op- erators in Hilbert spaces. Formally these operators have the following form

G_δ “ G₀`δ G₁,

whereG0 and G1 areδ´independent, densely defined and closed oper- ators. In this section we also introduce the notion of weak, strong and strict scale separation properties. These definitions corresponds to as- sumptions on G0 and G1 that enable an analysis of the limit process whenδgoes to zero.

‚ We then define the family of wave propagation problems under study.

In particular, a simplified formal definition of these problems is u:_δ`G^˚_δ G_δu_δ “0.

We provide existence, uniqueness and stability results for the solutions of these problems.

‚ In Section3we define the limit problem, i.e. the problem solved at the limit by the sequence of solutions of the family of the wave propagation problems.

We give in this section existence, uniqueness and stability results as well as other properties. In particular, we define the (self-explanatory) notion of propagating and non-propagating solutions.

‚ Section4is devoted to the convergence analysis of the solutions of the family of problems towards the limit problems.

‚ We first study the case where only the weak scale separation property holds. In that case we show a weak convergence property.

‚ We show strong convergence property in the cases where the strict or the strong scale separation property holds.

To easy the reading all the proofs of Section2and3are given Section5. Finally two appendices are devoted to more classical – yet not find in the literature – proofs. The first one is the proof of the existence, uniqueness and stability results for the family of wave propagation problems, and second one is a density proof that is required at some point in the analysis in order to show the convergence of solutions of the family of wave propagation problems to a limit problem.

2. Two-scale wave propagation problems.

2.1. Preliminary definition: Two-scale operators. We assume given Hilbert spaces

`H;p¨,¨qH

˘ and `

G;p¨,¨qG

˘

as well as a sequence of unbounded operatorsG_δ :DpG_δq ĂHÞÑG forδP p0,1q.

Definition 2.1. The sequence of operatorsGδ :DpGδq ĂHÞÑG is said to be a sequence of two-scale operators if

(2.1) Gδ “ G0`δ G1,

where the operatorsG0andG1areδ´independent, densely defined and closed operator fromHtoG.

In Definition 2.1 it is implicitly assumed thatG0`δG1 is closable hence Gδ is the closure of this operator. Without additional assumptions one can not deduce inter- esting properties from sequence of two-scale operators. In what follows we introduce three different properties, denoted respectively weak, strong and strict scale separa- tion properties. These properties describe in more details the relation between the operatorsG0 andG1. We begin with the notion of weak scale separation.

Definition 2.2. A sequence of two-scale operators Gδ satisfies the weak scale separation property if

(2.2) DpG0q XDpG1qdense in `

DpG0q;} ¨ }H` }G0 ¨ }G

˘

and (2.3)

# DpG^˚₀q XDpG^˚₁qdense in `

DpG^˚₀q;} ¨ }G` }G^˚₀ ¨ }H˘

, paq KerG^˚₀ XDpG^˚₁qdense in `

KerG^˚₀;} ¨ }_G˘

. pbq

Note that the definition above straightforwardly implies thatDpG₀q XDpG₁qis dense in DpG₀qfor the norm } ¨ }H henceDpG₀q XDpG₁qis dense inH, meaning thatG_δ is densely defined sinceDpG0q XDpG1qis a subspace ofDpGδq. We now explain the important consequences of the weak scale separation property.

Theorem 2.3. For any sequence of two-scale operators Gδ such that property (2.3a) holds, lettuδube a sequence inDpGδqsatisfying, for some scalarCindependent of δ,

(2.4) }uδ}H`δ^´1}Gδuδ}G ¬C,

then, up to a subsequence, there existsu₀PKerG₀ such that u_δ á

H u₀.

This property gives the intuitive result that, for a sequenceu_δ converging towardsu₀, if G₀`δ G₁u_δ goes to zero then in some sense G₀u_δ also vanishes and we expect at the limit thatG0u0“0.

To continue analyzing the benefit granted by the weak scale separation property we introduce the orthogonal projections operators in the kernel of G^˚₀ denoted Q0 and defined by,

Q₀PLpGq, Q²₀“Q₀, pg´Q₀g, hq_G “0 @ pg, hq PGˆKerG^˚₀.

Note that this definition make sense sinceG^˚₀ is densely defined and closed hence its kernel is a closed subspace ofG.

Lemma 2.4. For any sequence of two-scale operators G_δ satisfying (2.3) the op- eratorG^˚₁Q₀ is densely defined inG andQ₀G₁ is closable, densely defined in H.

The fact that Q0G1 is closable is essential in the analysis of problems involving the corresponding two-scale operator Gδ. The reason being that one can – thanks to the theorem below – construct aδ´independent space that embedsDpGδqand that is, in general, a proper subspaceH.

Theorem 2.5. For any sequence of two-scale operators Gδ satisfying (2.2) and (2.3)we have

(2.5) Q0Gδ Ă δ Q0G1

moreover

@uPDpGδq, }Q0G1u}_G ¬δ^´1}Gδu}_G

Note that the inclusion (2.5) implies first that DpGδq ĂDpQ0G1q, this latter space being a Banach space when equipped with its graph norm, second, that Q0Gδ and δ Q0G1 coincides onDpGδq.

Theorem 2.6. For any sequence of two-scale operators G_δ satisfying (2.2) and for any sequencetu_δuinDpG_δqsatisfying, for some scalarC independent of δ, (2.6) }u_δ}H`δ^´1}G_δu_δ}G¬C

and

(2.7) pδ^´1G_δu_δ, G₀vq_GÝÑ

δÑ00, @vPDpG₀q XDpG₁q, thenδ^´1pI´Q₀qG_δu_δ á

G 0.

To understand why the result above is important notice that, for anyuδ PDpGδqwe have, thanks to Theorem2.5,

(2.8) δ^´1Gδuδ“δ^´1`

Q0` pI´Q0q˘

Gδuδ “Q0G1uδ`δ^´1pI´Q0qGδuδ, hence, thanks to Theorem2.6the last term of (2.8) can be dealt with while the second one involves only aδ´independent operator.

We introduce now the property of strong scale separation.

Definition 2.7. A sequence of two-scale operators Gδ satisfies the strong scale separation property if

(2.9) DpG₀q XDpG₁qdense in `

DpG₀q;} ¨ }H` }G₀ ¨ }G

˘

and

(2.10) @ pu, vq PDpG₀q ˆDpG₁q pG₀u, G₁vq_G“0.

Many simplifications occur if the strong scale separation property holds as shown in the theorem below.

Theorem 2.8. Any sequence of two-scale operators Gδ that satisfies the strong scale separation property satisfies the weak two-scale separation property,

Gδ“G0`δG1 and Q0G1“G1.

Finally, we introduce the notion of strict scale separation property.

Definition 2.9. A sequence of two-scale operators Gδ satisfies the strict scale separation property if it satisfies

(2.11) @ pg, hq PDpG^˚₀q ˆDpG^˚₁q, pG^˚₀g, G^˚₁hq_H“0, and

(2.12) @ pu, vq PDpG₀q ˆDpG₁q pG₀u, G₁vq_G“0.

Theorem 2.10. Any sequence of two-scale operators Gδ that satisfies the strict scale separation property satisfies the strong scale separation property and

KerG0XDpG1qdense in`

KerG0;} ¨ }H

˘.

Remark 2.11. If a sequence of two-scale operators Gδ satisfies the weak scale separation property and if

@ pg, hq PDpG^˚₀q ˆDpG^˚₁q, pG^˚₀g, G^˚₁hqH“0,

then the operator Hδ “ G^˚₀`δG^˚₁ : DpHδq Ă G ÞÑ H is a sequence of two-scale operator satisfying the strong scale separation property (with the role of H and G reverse).

Remark 2.12. When the strict scale separation property holds one has DpG0q XDpG1q “`

KerG0XDpG1q˘

‘`

KerG1XDpG0q˘ where the orthogonality holds for the scalar productp¨,¨qH and

p¨,¨qH` pG0¨, G0¨qG` pG1¨, G1¨qG.

Such property shows that, when the strict scale separation, functions belonging to DpG0q XDpG1q (typically solutions of a given set of equations) can be naturally decomposed into two orthogonal elements that could potentially be computed inde- pendently.

2.2. A family of wave propagation problems.

Functional spaces. We assume given a vector spaceVδ (that depends onδ for the sake of generality) such that

(2.13) Vδ ĂH and Vδ is dense inH,

where the density holds with the norm } ¨ }H. We assume that V_δ is a Banach space when equipped with a norm denoted} ¨ }Vδ.Finally we introduce another space used later to define a bilinear form responsible for lower order effects (dissipation, couplings, ...) in the wave propagation problem. This space is independent ofδand denotedB, it is a dense subspace ofHand

VδĂBĂH.

The space B is a Hilbert space equipped with the scalar product p¨,¨qB and corre- sponding norm } ¨ }B. We assume that there exists C_I – independent of δ – such that

(2.14) @vPV_δ, }v}_B¬C_I}v}_Vδ and @vPB, }v}_H¬C_I}v}_B.

The family of problems. We introduce two bilinear forms denotedaδ andb0, a_δ :V_δˆV_δ ÞÑR, b₀:BˆBÞÑR.

pa_δq The bilinear forma_δis symmetric, nonnegative, continuous inVδand coercive.

More precisely, we assume that, for alluandv inVδ, (2.15) a_δpu, vq “a_δpv, uq, a_δpu, uq­0, moreover we assume that

(2.16) }u}²_Vδ “ pu, uq_Vδ with pu, vq_Vδ :“ pu, vq_H`aδpu, vq and therefore, the Cauchy-Schwartz inequality implies that

|a_δpu, vq|¬}u}_Vδ}v}_Vδ.

The spacepVδ;p¨,¨qV_δqis a Hilbert space, and, by inspection, one can see that the injection inHis continuous, indeed, by definition,

(2.17) }u}H¬}u}V_δ, @uPVδ.

pb0q The bilinear formb0 is nonnegative and continuous inB. More precisely for alluPVδ andvPVδ,

(2.18) 0¬b0pu, uq, |b0pu, vq|¬}u}_B}v}_B. For all scalarδP p0,1q, the abstract family of problems reads: find

u_δ PC¹pr0, Ts;Hq XC⁰pr0, Ts;Vδq solution to, for allvPVδ,

(2.19)

$

’’

’’

&

’’

’’

% d²

dt²puδ, vqH`aδpuδ, vq ` d

dtb0puδ, vq “ pfδ, vqH, D¹p0, Tq, u_δp0q “u⁰_δ,

u9δp0q “u¹_δ,

wherefδis a sufficiently smooth source term andpu⁰_δ, u¹_δqare sufficiently smooth initial data. Note that we consider for now generalδ-dependent initial data and source terms.

These quantities must be well-prepared in a sense given later.

Existence and uniqueness result. We introduce in this section the notion of weak solutions. These solutions are defined using minimal (to our knowledge) regu- larity assumptions on the data. More precisely, we assume that

(2.20) pu⁰_δ, u¹_δq PVδˆH and fδ PL¹`

0, T;H˘ .

Definition 2.13. A weak solution is a function u_δPL²`

0, T;Vδ

˘, that satisfies, for allvPH²`

0, T;H˘ XH¹`

0, T;Vδ

˘such thatvpTq “0andvpT9 q “ 0,

(2.21) żT

0

puδptq,:vptqqHdt` żT

0

aδpuδptq, vptqqdt´ żT

0

b0puδptq,vptqq9 dt

“ pu¹_δ, vp0qq_H`b0pu⁰_δ, vp0qq ´ pu⁰_δ,vp0qq9 _H

` żT

0

pf_δptq, vptqq_Hdt.

We have the following existence, uniqueness and stability results.

Theorem 2.14. There exists a unique weak solution uδ. It satisfies, up tp modi- fications on zero-measure sets,

uδ PC¹pr0, Ts;Hq XC⁰pr0, Ts;Vδq and, for all tP r0, Ts,

(2.22) a

Eδptq¬a

Eδp0q ` 1

?2 żt

0

}fδpτq}Hdτ,

where

(2.23) E_δptq “ 1

2

´

}u9_δptq}²_H`a<sub>δ</sub>`

u_δptq, u_δptq˘¯ . Moreover uδ is the unique solution to (2.19).

The stability result (2.22) give some first useful informations on the analysis of the family of problems (2.19). Indeed if

Eδp0q and żT

0

}fδptq}_Hdt

are bounded uniformly with respect to δ then so is the energy functional Eδptq. Al- though the energy functional corresponds only to aδ´dependent semi-norm for the so- lution it is possible to deduce that the norm inL²p0, T;Hqofuδ is uniformly bounded (ifu⁰_δ is uniformly bounded inH) henceuδ converges weakly, up to subsequences, to

some u0 in the space L²p0, T;Hq. Remark however that, first, even when the initial datau^δ₀ and u^δ₁ are independent on δ, the initial energy Eδp0qmay still depend on δ (see (2.23)), second, it is yet not clear at this stage, which equations, if any, are satis- fied by the weak limit. To clarify this we need additional assumption on the structure of the bilinear formaδ.

The two-scale structure. Our analysis of the sequence of solutions tu_δurely on the fundamental assumption that the bilinear forma_δ can be rewritten in such a way that the dependence inδis explicit and has the following specific structure.

Assumption 2.15. There exists a sequence of two-scale operators Gδ :DpGδq Ă HÑG satisfying, the weak scale separation property,DpGδq “Vδ and

(2.24) @ pu, vq PVδˆVδ, aδpu, vq “δ^´2`

Gδu, Gδv˘

G. 3. The limit problem.

Some notations: the spacesV₈ and V₀. For the sake of conciseness we intro- duce here several notations use extensively in what follows. We set

V₈:“DpG0q XDpG1q,

and define the following scalar product and norm: for alluandv inV₈, pu, vq_V8:“ pu, vq_H` pG0u, G0vq<sub>G</sub>` pG1u, G1vq_G, }u}²_V

8 :“ pu, uq_V8. We also introduce

V₀:“DpQ₀G₁q,

and, define the following scalar product and norm: for alluandv inV₀, (3.1) pu, vq_V0 :“ pu, vq_H` pQ₀G₁u, Q₀G₁vq_G, }u}²_V

0 :“ pu, uq_V0.

Proposition 3.1. The following inclusionsV₈ĂVδ ĂV0 hold. Moreover, (3.2) @uPV₈ }u}_Vδ ¬?

2δ^´1}u}_V8 and @uPVδ, }u}_V0 ¬}u}_Vδ.

Finally we introduce the following subspaces ofV₈ andV0, V₈⁰ :“V₈XKerG0 and V₀⁰:“V0XKerG0.

In order to study the asymptotic limit of the sequencetuδuwe use the following ad- ditional assumption.

Assumption 3.2. The following density properties hold (3.3) V₈⁰ is dense in pV₀⁰;} ¨ }_V0q.

Let us now comment Assumption3.2. One of the difficulty that is encountered when studying the variational formulation (2.19) is that, formally, to be able to pass to the limit in (2.19) one should use a test function that belongs to the δ-independent V₈⁰ hence obtaining a weak formulation in this space while the limit solution belongs (assuming the timetfixed) to the larger spaceV₀⁰.The density result (3.3) solves this lack of symmetry.

When dealing with dissipation or coupling terms, i.e., when the bilinear formb0is not trivial, one has to relate the spaceV₀⁰andB. Hopefully we have the following theorem.

Theorem 3.3. When Assumption3.2holds, thenV₀⁰ĂB and

@vPV₀⁰, }v}_B¬C_I}v}_V0.

Finally we introduce a closed subspace ofHdefined by H0“V₀^0}¨}H.

The spaceH0if the closure ofV₀⁰ for the norm ofH, it satisfiesH0ĂKerG₀ and is a Hilbert space when equipped with the scalar product and norm ofH.

We have now all the necessary material to define the limit problem.

Formal limit equation. We introduce the following symmetric, non-negative bilinear form

(3.4) @ pu, vq P V0ˆV0, a0pu, vq “ pQ0G1u, Q0G1vq_G. Now remark that with the definition (3.1) we have, for alluandv inV0, (3.5) }u}²_V0 “ }u}²_H`a0pu, uq and |a0pu, vq|¬}u}_V0}v}_V0,

hencea0 is continuous inV0and coercive. Therefore thanks to (3.5) it make sense to define the followingδ´independent wave propagation problem: find

u₀PC¹pr0, Ts;H0q XC⁰pr0, Ts;V₀⁰q solution to, for allvPV₀⁰,

(3.6)

$

’’

’’

&

’’

’’

% d²

dt²pu0, vq_H`a0pu0, vq ` d

dtb0pu0, vq “ pf0, vq_H, D¹p0, Tq, u0p0q “u⁰,

u90p0q “P0u¹,

where f₀ is a sufficiently smooth source term andpu⁰, u¹q sufficiently smooth initial data and whereP₀ is the orthogonal projection inH₀. It is defined by

(3.7) P0PLpHq, P₀²“P0, pu´P0u, vqH“0 @ pu, vq PHˆH0.

Note that the solution is sought in the subspace V₀⁰ of functions in V0. As shown below in Section4.1, under adequate assumptions, problem (3.6) turns out to be the limit problem satisfied by the weak limit of solutions to (2.19).

Existence and uniqueness results. We define weak solutions for the limit problem. We assume that the source termf0and initial data pu⁰, u¹qsatisfy

(3.8) pu⁰, u¹q PV₀⁰ˆH and f₀PL¹p0, T;Hq.

Existence, uniqueness and stability results for the limit problem are obtained as a special case of the theory presented Section2.2.

Definition 3.4. A weak solution is a function u0PL²`

0, T;V₀⁰˘ , that satisfies, for allvPH²`

0, T;H0

˘XH¹`

0, T;V₀⁰˘

such thatvpTq “0andvpT9 q “ 0,

(3.9) żT

0

pu0ptq,vptqq: _Hdt` żT

0

a0pu0ptq, vptqqdt´ żT

0

b0pu0ptq,vptqq9 dt

“ pu⁰,vp0qq9 _H´ pu¹, vp0qq_H´b₀pu⁰, vp0qq_H` żT

0

pf₀ptq, vptqq_Hdt.

As a straightforward consequence of Theorem2.14one can show the following result.

Proposition 3.5. There exists a unique limit weak solution u0. It satisfies, up to modifications on zero-measure sets,

u0PC¹pr0, Ts;H0q XC⁰pr0, Ts;V₀⁰q and, for all tP r0, Ts,

aE0ptq¬a

E0p0q ` 1

?2 żt

0

}f0pτq}_Hdτ where

E0ptq “1 2

´

}u9₀ptq}²_H` }Q₀G₁u₀ptq}²_G¯ . Moreover u₀ is the unique solution to (3.6).

The spaceR₀ and S₀. For the analysis of the limit problem we shall introduce the spaceR0 andS0 related to the kernel ofQ0G1. We define

(3.10) R0:“V₀⁰ X KerQ₀G₁“KerG₀XKerQ₀G₁.

Since by definitionQ0G1 is closed and densely defined inHits kernel is closed inH.

It is also closed in V₀⁰ since, for allrPKerQ0G1 we have }r}V₀ “ }r}H. We deduce then that the spaceR0 is a closed subspace ofV₀⁰. We can therefore defineS0as the orthogonal complement ofR0 inV₀⁰. We have

(3.11) V₀⁰“R0‘S0,

where the decomposition is orthogonal with respect to the scalar product inV0, more- over, by definition we have

@ pr, sq PR0ˆS0, 0“ pr, sq_V0“ pr, sq_H` pQ0G1r, Q0G1sq_G“ pr, sq_H, henceR0 and S0 are also orthogonal with respect to the scalar product in H. This means that we also have

H0“R0‘R^K₀ and S0ĂR^K₀

where the orthogonality relations stands with the scalar product inH.

The spaceR0 is called the space of non-propagating solutions whereas the space S0 is called the space of propagating solutions. This terminology is justified in the case whereb₀vanish or is proportional to the scalar product inH. Indeed in that case it is shown below that the projection of the limit solutionu₀in R₀satisfies a simple differential problem while its projection inS₀ satisfies an abstract wave propagation problem.

Reduced equation on the space of propagating solutions. To simplify this section we assume that initial data vanishes and that the source term belongs at almost all time toS0

(3.12) u⁰_δ “0, u¹_δ “0 and f0PL¹p0, T;S0q.

Thanks to the orthogonal decomposition (3.11) we deduce the following property.

Proposition 3.6. Any limit weak solutionu0 satisfies the decomposition u0“r0`s0

with

r₀PC¹pr0, Ts;R0q and s₀PC¹pr0, Ts;R^K₀q XC⁰pr0, Ts;S0q.

Our objective is now to obtain an equation only fors0 by eliminating the term r0 in (3.9). To do so, we choose as a test functionrPH²`

0, T;R0

˘such thatrpTq “0and rpT9 q “0in (3.9). We obtain, after integrating by parts the term involvingr,:

(3.13)

żT 0

pr90ptq,rptqq9 _Hdt` żT

0

b0pr0ptq,rptqq9 dt “ ´ żT

0

b0ps0ptq,rptqq9 dt where we have used the property thata₀pu₀ptq, rptqq “0, thatps₀ptq,:rptqq_H“0and f₀ PL¹p0, T;S₀q. Equation (3.13) can be written in a strong form in time, i.e. it is equivalent to, for allrPR0,

(3.14)

# pr90ptq, rq_H`b0pr0ptq, rq “ ´b0ps0ptq, rq, C⁰pr0, Tsq, r0p0q “0,

where r0p0q “ 0 is a consequence of (3.12) and Proposition 3.6. We recall now the properties of the bilinear formb0, we have, for all pu, vq PV₀⁰ˆV₀⁰

|b0pu, vq|¬}u}_B}u}_B ¬C_I²}u}_V0}u}_V0 and 0¬b0pu, uq,

Hence, from the bilinear formb0 we define bounded linear operators BR:R0ÞÑR0 and B_S^R:S0ÞÑR0

whereR0 andS0 are Hilbert spaces equipped with the scalar product inV0and the corresponding norm} ¨ }V0 (we recall that}r}_V0“ }r}_H forrPR₀) by

# @ pr,˜rq PR0ˆR0, pB_Rr,rq˜_V0 “ pB_Rr,˜rq_H“b0pr,rq,˜

@ ps, rq PS0ˆR0, pB^R_S s, rqV₀ “ pB^R_S s, rqH “b0ps, rq, whereB_R is a non-negative operator. Then equations (3.14) can be recast as (3.15)

# r90`B_Rr0“ ´B_S^Rs0, r0p0q “0,

where the first equation is written in C⁰pr0, Ts;R0q. From (3.15) and from [2] we deduce thatr₀ is given by,

(3.16) r0ptq “ ´ żt

0

e^BRpt´τqB_S^Rs0pτqdτ PC¹pr0, Ts;R0q,

wheree^´BRtis the uniformly continuous semi-group (see [2]) generated by the bounded operator´B_R.

Remark 3.7. Note that if a more general source term f0 PL¹p0, T;Hqis consid- ered then one can show that the first equation of (3.15) becomes

r90pτq `BRr0pτq “ ´B<sup>R</sup><sub>S</sub> s0pτq ` żt

0

QRfpτqdτ,

whereQ_Ris the orthogonal projection operatorR0with respect to the scalar product inH. Then observe that if B_S^R “0 (for instance either becauseb0 vanishes or is the scalar product in H) thenr0 satisfies a differential equation that is not an abstract wave propagation problem (we recall thatBR is a non-negative operator).

From the expression (3.16) we deduce that r90ptq “ΨtB_S^Rs0

withΨt:C⁰pr0, Ts;R0q ÑC⁰pr0, Ts;R0qgiven by

@rPC⁰pr0, Ts;R0q, Ψtr“BR

żt 0

e^BRpt´τqrpτqdτ´rptq.

With this expression and with the weak formulation (3.6) one can writes a formulation involving onlys0. Observing that, for allsPS0 we have

b0pr90ptq, sq “b0pΨtB_S^Rs0, sq and we can show that, for allsPS0,

(3.17)

$

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dt²ps0, sq_H`a0ps0, sq ` d

dtb0ps0, sq`b0pΨtB^R_Ss0, sq “ pf0, sq_H, D¹p0, Tq, s0p0q “0,

s90p0q “0.

This system corresponds to a wave equation with a non-local term in time that cor- responds to dissipative and other coupling effects. The non-local operator in time is in fact falsely non-local since we have seen that it is derived, rather simply, from a local in time evolution problem. Note that in application (see for instance [14]) this may not be as straightforwardly apparent. Note also that we have shown exis- tence of a solution for this problem by construction. Uniqueness of smooth solutions is obtained – by energy estimates – as a consequence of the following positivity property.

Theorem 3.8. For all s P H¹pr0, Ts;S0q such that sp0q “ 0, we have, for all tP r0, Ts,

żt 0

b0pspτq,9 spτ9 qqdτ` żt

0

b0pΨτB_S^Rs,spτqq9 _Hdτ­0.

Proof. The proof is done by rewinding some of the arguments given above to obtain (3.17). Assuming first thatsPC¹pr0, Ts;S0q, we have

(3.18) I “ żt

0

b0pspτq,9 spτ9 qqdτ´ żt

0

b0pΨτB_S^Rs,spτqq9 _H

“ żt

0

b0pspτq,9 spτqq9 dτ` żt

0

b0prpτq,spτqq9 Hdτ whererPC²pr0, Ts;R0qis defined by

(3.19) r9`B_Rr“ ´B_S^Rs and rp0q “0,

which implies rptq “9 ΨtB_S^Rs PC⁰pr0, Ts;R0q. From Eq. (3.19) we deduce that :r` B_Rr9“ ´B_S^Rs9 as well as the energy identity

(3.20) }rptq}9 ²_H` żt

0

b₀prpτq,9 rpτqq9 dτ“ ´ żt

0

b₀pspτq,9 rpτ9 qqdτ.

Using (3.20) in (3.18) we obtain I“

żt 0

b0prpτq `9 spτq,9 rpτ9 q `spτqq9 dτ` }rptq}9 ²_H­0.

The final statement of the theorem is obtained by density of C¹pr0, Ts;S0q in the spaceH¹pr0, Ts;S0q(c.f. [17] Chap. 1).

The strong and strict scale separation assumptions. To conclude this sec- tion, we summarize in the proposition below all the straightforward simplifications – consequences of Theorem 2.8 and Theorem 2.10– obtained when the sequence of two-scale operatorsG_δ satisfies the strong or the strict scale separation property.

Proposition 3.9. If the sequence of two-scale operators G_δ satisfies the strong scale separation property then

$

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V₈ “V_δ“DpG₀q X DpG₁q, V0“DpG1q,

V₈⁰ “V₀⁰“DpG1q X KerG0, R0“KerG1 X KerG0

and

(3.21) @ pu, vq PV0ˆV0, a0pu, vq “ pG1u, G1vqG. Moreover if the strict two-scale separation holds

(3.22)

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V₈“V₈⁰ ‘`

DpG0q X KerG1

˘, H0“KerG0,

R0“ t0u, S0“V₈⁰,

Note that the decomposition in (3.22)holds is orthogonal for the scalar product in H and inV₈.

The proposition above shows that Assumption3.2is automatically satisfied when the strong scale separation holds and, moreover, when the strict scale separation holds we have a simple characterization ofH0, namelyH0“KerG0.

4. Convergence analysis. In this Section we study the asymptotic limit, when δ goes to zero, of the family of theδ´dependent wave propagation problems (2.19) towards the solution of the limit problem (3.6). In Section 4.1 we investigate the condition on which the convergence is weak whereas in Section4.3 we study strong convergence properties.

4.1. Weak convergence. The objective of this section is to give the minimal assumption that guaranty that the sequencetuδuof weak solutions of problem (2.19) converges weakly towards the weak solution of the limit problem (3.6). In the sequel we assume that the weak scale separation holds and we shall also assume that the data are well-prepared as define below.

Assumption 4.1. Well-prepared data.The initial data and the source terms of problem (2.19)satisfy

pu⁰_δ, u¹_δq PVδˆH and fδ PL¹`

0, T;H˘ and we assume that there existspu⁰, u¹q PV0ˆHandf0PL¹`

0, T;H˘

such that

(4.1) u⁰_δ á

V0

u⁰, u¹_δ á

H u¹ and fδ á

L¹p0,T;Hqf0. Moreover we assume that there exists C0ą0 independent ofδ such that

(4.2) }G_δu⁰_δ}G ¬δ C₀.

It has to be noted that the only strong assumption here is onu⁰_δ. Indeed we impose that G0u⁰_δ vanishes sufficiently fast with δ going to zero. This can be interpreted as the assumption that the term u⁰_δ must be adapted to the structure of the solution u0ptq of the limit problem (3.6) that belongs to V₀⁰ “V0XKerG0. The first conse- quences of Assumption4.1are given below.

Lemma 4.2. If the data are well-prepared then u⁰PV₀⁰ and there exists a scalar C_uą0 independent of δsuch that the weak solutionsu_δ of problem (2.19)satisfies,

(4.3) sup

tPr0,Ts

`}u9δptq}<sub>H</sub>` }uδptq}_Vδ˘

¬Cu.

Proof. Since the weak convergence (4.1) hold we have that }u⁰_δ}V0, }u¹_δ}H and

żT 0

}fδptq}_Hdt,

are bounded sequences. Therefore, the first statement – u⁰ PV₀⁰ – is a consequence of the estimate (4.2) and Theorem2.3. Moreover, the second statement is a straight- forward consequence of the energy estimates given by (2.22) as well as the property that the energy at the initial timeE_δp0qsatisfies

E_δp0q “ }u¹_δ}²_H`a<sub>δ</sub>pu<sup>0</sup><sub>δ</sub>, u<sup>0</sup><sub>δ</sub>q¬}u<sup>1</sup><sub>δ</sub>}<sup>2</sup><sub>H</sub>`2δ^´2}G₀u⁰_δ}²_G`2}G₁u⁰_δ}²_G and is bounded uniformly.

Thanks to Lemma4.2 and in particular to estimate (4.3) we have the existence of a weak limit of the sequencetuδuinL²p0, T;Hq. This weak limit is still not character- ized as the solution of the limit problem (3.6). This is the object of the theorem below that is the main result of this section.

Theorem 4.3. Let the weak scale separation and Assumption 3.2 hold and as- sume the data of problem (2.19) are well-prepared. Then the weak solutions uδ of problem (2.19) satisfy,

uδ á

H¹p0,T;Hqu0 and uδ á

L²p0,T;V0q

u0,

whereu0is the unique weak solution of the limit problem(3.6)with initial datapu⁰, u¹q and source termf0.

Proof.

Step 1: Weak limit ofuδ.Thanks to the assumption that the data are well prepared there exists for each δa unique weak solution u_δ of problem (2.19) and the uniform estimate of the solutions (4.3) holds. In particular, one can show, using (3.2), that

sup

tPr0,Ts

}u9_δptq}²_H, sup

tPr0,Ts

}u_δptq}²_V

0 and sup

tPr0,Ts

δ^´2}G_δu_δptq}²_G

are bounded uniformly with respect to δ. From this estimate we deduce two results.

First, there exitsu0 PH¹p0, T;Hq XL²p0, T;V0qsuch that, up to a subsequence,uδ

weakly converges tou0. Second, using a similar reasoning used to prove Theorem2.3 we can prove thatu0PL²p0, T;KerG0qhenceu0PL²p0, T;V₀⁰q

Step 2: Weak limit of δ^´1G_δu_δ. Since δ^´1G_δu_δ is bounded uniformly in δ and tP r0, Ts, multiplying (2.21) byδone can see that

żT 0

pδ^´1G_δu_δptq, G₀vptqqdtÝÑ

δÑ00.

Applying Theorem2.6 (in fact one should use a time version of this theorem that is left to the reader) we obtain, sinceQ0G1uδ weakly converges toQ0G1u0, that

δ^´1Gδuδ á

L²p0,T;GqQ0G1u0.

Step 3: Limit equation. We now choose

(4.4) vPDpr0, Tq;V₈⁰q

in the variational formulation (2.21) and pass to the limit. We obtain thatu₀satisfies the limit weak variational formulation (3.9) for all v satisfying (4.4). Note that in particular we use the implication

uδ á

L²p0,T;V0q

u0 and żT

0

}uδptq}_Vδdt¬T Cu

ñ żT

0

b₀pu_δptq,vptqq9 dt ÝÑ

δÑ0

żT 0

b₀pu₀ptq,vptqq9 dt, that is a consequence of the property that Vδ is (uniformly) continuously embedded in B (see Eq. (2.14)). Finally using Assumption 3.2 one can show that the space Dpr0, Tq;V₈⁰qis dense in the space

W0:“ tvPH²`

r0, Ts;H0

˘XH¹`

r0, Ts;V₀⁰˘

| vpTq “9 0, vpTq “0u.

Thanks to this density result we can then show thatu0satisfies the limit weak varia- tional formulation (3.9) for all test function inW0 (the proof of this density result is given in Appendix7for the sake of completeness).

Conclusion.We have shown that sequenceu_δconverges weakly, up to a subsequence to a functionu₀ PL²p0, T;V₀⁰qthat satisfies the weak variational formulation (3.9).

The function u₀ is therefore the unique weak solution of the limit problem corre- sponding to datapu⁰, u¹qandf₀. This last statement implies that all the converging subsequences converge to the same elementu₀.

Generalization. The weak convergence that we have shown can be easily extended to more general situations that correspond to low order perturbation in δ of the equations. In particular the same convergence result (towards the solution of the same limit problem) can be proven in the case where, instead of Assumption2.15one has,

@ pu, vq PVδˆVδ, aδpu, vq “δ^´2`

pI`γapδqSδqGδu, Gδv˘

G,

where Sδ is a bounded (uniformly with respect to δ) self-adjoint operator in G and γapδq a continuous scalar function vanishing when δ goes to zero. Of course, in the same way, one could define, aδ´dependent perturbation in of b0puδ, vq, namely

@ pu, vq PVδˆVδ, bδpu, vq “b0pu, vq `γbpδqb1,δpu, vq

whereγbpδqis another a continuous scalar function vanishing whenδgoes to zero and b1,δ a uniformly (with respect toδ) continuous bilinear form in Bsuch that

@ pu, vq PVδˆVδ, 0¬bδpu, uq, |bδpu, vq|¬}u}_B}v}_B. We could have then considered instead of (2.19) the following equation

(4.5)

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dt²ppI`γmpδqMδquδ, vq<sub>H</sub>`aδpuδ, vq ` d

dtbδpuδ, vq “ pfδ, vq_H, D¹p0, Tq, uδp0q “u⁰_δ,

u9δp0q “u¹_δ,