Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions

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Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric

conditions

Rayan Nasser, Nahla Noun, Ali Wehbe

To cite this version:

Rayan Nasser, Nahla Noun, Ali Wehbe. Stabilization of the wave equations with localized Kelvin–

Voigt type damping under optimal geometric conditions. Comptes Rendus. Mathématique, Académie

des sciences (Paris), 2019, 357 (3), pp.272-277. �10.1016/j.crma.2019.01.005�. �hal-02098422�

HAL Id: hal-02098422

https://hal-confremo.archives-ouvertes.fr/hal-02098422

Submitted on 24 Jun 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric

conditions

C Acad, • Paris, Rayan Nasser, Nahla Noun, Ali Wehbe

To cite this version:

C Acad, • Paris, Rayan Nasser, Nahla Noun, Ali Wehbe. Stabilization of the wave equations with

localized Kelvin–Voigt type damping under optimal geometric conditions. Comptes Rendus Mathé-

matique, Elsevier Masson, 2019, 357 (3), pp.272-277. �10.1016/j.crma.2019.01.005�. �hal-02098422�

Partial differential equations/Optimal control

Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions

Stabilisation d’une équation des ondes avec un amortissement de type Kelvin–Voigt localisé sous conditions géométriques optimales

Rayan Nasser^a,c, Nahla Noun^b, Ali Wehbe^b

aUniversitélibanaise,EDST&Hadath,Beyrouth,Liban

bUniversitélibanaise,Facultédessciences1etEDST&Hadath,Beyrouth,Liban cUniversitédeBretagneoccidentale,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received6October2018

Acceptedafterrevision21January2019 Availableonlinexxxx

PresentedbytheEditorialBoard

The purpose of thisnote is to investigate the stabilization of the wave equation with Kelvin–Voigt damping in a bounded domain. Damping is localized via a non-smooth coefficientinasuitablesubdomain.We proveapolynomialstabilityresult inanyspace dimension, provided that the damping region satisfies some geometricconditions. The main noveltyof thisnoteis that thegeometric situationscovered hereare richer than thatconsideredin[25],[22],[16] andincludeinparticularanexamplewherethedamping regionisnotlocalizedinaneighborhoodofthewholeorapartoftheboundary.

r é s um é

Nous nous intéressons à l’étude de la stabilisation d’une équation des ondes avec un amortissementdetypeKelvin–Voigtdansundomaineborné.L’amortissementestlocalisé viaun coefficientsingulier dans unepartie dudomaine. Nousmontrons un résultatde stabilisationpolynomialeentoutedimension d’espacedèsquelarégiond’amortissement satisfaitcertainesconditionsgéométriques.Laprincipalenouveautédecettenoteestque lessituationsgéométriquescouvertesicisontplusrichesquecellesconsidéréesdans[25], [22], [16] et incluent notamment un exemple où la région d’amortissement n’est pas localiséedansunvoisinagedelatotalitéoud’unepartiedelafrontière.

E-mailaddresses: [email protected] (R. Nasser),[email protected] (N. Noun),[email protected] (A. Wehbe).

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1. Introduction

Localviscoelasticdampingisanaturalphenomenonof bodiesarisingfromasolidthathaveonepartmadeofviscoelastic material,andtheothermadeofelasticmaterial.Let⊂R^{N beanonemptyboundedopensetwithboundary}ofclassC². WeconsiderthewaveequationwithlocallydistributedKelvin–Voigt-typedampinggiveninthefollowingequation:

⎧⎪

⎨

⎪⎩

ρ(x)utt(x,t)=^div(a(x)∇^u+^b(x)∇^ut) in×R⁺,

u(x,t)=^{0 on}×R⁺,

u(x,0)=^u0(x),ut(x,0)=^u1(x) in,

(1)

whereweassumethatthecoefficientfunctions ρ,a,b∈^L∞()and ρ(x)≥ρ0>0,a(x)≥^a0>0 andb(x)≥^{0 forallx}∈. In1988,F.HuangprovedthatwhentheKelvin–Voigtdampingdiv(b(x)∇^ut)isgloballydistributed,i.e. b(x)≥^b0>0 for almost every xin,thecorrespondingsemigroupofsystem(1) isnotonlyexponentially stable,butalsoisanalytic(see [8]).Thus,Kelvin–Voigtdamping isstrongerthan theviscous dampingb(x)u_t inthiscase. Indeed,in[10], itwas proved that the semigroupcorresponding to the systemof wave equations with globalviscous damping is exponentially stable butnot analytic.However, thisresultisstill trueifviscousdampingislocalized;via asmooth oranon-smoothdamping coefficient, inasuitable subdomainsatisfyingtheGeometric ControlCondition(GCC inshort)introducedby C.Bardos,G.

Lebeau,andJ.Rauchin[2] (seealso[10]).Nevertheless,when viscoelasticdampingis distributedlocally,thesituation is moredelicateandsuchcomparisonbetweenviscousandviscoelasticdampingisnotvalidanymore.Infact,in1998,K.Liu andZ.Liuconsidered aone-dimensional waveequation withKelvin–Voigtdampingdistributedlocallyonanysubinterval of theregionoccupiedby thebeam, wherethedampingcoefficient isthecharacteristicfunction ofthesubinterval.They proved that the semigroupassociated with the equation forthe transversal motionof the beamis exponentially stable, althoughthesemigroupassociatedwiththeequation forthelongitudinalmotionofthebeamisnot(see[13]).Thisshows that Kelvin–VoigtdampingdoesnotobeytheGCC.Thissurprisingresult,duetothediscontinuityofthematerials andthe unboundednessofviscoelasticdamping,motivatedthestudyofelasticsystemswithlocalKelvin–Voigtdamping.Later,inthe one-dimensionalcase,itwasfoundthatthesmoothnessofthedampingcoefficientattheinterfaceisacriticalfactorforthe stabilityandtheregularityofthesolutions(see[7,14,15,17,18,23]).However,thereareonlyasmall numberofpublications onthecorresponding N-dimensionalcase.In2006,K.LiuandB.Raoconsideredthisprobleminthe N-dimensionalspace where the dampingregion is a neighborhood (in ) ofthe entireboundary (see [16]). Theyproved that the energy of the system goes exponentially to zero ast goes to infinity for all usual initial data by assuming that the damping coefficient b satisfies b∈^C1,1(), b∈^L∞() and |∇^b(x)|²≤^M0b(x) foralmost every x in , where M0 isa positive constant. Alsoin [19], underthe same assumption on b, S. Nicaise and C. Pignotti established the exponential stability of the wave equation with local Kelvin–Voigt dampinglocalized around a part of the boundary andan extra boundary dampingwithtime delaywheretheyaddedan appropriategeometriccondition(section3.2(Q4)).Lateron,M.Cavalcanti, V. Cavalcanti and L. Tebou showed the exponential decay of the energy of a wave equation with two types of locally distributedmechanisms;africtionaldampingandaKelvin–Voigt-typedampingwherethelocationofeachdampingissuch that noneofthemaloneis abletoexponentiallystabilizethe system(see [6]).Underan appropriategeometric condition (PMGC) on a subset ω of ⊂R^{N wherethe} dissipation is effective, they proved that the energy of thesystem decays polynomially astype ¹_t in theabsenceofregularity oftheKelvin–Voigtdampingcoefficient b.However, they established exponential stability when this coefficient is smooth. In [1], K.Ammari, F. Hassine, andL. Robbianio considered a wave equationwithKelvin–Voigtdampinglocalizedinasubdomain ωfarawayfromtheboundarywithoutgeometricconditions.

Theyestablished alogarithmic energydecayrateforsmooth initialdata.Ontheother hand,in[22] L.Tebou studiedthe stabilizationofthewaveequationwithKelvin–Voigtdamping.Heestablishedpolynomialenergydecayoftype ¹_t provided that the dampingregion islocalized in a neighborhood of a part of the boundaryand verifies the Piecewise Multiplier Geometric Condition(PMGCinshort)introducedby K.Liu[12]. Moreover,Q.Zhangin[25] consideredthewaveequation withKelvin–Voigtdampinginanonemptyboundedconvexdomain withpartition=1∪2 wheretheviscoelastic dampingislocalizedin1.Undertheconditionthatthedampingcoefficientb isnon-smooth,sheestablishedapolynomial energydecayrateoftype ¹_t forsmoothinitialdatainthefollowingtwocases:(1)thedampingregion1isaneighborhood of theentireboundaryof;(2)⊂R^N (N=2 or3), ∂1 and∂2 areeitherconvexcurvilinearpolygonsorcurved plane polyhedra, thedamping region1 is aneighborhood ofapart 1 = ∅^{ofthe boundary}andm(x)·ν2≤^{0 where} m(x)=^x−^x0 forx0 fixedinR^N(N=²,3)forallx∈2=\1.So,severalimportantgeometricsituationsarenotcovered by thepreviouspapers (seeforinstanceFig.1-c, Fig.2,Fig.3) andtheproblemoftheenergydecayrateisstill open.So, ouraimistoanswerthisopenproblem.

Inthisnote,weconsiderthestabilizationofthewave equationwithKelvin–Voigtdampinginaboundeddomain of classC² withnon-smoothdampingcoefficient.Thesystemisgivenby(1).Weestablishapolynomialenergydecayestimate of type ¹_t for smooth initial data provided that the damping coefficient b satisfies the localization condition (LA) (see below)andthedampingregion ωsatisfiesoneofthegeometricconditions(A1)or(A2)(seebelow).Thefrequencydomain approach andthepiecewise multipliermethodare used.Toourknowledge,theresultofTheorem3.5isnew.Indeed,the geometricsituationscoveredbythistheoremarericherthanthatconsideredin[25],[22],[16] andincludeinparticularan exampleofdampingregionfarawayfromtheboundary.

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2. Well-posednessandstrongstability

LetusdefinetheenergyspaceH=^H1₀()×^L2()equippedwiththefollowinginnerproduct

((u,v), (^u˜,^v˜))_H=

(a∇^u·∇ ˜^u+ρv^v˜)dx.

Let(u,u_t)bearegularsolutiontothesystem(1).Itsassociatedenergyisdefinedby E(t)=

(a|∇^u|²+ρ_|ut|²)dx. (2)

Astraightforwardcomputation givesthat E(t)= −

b(x)|∇^ut|^2dx≤⁰. (3)

Consequently,system(1) isdissipativeinthesensethatitsenergyisnon-increasingwithrespecttot.SettingU=(u,u_t), system(1) maybe recastas: U=^{AU in}(0,+∞), U(0)=(u0,u1), wherethe unboundedoperator A:^D(A)−→H^is givenby

D(A)=

(u,v)∈H: ^v∈^H1₀(),div(a∇^u+^b∇^v)∈^L2()

, A(u,v)= ^v,¹

ρ^{div(a∇u+b∇v)}

.

Notingthatduetothefactthatb(x)≥^{0,theoperator}A^ism-dissipativeinH^{andgeneratesaC}0-semigroupofcontractions e^tA.So,system(1) iswell-posedinH^(see[16]).

Inaddition,if ω = ∅^{andbsatisfiesthefollowing}localizationassumption

∃^b0>0:^b(x)≥^b0 ∀^x∈ω, (LA)

thensystem(1) isstronglystable(see[1],Theorem2.2)i.e.

t→+∞lim ^etA(u₀,u₁) =⁰, ∀(u₀,u₁)∈H.

So,ouraimistostudytheenergydecayrate.

3. Polynomialenergydecayrate

Q. Zhangprovedin[24] thatsystem(1) isnotuniformly(exponentially)stableinanygeometry.So,itisnaturaltohope forapolynomialstabilityundersomeconsiderationsthatrepresentthemaingoalofthisnote.

Beforestatingourresults,werecalltheGeometricControlCondition(GCCinshort)introducedbyJ.RauchandM.Taylor in[21] formanifoldswithoutboundariesandby C.Bardos,G.LebeauandJ.Rauchin[2] (seealso[10])fordomains with boundaries.

Definition3.1.Forasubset ωof andT>0,weshall saythat(ω,T)satisfiesthe GeometricControlConditionifevery geodesictravelingatspeedoneinmeets ωintimet<T.

Wealsointroducethefollowinggeometriccondition:

Definition3.2.Forasubset ωof,weshallsaythat ωsatisfiesStrictlytheGeometricControlCondition(SGCC inshort)if thereexistsanopensubsetωincludedstrictlyin ω(i.e.ω_⊂ω)andsatisfyingthe GCC.

Forthestudyoftheenergydecayrate,weneedthefollowinggeometricassumptions:

(A1) theopensubset ωverifiestheGCCandmeas(ω_∩)>0, (A2) theopensubset ωverifiestheSGCC.

Remark3.3.Itiseasytoseethat,if ωverifiesthe SGCC,thenitverifiesthe GCC.The converseofthisimplicationisfalse (seeFig.1-c).

Thereareseveralgeometriesthatverifythepreviousassumptions.Forexample:

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Fig. 1.Elastic–viscoelastic waves interaction model satisfying the assumption (A1).

Fig. 2.A model satisfying assumption (A2).

Fig. 3.A model satisfying both (A1) and (A2).

Remark3.4.The PMGC introduced in [12] isa generalization of the -condition introduced in [11] and is much more restrictivethanthe GCC.Forexample,inFig.1,weconsiderthecasewhereisadiskandwedrawthreedifferentsubsets in.The-conditionisonlysatisfiedby ω0.The PMGC issatisfiedby ω0and ω1.However, ω2 doesnotsatisfyeitherthe PMGC orthe-condition.Finally,the GCC issatisfiedbythethreedifferentsubsetsof.

Now,weareinpositiontostateourmainresult.

Theorem3.5(Polynomialdecayrate).Assumethatcondition(L A)holds.Assumealsothatassumption(A1)orassumption(A2) holds.Then,forallinitialdataU0∈^D(A),thereexistsaconstantC>0independentofU0suchthattheenergyofsystem(1) satisfies thefollowingestimation

E(t,U)≤C 1

t ||U0||²_D(A), ∀t>0. (4)

Remark3.6.i)TheresultofTheorem3.5generalizesthatof[16],[22],and[25].Indeed,thegeometricsituationscovered by thistheoremarericherthanthatconsideredinthepreviousreferences.Inaddition,unliketheresultofTheorem4.1in [25],ourresultholdsforall N≥^{2 andfornon-convexdomains.}

ii) Itisunknown whetherthepolynomialdecayrateobtainedin(4) isoptimalinthesensethat,forany ε>0,wecan not expectthedecayrateoftype _t1¹+ε forallinitialdata U0∈^D(A).From ourpointofview,theenergydecayrate(4) is notoptimal,andweconjectureanoptimaldecayoftype _t¹₂.

IdeasforprovingTheorem3.5undertheassumption(A1). For the proof of Theorem 3.5, we use a frequency-domain approach;namely,weuseTheorem2.4of[5] (seealso[3,4,17])thatwepartiallyrecall.

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Theorem3.7.Let(T(t))t≥⁰beaboundedC0-semigrouponaHilbertspaceH withgeneratorA suchthatiR⊂ρ(A).Thenforafixed >0thefollowingconditionsareequivalent

(^is−^A)⁻¹ =^O(|^s|),s→ ∞, ⁽⁵⁾

T(t)A⁻¹ =O(t^−1/),t→ ∞. (6)

Since the resolventof theoperator A ^{is not compact inthe energy space}H ^{(see [16]) and0}∈ρ(A)^{, then toprove} iR⊂ρ(A)isequivalent to provethat (iβI−A)isbijective intheenergy spaceH^forall β∈R^{. Thislast isproven in} [1] according toa uniquecontinuation theoremandFredholm’salternative.Then, theproof ofTheorem3.5isreducedto show thatcondition (5) holdswith=^{2.Thisischeckedbyusinga}contradictionargument.Indeed,assumethatit does nothold,thenthereexistasequenceβn∈R^andasequence(un,vn)∈^D(A)suchthat

|β_n| → +∞, ||(u_n,v_n)||H=¹, (7)

β_n²(iβnI−A)(un,vn)=(fn,gn)→^{0 in}H. (8)

Ouraimistoshowthat||(^un,v_n)||H→^{0.Thisconditionpermitstoconcludea}contradictionwith(7).Theproofisdivided intoseveralsteps.

Step1.(Localasymptoticestimationofβnv_n).First,using(7) and(8),wehavethefollowingestimation

|βnu_n|^2dx=^O(1). (9)

Multiplying(8) byUn=(un,vn)inH,weget Re(β_n²(iβnI−A)U_n,U_n)_H= −

b|βn∇^vn|²=^o(1). (10)

Itfollowsfromthelocalizationassumption(LA)that

ω

|βn∇vn|²dx=o(1). (11)

Sinceassumption(A1)holds,thenusingPoincaré’sinequalityandequation(11),weobtain

ω

|β_nv_n|^2dx=^o(1). (12)

Theaimofstep1isachieved.

Now,writingequation(8) inadetailedform:

iβnu_n−^vn= ^fn

β_n²→^{0 inH1}₀(), (13)

iβnvn−¹

ρ^{div(a∇un+b∇vn)=}

g_n

β_n²→^{0 inL2}(). (14)

Step2.(Localasymptoticestimationofβnun).Multiplyingequation(13) by iβnun,integratingover ωandusingestimation (12),weget

ω

|βnun|²=o(1). (15)

Step3.(Themultiplier ϕn).Now,forallβn∈R^,let ϕn∈^H2()∩^H1₀()bethesolutiontothefollowingsystem

ρβ_n²ϕn+^div(a∇ϕn)−ⁱβn(1ωb)(x)ϕn=^un, in,

ϕn=0, on ⁽¹⁶⁾

where (un,vn) is solution to (13)–(14). Since ω satisfies the GCC, then the wave equation with local viscous damping (1ωb)(x)ϕt isexponentiallystable(see[10])and,followingHuang[9] andPruss[20],theresolventofitsassociatedoperator Aaux:^D(Aaux)−→^H₀¹()×^L2()definedby: