hal-00260227, version 1 - 3 Mar 2008StabilityofmultipeakonsKhaledEl DikaandLucMolinetL.A.G.A.,InstitutGalilee,UniversiteParis-Nord,93430 Villetaneuse,France.khaled@math.u-paris13.frmolinet@math.u-paris13.frAbstract. The Camassa-Holmequationpossesses well-knownpeakedsolitarywaves that are calledpeakons. Their orbital stability has beenestablishedbyConstantinandStraussin[6]. Weproveherethestabilityof orderedtrainsof peakons. Wealsoestablisharesultonthestabilityofmultipeakons.1 IntroductionTheCamassa-Holmequation(C-H), 0,ut utxx= 2ux 3uux + 2uxuxx +uuxxx, (t, x) IR2, (1)can be derived as a model for the propagation of unidirectional shalow waterwaves over aat bottombywritingtheGreen-Naghdi equations inLie-PoissonHamiltonianform and then making an asymptoticexpansion whichkeepsthe Hamiltonianstructure ([3],[19]). It was alsofound independentlybyDai [10] asamodel fornonlinearwavesincylindrical hyperelasticrodsandwas, infact, rstdiscoveredbythemethodof recursiveoperator byFokasandFuchsteiner[16]asanexampleofbi-Hamiltonianequation.(C-H)iscompletelyintegrable(see[3],[4]). ItpossessesamongothersthefollowinginvariantsE(v) =_IRv2(x) +v2x(x) dxandF(v) =_IRv3(x) +v(x)v2x(x) +2v2(x) dx(2)andcanbewritteninHamiltonianformastE(u) = xF(u) . (3)For>0itpossessessmoothpositivesolitarywaves,cwithspeedc>2, theirorbital stabilityhasbeenprovedin[7] byapplyingtheclassical1spectralmethodinitiatedbyBenjamin[2](seealso[17]). In[15],followingthegeneralmethoddevelopedin[20](seealso[14]),theauthors provedthestabilityoforderedtrainsofsuchsolitarywaves. Itisworthrecallingthatthis general methodrequires principallytwoingredients : Apropertyofalmostmonotonicitywhichsaysthatforasolutioncloseto,c, thepartoftheenergytravelingattherightof,c( ct)isalmosttimedecreasing;Adynamical proof of thestabilityof thesolitarywaveusingthespectralapproach(asin[2]or[17]forinstance).InthispaperweconsidertheCamassa-Holmequationinthecase=0,thatisut utxx= 3uux + 2uxuxx +uuxxx, (t, x) IR2. (4)Henceforth, wereferto(4)astheCamassa-Holmequation(C-H). (4)pos-sessesalsosolitarywavesbuttheyarenonsmoothandarecalledpeakons.Theyaregivenbyu(t, x) = c(x ct) = c(x ct) = ce|xct|, c IR.Theirstabilityseemsnottoenterthegeneral frameworkmentionedabove(seethebeginningof Section3for further commentaries onthis aspect).However,ConstantinandStrauss[6]succeededinprovingtheirorbitalsta-bilitybyadirect approach. Inthis work, followingthegeneral strategyinitiatedin[20](notethatduetothereasonsmentionedabove, thegeneralmethod of [20] is not directly applicable here ), we combine the monotonicityresultprovedin[14] withlocalizedversionsoftheestimatesestablishedin[6]toderivethestabilityofthetrainsofpeakons.Beforestatingthemainresultwehavetointroducethefunctionspacewhere will live our class of solutions to the equation. For Ia nite or inniteintervalofIR,wedenotebyY (I)thefunctionspace1Y (I) :=_u C(I; H1(IR)) L(I; W1,1(IR)), ux L(I; BV (IR))_. (5)Wearenowreadytostateourmainresult.Theorem1.1Let begivenNvelocitiesc1, .., cNsuchthat 0 0 and0> 0 such thatif u Y ([0, T[),with0 < T ,isasolutionof(C-H)satisfying|u0 N

j=1cj( z0j)|H1 2(6)1W1,1(IR)isthespaceof L1(IR)functionswithderivativesinL1(IR)andBV (IR)isthespaceoffunctionwithboundedvariation2for some0 L/2, t [0, T[ . (8)As discoveredbyCamassaandHolm[3], (C-H) possessesalsospecialsolutionscalledmultipeakonsgivenbyu(t, x) =N

i=1pj(t)e|xqj(t)|,where(pj(t), qj(t))satisfythedierential system(60). In[1](seealso[3]),theasymptoticbehaviorof themultipeakonsisstudied. Inparticular, thelimitsasttendsto+and of pi(t)and qi(t)aredetermined. Com-biningtheseasymptoticswiththeprecedingtheoremwegetthefollowingresultonthestabilityofthevariety ^ofH1(IR)denedby^:=_v =N

i=1pje|qj|,(p1, .., pN) (IR+)N,q1< q2< .. < qN_ .Corollary1.1Let begivenNpositivereal numbersp01, .., p0NandNrealnumbersq010thereexists>0suchthatifu0 H1(IR)satises2m0:= u0u0,xx /+(IR)with|m0|M B and |u0 N

j=1p0j exp( q0j)|H1 (9)thent IR, infP(IR+)N,QIRN|u(t, ) N

j=1pj exp( qj)|H1 . (10)2M(IR) is the space of Radon measures on IR with bounded total variation and M+(IR)isthesubsetofnon-negativemeasures3Moreover,thereexistsT> 0suchthatt T, infQG|u(t, ) N

j=1jexp( qj)|H1 (11)andt T, infQG|u(t, ) N

j=1N+1jexp( qj)|H1 , (12)where (:= Q IRN,q1 0suchthatif u C([0, T[; H1(IR))isasolutionof (4)suchthat E(u(t))andF(u(t))areconservedquantitieson[0, T[and |u(0) c|H1 2,then|u(t, ) c( r(t))|H1 C, t [0, T[, (16)wherer(t) IRisanypointwherethefunctionu(t, )attainsitsmaximum.The proof of this theorem is principally based on the following lemma of [6].Lemma3.1Foranyu H1(IR)and IR,E(u) E(c) = |u c( )|2H1+ 4c(u() c). (17)Foranyu H1(IR),letM= maxxIRu(x),thenF(u)ME(u) 23M3. (18)Remark3.1It is worthnoticingthat (17) ensures that theminimumoftheH1-distancebetweenuand c( ), IRisexactlyreachedatanypointwhereuattainsitsmaximumonIR.Proof of Theorem3.1Letu C([0, T[; H1(IR))beasolutionof(4)with|u(0) c|H12andlet(t) IRbesuchthatu(t, (t)) = maxIRu(t, ).By the remark above,t |u(t) c( (t))|H1 is continuous on [0, T[ and|u(0) c( (0))|H12. Moreover,asshownin[6],itisnotohardto6checkthatforanyv H1(IR)suchthat |u c|H1< forsome< 1,itholds[E(u) E(c)[ < 4 cand [F(u) F(c)[ < 10 c. (19)Fromtheconservationlawsitfollowsthatforanyt [0, T[[E(u(t)) E(c)[ < 4 c2and [F(u(t)) F(c)[ < 10 c2. (20)Therefore, by a classical continuity argument, it suces to prove that for anyv H1(IR) satisfying (20) and |vc()|H11/4, with v() = maxIRv,itholdsactually|v c( )|H1 .SettingM=v()and=c M=c v(), wenoticethat(17)ensuresthatfor 0,|v c( )|2H1 E(u0) E(c)2.Hencetoprovethestabilityitremainstoexaminethecase>0, thatisthemaximum ofthe function u is less thanthe maximum of the peakonc.SubstitutingMbyc in(18),using(20)andthatE(c) = 2c2andF(c) =43c3, (21)onecaneasilycheckthat43c3O(2)(c )(2c2+O(2)) 23(c )3whichleadsto2(c /3)O(2) . (22)Ontheotherhand, onaccountofthehypothesis |v c( )|H1 1/4andof thecontinuousembeddingof H1(IR)intoL(IR), itholds 0 and L > 0 we dene the following neighborhood of all the sums ofN peakons of speed c1, .., cNwith spatial shifts xjthat satised xjxj1 L.U(, L) =_u H1(IR), infxjxj1>L|u N

j=1cj( xj)|H1< _ . (23)7Bythecontinuityofthemapt u(t)from[0, T[intoH1(IR),toproveTheorem1.1itsucestoprovethatthereexistA> 0, 0> 0andL0>0suchthat L>L0and0 0,L0> 0andC0> 0suchthatforall 00largeenough, D(y1,..,yN)Y (0, .., 0, RZ) =D+P where Dis aninvertible diagonal matrix with |D1|(c1)2and|P|O(eL/4). Hence there exists L0>0 such that for L >L0,D(y1,..,yN)Y (0, .., 0, RZ)isinvertiblewithaninversematrixofnormsmallerthan2 (c1)2. From the implicit function theorem we deduce that there ex-ists0>0andC1functions(y1, .., yN)fromB(RZ, 0)toaneighborhoodof(0, .., 0)whichareuniquelydeterminedsuchthatY (y1, .., yN, u) = 0forallu B(RZ, 0) .In particular, there exits C0> 0 such that if u B(RZ, ), with 0 < 0,thenN

i=1[yi(u)[ C0; . (34)Note that 0andC0only dependonc1andL0andnot onthe point(z1, .., zN). For u B(RZ, 0)weset xi(u)=zi+ yi(u). Assumingthat0 L08C0,( x1, .., xN)arethusC1-functionsonB(RZ, )satisfying xj(u) xj1(u) > L/2 2C0> L/4 . (35)For L L0and00onlydependingonc1suchthat if0 < < 0andL L0thenforany4 K L1/2,Ij,K(t) Ij,K(0) O(e0L8K), j 2, .., N, t [0, t0] . (42)4.3 Alocalizedandaglobal estimateWedenethefunctioni= i(t, x)by1= 1 2,K= 1 K( y2(t)),N= N,K= K( yN(t))andfori = 2, .., N 1i= i,K i+1,K= K( yi(t)) K( yi+1(t)) ,whereKandtheyisaredenedinSection4.2. ItiseasytocheckthatN

i=1i,K 1. WetakeL > 0andL/K> 0largeenoughsothatisatises[1 i,K[ 4eL4Kon[ xi L/4, xi +L/4] (43)and[i,K[ 4eL4Kon[ xj L/4, xj+L/4]wheneverj ,= i . (44)We will use the following localized versionof EandF denedfor i 1, .., N, byEti(u) =_IRi(t)(u2+u2x)andFti(u) =_IRi(t)(u3+uu2x) . (45)PleasenotethathenceforthwetakeK= L1/2/8.Thefollowinglemmagivesalocalizedversionof(18). Notethatthefunc-tionalsEiandFidonotdependontimeinthestatementbelowsincewex x1< .. 0independentofAsuchthat|u N

j=1cj( xj)|H1< C(1/2+L1/8),15sothatonecantakeA = 2Ctoconcludetheproof(Recallthatwealreadyknowfrom(28)-(30)thatxi xi1 2L/3fori 2, .., N). Letusprove(53). From(46)bytakingthesumoverionegets:F(u(t0)) =N

i=1Fi(u(t0)) N

i=1MiEi(u(t0)) 23N

i=1M3i+O(L1/2)Settingt00F(u) = F(u(t0)) F(u(0))andt00E(u) = E(u(t0)) E(u(0)),thisimplies0 = t00F(u) =N

i=1t00Fi(u) N

i=1Mit0Ei(u) 2/3N

i=1M3i(54)+N

i=1(Fi(u0) +MiEi(u0)) +O(L1/2)By(6), theexponential decayofthecisandtheis, andthedenitionofEiandFi,itiseasytocheckthat[Ei(u0) E(ci[ +[Fi(u0) F(ci[ O(2) +O(eL), i 1, .., N .SettingM0=0andusing(21), onethusndsafterhavingsubstitutedMibyciithatN

i=1(Fi(u0)+MiEi(u0)2/3M3i ) = 2N

i=1(ci2i +133i)+O(2)+O(eL) .(55)Notethatby(51) andthecontinuousembeddingof H1(IR)intoL(IR),Mi= ci +O() +O(eL/8),andthus0 < M1 0andq1< q2< qN(61)then(61)remainstrueforall time. Inparticular, underthesehypothesesthedierent peakons never overlapeachothers. For example, if alargerpeakonfollowsasmallerone, itwill comeclosetothislastoneandthentransferpartof itsenergytoit. Inthisway, thesmalleronewill becomethelarger oneandthetwopeakonswill bewell ordered. In[1] (seealso[3]), usingtheintegrabilityof(4), Bealsetal establishedaformulafortheasymptoticsoftheqisandthepis. Inparticular,theyprovethefollowinglimitsforthepiand qi,i 1, .., N,limt+pi(t) = limt+ qi(t) = i(62)andlimtpi(t) = limt qi(t) = N+1i, (63)where 0 < 10. FromtheasymptoticsabovethereexistsT> 0suchthatqi(T) qi1(T) > Landqi(T) qi1(T) > L (66)withL > max_L0, (2A)8_ . (67)Fromthelastassertionof Theorem2.1, foranygivenB>0, thereexists > 0suchthatifu0satises(9)thenforallt [T, T],___u(t) N

i=1pi(t)e|xqi(t)|___H1_2A_4. (68)Atthisstage, itiscrucial toremarkthatsince(4)isinvariantunderthetransformation(t, x) (t, x), Theorem 1.1remains true when replacingt by t, z0jby z0jand xj(t) by xj(t). This gives a stability result in thepastfortrainsofpeakons thatareorderedinthe inverseorderwithrespecttoTheorem1.1.Combining(66), (68), Theorem1.1andtheremarkabove, therstpartofthecorollayfollows.18Finally,from(62)-(63), wecanalsoassumethat[pi(T) i[ 1100N_2A_4and [pi(T) Ni[ 1100N_2A_4sothat___u(T)N

i=1ie|xqi(T)|___H1_2A_4and___u(T)N

i=1Nie|xqi(T)|___H1_2A_4.Thiscompletestheproofofthecorollary.6 AppendixProof of Lemma 4.2. Let us assume that uis smoothsince the caseu Y ([0, T[)followsbymodifyingslightlythearguments(seeRemark3.2of [14]). From(13), itisnottoohardtocheckthatforanysmoothspacefunctiong,thefolllowingdierentialidentityontheweightedenergyholds:ddt_IR(u2+u2x)g dx =_IR(u3+ 4uu2x)gdx_IRu3gdx _IRug(1 2x)1(2u2+u2x) dx. (69)Applying(69)withg= j,Konegetsddt_IRj,K(u2+u2x) dx = yj_IRj,K(u2+u2x) +_IRj,K(u3+ 4uu2x) dx_IRj,Ku3dx _IRj,Ku(1 2x)1(2u2+u2x) dx c12_IRj,K(u2+u2x) +J1 +J2 +J3. (70)Weclaimthatfori 1, 2, 3,itholdsJi c18_IRj,K(u2+u2x) +CKe1K(0t+L/8). (71)TohandlewithJ1wedivideIRintotworegionsDjandDcjwithDj= [ xj1(t) +L/4, xj(t) L/4]19Firstsincefrom(28),forx Dcj,[x yj(t)[ xj(t) xj1(t)2L/4 cj cj12t +L/8 ,weinferfromthedenitionofinSection4.2that_Dcjj,K(u3+ 4uu2x) CK |u0|3H1e1K(0t+L/8).Ontheotherhand,onDjwenotice,accordingto(27),that|u(t)|LDjN

i=1|ci( xi(t))|L(Dj) +|u N

i=1ci( xi(t))|L(Dj) C eL/8+O() . (72)Therefore,forsmallenoughandLlargeenoughitholdsJ1 c18_IRj,K(u2+u2x) +CKe1K(0t+L/8).Since J2can be handled in exactly the same way,it remains to treat J3. Forthis,werstnoticeasabovethat_Dcjuj,K(1 2x)1(2u2+u2x) 2|u|supxDcj[j,K(x yj(t))[_IRe|x| (u2+u2x) dxCK|u0|3H1 e1K(0t+L/8), (73)sincef L1(IR), (1 2x)1f=12e|x| f. (74)NowintheregionDj, noticingthatj,Kandu2+ u2x/2arenon-negative,weget_Djuj,K(1 2x)1(2u2+u2x) |u(t)|L(Dj)_Djj,K((1 2x)1(2u2+u2x) |u(t)|L(Dj)_IR(2u2+u2x)(1 2x)1j,K. (75)20On the other hand, from the denitionof inSection4.2and (74)we inferthatforK 4,(1 2x)j,K (1 10K2)j,K (1 2x)1j,K (1 10K2)1j,K.Therefore,takingK 4andusing(72)wededuceforsmallenoughandLlargeenoughthat_DjuK(1 2x)1(2u2+u2x) c18_IR(u2+u2x)K. (76)Thiscompletestheproofof(71). Gathering(70)and(71)weinferthatddt_IRj,K(u2+u2x) dx c18_IRj,K(u2+u2x) +CK|u0|3H1 e1K(0t+L/8).Integratingthisinequalitybetween0andt,(42)follows.References[1] R. Beals, D.H. SattingerandJ. Szmigielski,Multipeakonsandtheclassicalmomentproblem.Adv.Math.154(2000),no.2,229257.[2] T.B.Benjamin, Thestabilityofsolitarywaves.Proc.Roy.Soc.Lon-donSer.A328(1972), 153 183.[3] R. CamassaandD. Holm, Anintegrable shallowwater equationwithpeakedsolitons,Phys.rev.Lett. 71(1993), 16611664.[4] R. Camassa, D. HolmandJ. Hyman, Annewintegrableshallowwaterequation,Adv.Appl.Mech.31(1994)[5] A. ConstantinandJ. Escher,Globalexistenceandblow-upforashallowwaterequation, Annali Sc. Norm. Sup. Pisa26(1998), 303328.[6] A. ConstantinandW. Strauss, Stabilityof peakons, Commun.PureAppl.Math. 53(2000), 603-610.[7] A. ConstantinandW. Strauss, Stabilityof the Camassa-Holmsolitons,J.NonlinearSci.12(2002), 415-422.[8] A.ConstantinandL.Molinet, Global weak solutions for a shallowwaterequation,Comm.Math.Phys.211(2000),4561.21[9] A. ConstantinandL. Molinet,Orbitalstabilityofsolitarywavesforashallowwaterequation,PhysicaD157(2001), 75-89.[10] H.-H. Dai, Model equations for nonlinear dispersive waves incom-pressibleMooney-Rivlinrod,ActaMech. 127(1998),293308.[11] R.Danchin, A few remarks on the Camassa-Holm equation,Di. Int.Equ.14(2001), 953980[12] K. ElDika, Smoothingeect of thegeneralizedBBMequationforlocalizedsolutions moving to the right. Discr. Cont. Dyn. Syst. 12(2005), 973-982.[13] K. ElDika,AsymptoticstabilityofsolitarywavesfortheBenjamin-Bona-Mahony,Discr.Cont.Dyn.Syst. 13(2005),583-622.[14] K. ElDikaandY. Martel, Stabilityof NsolitarywavesforthegeneralizedBBMequations, Dyn. Partial Dier. Equ. 1(2004), 401-437.[15] K. El Dika andL. Molinet, Exponential decayof H1-localizedsolutions and stability of the train of Nsolitary waves for the Camassa-Holmequation.Phil.Trans.R.Soc.A.365(2007), 23132331[16] A. S. Fokas and B. Fuchssteiner, Symplectic structures, theirBacklund transformation and hereditary symmetries, Physica D4(1981), 4766.[17] M.Grillakis,J.ShatahandW.Strauss, Stabilitytheory ofsoli-tarywaves inthepresenceof symmetry, J. Funct. Anal. 74(1987),160-197.[18] H.HoldenandX.Raynaud, A convergent numerical scheme for theCamassa-Holm equation based on multipeakons, Discrete Contin. Dyn.Syst.14(2006),no.3,505523.[19] R.S.Johnson, Camassa-Holm,Korteweg-deVriesandrelatedmodelsforwaterwaves,J.FluidMech. 455(2002), 6382.[20] Y. Martel,F.MerleandT-p. TsaiStabilityandasymptoticsta-bility in the energy space of the sum of Nsolitons for subcritical gKdVequations.Comm.Math.Phys.231(2002), 347373.[21] L.Molinet Onwell-posednessresultsforCamassa-Holmequationontheline: asurvey.J.NonlinearMath.Phys.11(2004),521533.22