Marco Sammartino

Maria Carmela Lombardo

Vincenzo Sciacca

Comptes Rendus Mathématique

1/nUsing theanalytices dese solutionlytique a,Théorèmel’equa-rème dee directeltat est leedia”.acca).C. R. Acad. Sci. Paris, Ser. I 341 (2005) 659–664http://france.elsevier.com/direct/CRASSPartial Differential EquationsA Note on the analytic solutions of the Camassa–Holm equatio✩Maria Carmela Lombardo, Marco Sammartino, Vincenzo SciaccaDepartment of Mathematics, University of Palermo, via Archirafi 34, 90123 Palermo, ItalyReceived 11 August 2005; accepted 11 October 2005Available online 8 November 2005Presented by Yvonne Choquet-BruhatAbstractIn this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces.Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a uniquesolution. Moreover, if the initial data is real analytic, belongs toHs(R) with s > 3/2,‖u0‖L1 < ∞ andu0 −u0xx does not changsign, we prove that the solution stays analytic globally in time.To cite this article: M.C. Lombardo et al., C. R. Acad. Sci. Paris,Ser. I 341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméSolutions analytiques de l’équation de Camassa–Holm.On étude ici l’équation de Camassa–Holm dans les espacefonctions analytiques. On montre que, si les données initiales sont analytiques, il existe, localement dans le temp, ununique analytique.En outre si la la donnée initiale analytique0(x) est bornée dansL1, appartient àHs(R) s > 3/2 et satisfait la conditionu0 − u0xx  0, la solution résulte analytique globalement dans le temp.Pour citer cet article : M.C. Lombardo et al., C. R. Acad.Sci. Paris, Ser. I 341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeDans cette Note nous montrons que l’equation de Camassa–Holm (1) avec des données initiales anapour un temp dependant des données, une unique solution analytique. Notre preuve est basée sur lede Cauchy–Kowalewski dans sa formulation abstraite, voir le Théorème 2.5. En effet on peut prouver quetion de Camassa–Holm, écrite dans sa forme pseudo-différentielle (5), satisfait les hypotheses du ThéoCauchy–Kowalewski (l’opérateure est quasi contractif dans les normes analytiques). Cette propriété est unconséquence de l’estimation de Cauchy pour la dérivée d’une fonction analytique. Donc, notre premier résusuivant :✩ Work supported by the PRIN grant “Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous ME-mail addresses: lombardo@math.unipa.it (M.C. Lombardo), marco@math.unipa.it (M. Sammartino), sciacca@math.unipa.it (V. Sci1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.10.006660 M.C. Lombardo et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 659–664n connut sur uner lest avec unly:of the[1].edness ofez-Blancovelop aortfslewskiy. UsingThéorème 0.1.On suppose que la donnée initiale de l’equation de Camassa–Holm (1) u0 ∈ H 1,ρ0 avec ρ0 > 0. Alorsils existent ρ > 0, β > 0 et T > 0 telles que l’Éq. (1) admet une unique solution u ∈ H 1,ρβ,T .Notre deuxième résultat concerne l’existence globale dans le temp des solutions analytiques. Il est bie(voire, e.g., [17]) que, si la donnée initiale est tel que le potentiel(1 − ∂2x )u0  0 (ou 0), la solution de l’equationde Camassa–Holm est globale. Nous montrons le même résultat pour les solutions analytiques réelles.Théorème 0.2.On suppose que la donnée initiale de l’equation de Camassa–Holm (1) u0 ∈ H 1,ρ avec ρ > 0. En pluson suppose que u0 ∈ L1 et que (1− ∂2x )u0  0 (ou  0). Alors pour tout T > 0 il existe une solution unique analytiqueréelle u(x, t) avec t ∈ [0, T ].La preuve du théorème est basée sur la construction des approximations de Galerkin de la solution, eestimation a priori globale de la solution dans les espacesHr,ρ . L’estimation a priori est dans (14) et est basée suestimations du Lemma 3.2.Le sens de l’estimation a priori est que la region du plane complexe où la solution est analytique se réduitaux exponential et donc il y a toujours une bande d’analyticité de la solution.1. IntroductionIn recent years much attention has been devoted to the study of the Camassa–Holm equation (CH), nameut − uxxt + 3uux = 2uxuxx + uuxxx, (1)whereu = u(x, t) with x ∈ R and t  0. Eq. (1) was first introduced by Fokas and Fuchssteiner [11] as partstudy of some general aspects of soliton theory and later Camassa and Holm gave a physical derivation of itDuring the past seven years there has been an extensive investigation about the problem of the well-posthe Camassa–Holm equation. Local well-posedness of classical solutions in the Sobolev spaceH 3 was first proved in[5] and [4]. The result was later improved for initial data inHs with s > 3/2 through different methods, both in thperiodic and non periodic cases (see for example [15,17,7,12,16,8] and references therein).The existence of global weak solutions was obtained in [6] with initial datau0 ∈ H 1(R) such thatu0 − u0xx is apositive regular Borel measure onR with total bounded variation.Global well-posedness of classical solutions to the Camassa–Holm equation was proved by Rodríguein [17].On the other hand it is well known that, under proper conditions, the solutions of the CH equations can desingularity in finite time [1,2,5,3].In this Note first we shall prove that, if the initial datumu0 is analytic then the solution stays analytic for a shtime:Theorem 1.1.Let u0 ∈ H 1,ρ0 be the initial datum of the Camassa–Holm equation (1). Then there exists β > 0 suchthat for any ρ with 0 < ρ < ρ0 there exists a unique continuously differentiable (w.r.t. time) solution u of the Camassa–Holm equation with the following property:• u(·, t) ∈ H 1,ρ and ∂tu(·, t) ∈ H 1,ρ when t ∈ [0, ρ0−ρβ ].For the definition of the function spacesH 1,ρ0 see Section 2 below. Roughly speakingH 1,ρ0 is the space ofunctionsf (x) analytic w.r.tx in a strip of widthρ0 in the complex planeC. The spaceH1,ρβ,T is the space of functionf (x, t), analytic w.r.t.x in a strip of the complex plane whose width shrinks linearly in time at speedβ. The abovetheorem says the solutionu ∈ H 1,ρβ,T . The proof of the above theorem is based on the abstract Cauchy–KowaTheorem in Banach spaces in the form given by Safonov in [18].A similar result (for the CH equation in periodic domain) appeared in [13].Secondly, we state a global-in-time result. If the initial data belongs toHs(R) with s > 3/2 andu0 − ∂2xu0 doesnot change sign, we prove that time analyticity of the solutions to the Camassa–Holm equation holds globallM.C. Lombardo et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 659–664 661that thetheeLet usennse-some a priori estimates and the result of global well-posedness given in Theorem 3.1, we are able to provesolution is analytic, globally in time, with values in a Gevrey class of functions. Our result is the following:Theorem 1.2.Suppose the initial datum of the Camassa–Holm equation satisfy u0 ∈ Hr,ρ with r > 3/2, |u0|L1 < ∞,u0 − ∂2xu0  0 (or  0). Then the unique solution u(x, t) belongs to the Gevrey class of index 1 globally in time.In the proof of global analyticity of the solutions we shall closely follow the ideas of [14,10,9].In this Note we consider the CH equation inR. With minor modification one can obtain the same results forCH equation in a periodic domain.2. Local in time analyticityIn the sequel we shall be dealing with functions that are analytic in the complex variablex. W hence introduce thstrip in the complex plane.D(ρ) = R × (−ρ,ρ) = {x ∈ C: x ∈ (−ρ,ρ)}.TheL2 integration is performed along the following path:Γ (b) = {x ∈ C: x = b}.We can now introduce the function spaces where local in time analyticity of the CH equation will be proved.first introduce the function spaces of the initial datum.Definition 2.1.H 0,ρ is the set of all complex functionsf (x) such that• f is analytic inD(ρ);• f ∈ L2(Γ (x)) for x ∈ (−ρ,ρ); i.e. if x is inside(−ρ,ρ), thenf (x + ix) is a square integrable functioof x;• |f |ρ = supx∈(−ρ,ρ) ‖f (· + ix)‖L2(Γ (x)) < ∞.Definition 2.2.Hk,ρ is the set of all complex functionsf (x) such that• ∂jx f ∈ H 0,ρ for 0 j  k;• ‖f ‖k,ρ ≡ ∑0jk |∂jx f |ρ < ∞.This norm is equivalent to:‖f ‖k,ρ =[∫e2ρ|ξ |(1+ |ξ |2)k∣∣f̂ (ξ)∣∣2 dξ]1/2, (2)wheref̂ (ξ) if the Fourier transform of . With the above expression of the norm one can defineHr,ρ with r ∈ R+.This will be useful in Section 3.We now introduce the dependence on the time variable.Definition 2.3.The spaceH 1,ρβ,T is defined as the space of all the functionsf (x, t) such that:• f ∈ C1([0, T ],H 1,ρ).• |f |1,ρ,β,T = sup0tT |f (t, ·)|0,ρ−βt + sup0tT |∂tf (t, ·)|0,ρ−βt + sup0tT |∂xf (t, ·)|0,ρ−βt < ∞.One can analogously define the spaceHk,ρβ,T . We shall also need the following estimate which is a direct coquence of the Cauchy estimate on the derivative of analytic functions:662 M.C. Lombardo et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 659–664ee thatlication(b)d).Lemma 2.4.Let u,w ∈ H 1,ρ′′ . If ρ′ < ρ′′ then‖u∂xu − w∂xw‖1,ρ′  c‖u − w‖1,ρ′′ρ′′ − ρ′ . (3)We now write the Camassa–Holm in a form suitable for the application of the ACK Theorem. It is easy to sthe CH can be written in the form:(1− ∂2x)(ut + uux) = −2uux − uxuxx = −∂x(u2 + 12u2x).Thus one can write the CH equation in the pseudodifferential form:ut + uux = − iξ1+ ξ2(u2 + 12u2x), (4)whereξ is the dual Fourier variable ofx. With an integration in time one gets:u = F(t, u) whereF(t, u) ≡ u0 −t∫0dt ′[uux + iξ1+ ξ2(u2 + 12u2x)]. (5)We now state the ACK Theorem in the form given by Safonov in [18]. Consider the problem:u = F(t, u). (6)Theorem 2.5(ACK). Suppose that ∃R > 0, ρ0 > 0, and β0 > 0 such that if 0 < t  ρ0/β0, the following propertieshold:(a) ∀0< ρ′ < ρ  ρ0 and ∀u s.t. {u ∈ Xρ : sup0tT |u(t)|ρ  R} the map F(t, u) : [0, T ] → Xρ′ is continuous.(b) ∀0< ρ < ρ0 the function F(t,0) : [0, ρ0/β0] → {u ∈ Xρ : sup0tT |u(t)|ρ  R} is continuous and∣∣F(t,0)∣∣ρ R0 < R. (7)(c) ∀0< ρ′ < ρ(s) < ρ0 and ∀u and w ∈ {u ∈ Xρ : sup0tT |u(t)|ρ−β0t  R},∣∣F(t, u) − F(t,w)∣∣ρ′  Ct∫0ds( |u − w|ρ(s)ρ(s) − ρ′). (8)Then ∃β > β0 such that ∀0 < ρ < ρ0 Eq. (6) has a unique solution u(t) ∈ Xρ with t ∈ [0, (ρ0 − ρ)/β]; moreoversupρ<ρ0−βt |u(t)|ρ  R.The proof of the short time existence of analytic solutions of the CH equation easily follows as a simple appof the ACK Theorem. In fact it is obvious that the operatorF(u, t) given by (5) satisfies the hypotheses (a) andof the ACK Theorem. Regarding hypothesis (c), considering first the norm|F(u, t)|1,ρ′ , the bound is easily achieveby using Lemma 2.4. Finally, the bound on|∂xF (u, t)|1,ρ′ is obtained using the fact that the operatorξ2/(1 + ξ2) isbounded.The proof of Theorem 1.1 (or the equivalent form Theorem 0.1) is thus achieved.3. Global in time analyticityThe function space we shall deal with in this section is the spaceHr,ρ , r ∈ R+, r > 3/2, endowed with the norm (2One can see [14] that the spaceG1 of Gevrey functions of index 1 can be recovered as:G1 =⋃ρ>0Hr,ρ.Here we state a global existence theorem in Sobolev spaces [17], that will be useful in the sequel:M.C. Lombardo et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 659–664 663), theridthng thedolution,Theorem 3.1(Rodríguez-Blanco). Suppose that u0 ∈ Hs(R), s > 3/2, with (1 − ∂2x )u0  0 and that u0 ∈ L1(R).Then Eq. (1) is globally well posed in Hs(R).To prove our result, Theorem 1.2, the crucial step is the global estimate onu in the spaceHr,ρ , with ρ dependingon time t . We will show that, ifρ decreases to 0 exponentially fast with the appropriate rate (see (13) belownorm ofu in Hr,ρ is bounded in[0, T ] for anyT . To simplify the notation we shall formally derive the estimate foubut all the computations can be made rigorous by considering a Galerkin approximation.Consider the solutionu of the CH equation (4). Using the usual notation for the scalar product inL2, and denotingwith ‖ · ‖ and‖ · ‖Hr the norms inL2 andHr , one can write:12ddt∥∥|ξ |r eρ|ξ |u∥∥2L2= ρ̇〈|ξ |r+1 eρ|ξ |u, |ξ |r eρ|ξ |u〉 + 〈|ξ |r eρ|ξ |∂tu, |ξ |r eρ|ξ |u〉= ρ̇∥∥|ξ |r+1/2 eρ|ξ |u∥∥2 − 〈|ξ |r eρ|ξ |uux, |ξ |r eρ|ξ |u〉−〈|ξ |r eρ|ξ | iξ1+ ξ2(u2 + u2x2), |ξ |r eρ|ξ |u〉. (9)Notice how in the above calculation we have takenρ (i.e. the rate of decay of the spectrum or, equivalently, the wof the strip of analyticity) as a function of time. Now we state our basic estimates:Lemma 3.2.Let u ∈D(|ξ |r+1/2 eρ|ξ |) with r > 3/2, then:∣∣〈|ξ |r eρ|ξ |uux, |ξ |r eρ|ξ |u〉∣∣  c‖u‖Hr (‖u‖2Hr + 2ρ∥∥|ξ |r+1/2 eρ|ξ |u∥∥2), (10)∣∣∣∣〈|ξ |r eρ|ξ | iξ1+ ξ2(u2 + u2x2), |ξ |r eρ|ξ |u〉∣∣∣∣  c‖u‖Hr (‖u‖2Hr + 2ρ∥∥|ξ |r+1/2 eρ|ξ |u∥∥2). (11)Proof. The proof of the estimate (10) is standard. In estimate (11) the most difficult term is the term involviderivative ofu. The proof of the estimate is the following:∣∣∣∣〈|ξ |r eρ|ξ | iξ1+ ξ2u2x, |ξ |r eρ|ξ |u〉∣∣∣∣  ∣∣〈|ξ |r−1 eρ|ξ |u2x, |ξ |r eρ|ξ |u〉∣∣  ‖ux‖L∞ ∣∣〈|ξ |r−1 eρ|ξ |ux, |ξ |r eρ|ξ |u〉∣∣= ‖ux‖L∞∫R|ξ |2r e2ρ|ξ |∣∣û(ξ)∣∣2 dξ ‖ux‖L∞∫R|ξ |2r(1+ 2ρ|ξ |e2ρ|ξ |)∣∣û(ξ)∣∣2 dξ c1‖u‖Hr(∥∥|ξ |ru∥∥2 + 2ρ∥∥|ξ |r+1/2 eρ|ξ |u∥∥2) c1‖u‖Hr(‖u‖2Hr + 2ρ∥∥|ξ |r+1/2 eρ|ξ |u∥∥2).In the fourth inequality we have used the fact that ey  1 + y ey for y  0 while in the fifth inequality we have usethe Sobolev estimate on‖ux‖L∞ . Using Lemma 3.2 in (9) we get:12ddt∥∥|ξ |r eρ|ξ |u∥∥2  c(ρ̇ + 4ρ‖u‖Hr )∥∥|ξ |r+1/2 eρ|ξ |u∥∥2 + 2c‖u‖3Hr . (12)By virtue of the proof of Theorem 3.1 (Section 4 in [17]), which guarantees the existence of a unique global sthere exists a functionθ(t), with θ3 integrable in[0, T ] for eachT > 0, such that‖u‖Hr  θ(t), ∀t ∈ [0,∞). So,provided that:ρ̇ + 6ρ‖u‖Hr  0, which is insured by ρ(t)  ρ(0)exp(−4‖u‖Hr t), (13)664 M.C. Lombardo et al. / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 659–664ticpaper..I (1998)on, Comm.er. Funct.7–66.21–831.ial Equa-7.we get the bound int ∈ [0, T ]:∥∥|ξ |r eρ|ξ |u∥∥2  ∥∥|ξ |r eρ(0)|ξ |u0∥∥2 + 2ct∫0θ3(t ′)dt ′. (14)We stress once again that such a bound can be obtained for anyT > 0. The rigorous construction of global analysolutions is now standard and the proof of Theorem 1.2 is achieved. The details will appear in a forthcomingReferences[1] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (11) (1993) 1661–1664.[2] A. Constantin, On the Cauchy problem for the periodic Camassa–Holm equation, J. Differential Equations 141 (2) (1997) 218–235[3] A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci. 10 (2000) 391–399.[4] A. Constantin, J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) XXV303–328.[5] A. Constantin, J. 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A Note on the analytic solutions of the Camassa–Holm equation

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A Note on the analytic solutions of the Camassa–Holm equation

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