NON-UNIFORM CONTINUITY OF THE FLOW MAP FOR AN EVOLUTION EQUATION MODELING SHALLOW WATER WAVES OF MODERATE AMPLITUDE

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1 This is a preprint of: Non-uniform continuity of the flow map for an evolution equation moeling shallow water waves of moerate amplitue, Nilay Duruk-Mutlubas, Anna Geyer, Bogan-Vasile Matioc, Nonlinear Anal. Real Worl Appl., vol. 1, , DOI: [ /j.nonrwa ] NON-UNIFORM CONTINUITY OF THE FLOW MAP FOR AN EVOLUTION EQUATION MODELING HALLOW WATER WAVE OF MODERATE AMPLITUDE NILAY DURUK MUTLUBAŞ, ANNA GEYER, AND BOGDAN VAILE MATIOC Abstract. We prove that the flow map associate to a moel equation for surface waves of moerate amplitue in shallow water is not uniformly continuous in the obolev space H s with s > 3/2. The main iea is to consier two suitable sequences of smooth initial ata whose ifference converges to zero in H s, but such that neither of them is convergent. Our main theorem shows that the exact solutions corresponing to these sequences of ata are uniformly boune in H s on a uniform existence interval, but the ifference of the two solution sequences is boune away from zero in H s at any positive time in this interval. The result is obtaine by approximating the solutions corresponing to these initial ata by explicit formulae an by estimating the approximation error in suitable obolev norms. 1. Introuction an the main result We consier a moel equation for surface waves of moerate amplitue in shallow water ut + ux + 6uux 6u2 ux + 12u3 ux + uxxx uxxt + 14uuxxx + 28ux uxx = 0, 1 which arises as an approximation of the Euler equations in the context of homogenous, invisci gravity water waves. In recent years, several nonlinear moels have been propose in orer to unerstan some important aspects of water waves, like wave breaking or solitary waves. One of the most prominent examples is the Camassa-Holm CH equation [3], which is an integrable, infinite-imensional Hamiltonian system [1, 4, ]. The relevance of the CH equation as a moel for the propagation of shallow water waves was iscusse by Johnson [19], where it is shown that it escribes the horizontal component of the velocity fiel at a certain epth within the flui; see also [5]. Builing upon the ieas presente in [19], Constantin an Lannes [8] have recently erive the evolution equation 1 as a moel for the motion of the free surface of the wave, an they evince that 1 approximates the governing equations to the same orer as the CH equation. Besies eriving 1, the authors of [8] also establish the local well-poseness results for the Cauchy problem associate to 1. Relying on a semigroup approach ue to Kato [21], Duruk [10] has shown that this feature hols for a larger class of initial ata, as well as for solutions which are spatially perioic [11]. The well-poseness in the context of Besov spaces together with the regularity an the persistance properties of strong solutions are stuie in [26]. imilarly to the CH equation, cf. [6, 25], the moel equation 1 can also capture the phenomenon of wave breaking: for certain initial ata the solution remains boune, but its slope becomes unboune in finite time cf. [8, 11]. Unlike for the CH equation, which is known to posses global solutions, cf. [2, 6], it is not apparent how to control the solutions of 1 globally, ue to the fact that this equation involves higher orer nonlinearities in u Date: December 11, Mathematics ubject Classification. 35B30, 35G25, 35L05. Key wors an phrases. Camassa-Holm equation; flow map; non-uniform continuity; water waves. 1

2 2 N. DURUK MUTLUBAŞ, A. GEYER, AND B. V. MATIOC an its erivatives than the CH equation. On the other han, if one passes to a moving frame, it can be shown that there exist solitary travelling wave solutions ecaying at infinity [14]. Their orbital stability has been recently stuie in [9] using an approach propose by Grillakis, hatah an trauss [15], which takes avantage of the Hamiltonian structure of 1. In the present paper, we consier the Cauchy problem associate to 1 in the setting of perioic functions. From the local well-poseness results [11, 10], we know that its solutions epen continuously on their corresponing initial ata in obolev spaces H s with s > 3/2. Our main result states that this epenence is not uniformly continuous. This property was only recently shown to hol true for the CH equation [16, 1], an was subsequently confirme also for the Euler equations [18] an for several relate hyperbolic problems such as the µ b equation [23], the hyperelastic ro equation [20], for a moifie CH system [24], an for the moifie CH equation [13]. The main ifficulty we encounter compare to all these references is that, as mentione before, our equation has a higher egree of nonlinearity. Nevertheless, we were able two fin two sequences of smooth initial ata whose ifference converges to zero in H s, but such that none of them is convergent, with the corresponing solutions of 1 being uniformly boune on a common nonempty interval of existence. Approximating these solutions by explicit formulae, we then successively estimate the error in suitable obolev norms an use well-known interpolation properties of the obolev spaces an commutator estimates to show that at any time of the common existence interval the ifference of the two sequences of exact solutions is boune from below in the H s -norm by a positive constant. More precisely, enoting by u ; u 0, the unique solution of 1 corresponing to the initial ata u 0 H s with s > 3/2, cf. Theorem 2.1, our main result states: Theorem 1.1 Non-uniform continuity of the flow map. For s > 3/2, the flow map u 0 u ; u 0 : H s C[0, T, H s C 1 [0, T, H s 1 associate to the evolution equation 1 is continuous, but it is not uniformly continuous. More precisely, there exist two sequences of solutions u n n, ũ n n C[0, T u ], H s C 1 [0, T u ], H s 1, where T u > 0, an a positive constant C > 0 with the following properties: but sup max u n H s + ũ n H s C, [0,T u] n N lim u n0 ũ n 0 H s = 0, n lim inf n u nt ũ n t H s C 1 sint for t 0, T u ]. The structure of the paper is as follows: In Theorem 2.1 we recall some properties concerning the well-poseness of 1 from [11] an etermine a lower boun on the existence time of the solution in H s in terms of the initial ata. Then, we introuce two sequences of approximate solutions u n, ω { 1, 1}, an compute the approximation error in Lemma 3.1. The corresponing solutions u of 1 etermine by the initial ata u 0 are then shown to be uniformly boune on a common interval of existence, the absolute error u u being compute in ifferent obolev norms, cf. Lemmas We en the paper with the proof of the main result.

3 NON-UNIFORM CONTINUITY OF THE FLOW MAP 3 Notation. Throughout this paper, we shall enote by C positive constants which may epen only upon s. Furthermore, H r := H r, with r R, is the L2 base obolev space on the circle := R/Z. Given r R, we let Λr := 1 x2 r/2 enote the Fourier multiplier with symbol 1 + k 2 r/2 k Z. It is well-known that Λr : H q H q r is m := W m, an isometric isomorphism for all q, r R. Furthermore, the Banach space W m N, consisting of all boune functions which possess boune weak erivatives of orer less than or equal to n, is enowe with the usual norm. ome useful estimates. The following commutator estimates play a crucial role in our analysis: [Λr, f ]g L2 Cr fx L Λr 1 g L2 + Λr f L2 g L for all r > 3/2, 2 [Λσ x, f ]g L2 C f H s g H σ for s > 3/2 an 1 + σ [0, s]. 3 They hol for all functions f, g an for the commutator [, T ] := T T. The Caleron-Coifman-Meyer estimate 3 follows from Proposition 4.2 in Taylor [30]. The estimate 2 is ue to Kato an Ponce [22, 29]. Aitionally, we shall use the following multiplier inequality C an f H t, g f g H t C f H t g H r H r, cf. e.g. [28]. for t r, r > 1/ The local well-poseness result Using the above notation, we observe that the evolution problem associate to 1 can be renere as the following Cauchy problem: { ut = ux + 14uux + x Λ 2 R for t > 0, 5 u0 = u0, where R := Ru is efine as Ru := u2x 3u4 + 2u3 10u2 2u. 6 Relying upon the local well-poseness results establishe in [11] for the quasilinear Cauchy problem 5, we etermine in the following theorem a lower boun for the maximal existence time of the solutions in terms of obolev norms of the initial ata. Aitionally, we obtain a boun on the H s -norm of the local strong solutions on this particular existence interval. Theorem 2.1. Let s > 3/2 be given. Then, we have: i The problem 5 possesses for each u0 H s a unique maximal solution u ; u0 C[0, T, H s C 1 [0, T, H s 1, whereby T = T u0. Moreover, the flow map u0 u ; u0 : H s C[0, T, H s C 1 [0, T, H s 1 is continuous. ii Given u0 H s, the maximal existence time of the solution u ; u0 of 5 satisfies T > T0 := u0 5H 1 2C 1 + u0 5H 1 u0 3H s where C is a positive constant. iii We have ut H s 2 u0 H s for all t [0, T0 ]. 8

4 4 N. DURUK MUTLUBAŞ, A. GEYER, AND B. V. MATIOC Before proceeing with the proof, one can show by using integration by parts shows [8, 11] that the H 1 -norm of the solutions of 1 is preserve in time when s 2. Base upon this observation an relying on Theorem 2.1 i, we then fin that ut H 1 = u0 H 1 for all u0 H s, s > 3/2, an 0 t < T. Proof of Theorem 2.1. The assertion i follows from the local well-poseness results establishe in [11]. For ii, we first pick u0 H s with u0 = 0 an enote by T the maximal existence time of the associate solution u = u ; u0. In orer to etermine a lower boun for T, we first show that u 2H s satisfies a ifferential inequality. We procee as in [2, 29] an pick a Frierichs mollifier 1 Jε OP, ε 0, 1. ince Jε is itself a Fourier multiplier, the time evolution of the H s -norm of Jε u is given by 1 s 1 Jε u 2H s = Λ Jε u 2L2 = Λs Jε uλs Jε ut x = I1 + I2, 2 t 2 t where I1 := s s Λ Jε uλ Jε uux x, I2 := Λs Jε uλs Jε x Λ 2 R x. The latter equality is base on the observation that s s Λ Jε uλ Jε ux x = Λs Jε u x Λs Jε u x = 0. To estimate the first term, we use the following boun which was erive in Taylor [29], by means of the Kato-Ponce estimate 2: 2 I1 C u W 1 u H s. Employing the Cauchy-chwartz inequality an the algebra property of H r, r > 1/2, the term I2 can be estimate as follows: I2 Jε Λs u L2 Jε Λs x Λ 2 R L2 C u H s R H s 1 C1 + u 5H s. Finally, we combine these estimates an let ε ten to 0 to fin that u 2H s C1 + u 5H s for all t [0, T. 9 t Recalling that the H 1 -norm of u is preserve in time, we get u0 H 1 ut H s for all t [0, T, an together with 9 we fin that We conclue that ut H s 1 + u0 5H 1 u 2H s C u 5H s t u0 5H 1 u0 H s 1 C 1+ u0 5 H1 u0 5 1 H u0 3H s t 1/3 for all t [0, T. for all t < max { 10 } u0 5H 1, T. C 1 + u0 5H 1 u0 3H s It follows that the constant T0 efine by the relation is a lower boun for T, an that ut H s 2 u0 H s for all t T0. This proves the claim. 1Choosing ρ C R with supp ρ 1/2, 1/2 an setting ρ x := ε 1 ρx/ε for ε 0, 1 an x R, ε 0 the mollifier Jε is efine by Jε u := ρε u for all u L2. For every ε 0, 1 an s > 0, we have that Jε : L2 H k is continuous, Jε u u H s ε 0 0 for all u H s, an Jε : L2 L2 is a contraction. Being a Fourier multiplier, Jε commutes with t, x, an Λs, s > 0, cf. e.g [12].

5 NON-UNIFORM CONTINUITY OF THE FLOW MAP 5 3. Approximate solutions for the evolution equation In the following we consier approximate solutions of the evolution equation 1 of the form ωn 1 1 n s cosnx + ωt, 11 u t, x := 14 where ω { 1, 1} an n N \ {0}. When n is very large, the term involving the cosine has a high spatial frequency whereas the other term is constant. Before we estimate the error of these approximate solutions, observe that sinnx α H σ = cosnx α H σ = π1 + n2 σ/2 12 for all α, σ R an n N \ {0}. Inee, the functions ϕn := ein / 2π, n Z, form an orthonormal basis of L2, an therefore a irect computation see also [1, Lemma 1] shows that 2π 2π n2 σ cosnx α 2H σ = cosnx αe inx x + cosnx αeinx x 2π n2 σ iα 2 2 πe = π1 + n2 σ. = + πeiα 2π We emphasize that in contrast to [1], ue to aitional terms appearing in 1 the precise computation of 12 is very important when estimating the norm of u in H s. In view of 12 an noting that 1 H σ = 2π, we obtain the boun u H σ C1 + nσ s for all σ > 0, ω { 1, 1}, an n N \ {0}. ubstituting the approximate solution for the error is foun: u 13 into the equation 1 the following expression E := u u ux x Λ 2 Ru x 14u t 14 Lemma 3.1 Estimating the error of approximate solutions. Given s > 3/2, there is a positive constant C such that { n 2s+1+σ, if 3/2 < s < 2, E H σ C 15 n s 1+σ, if s 2, for all 1/2 < σ 1, ω { 1, 1}, an n N \ {0}. Proof. Observe that E = E1 E2, where n 2s+1 E1 :=u u ux = sin2nx + ωt, x 14u t 28 3 E2 :=Λ 2 14u ux + 6u 2 u ux 2u. x uxx 12u x 20u x Recalling 12 an the fact that u H 1 C for all n 1, ω { 1, 1}, cf. 13, we obtain that E1 H σ Cn 2s+1+σ 3 E2 H σ C u ux H σ 2 + u 2 u x uxx H σ 2 + u x H σ 2 + u ux H σ 2 + ux H σ 2 [ 2s+3 C n sin2nx + ωt H σ 2 3 ] + u H 1 + u 2H 1 + u H u x H σ 2 Cn 2s+1+σ + n s 1+σ, 16 1

6 6 N. DURUK MUTLUBAŞ, A. GEYER, AND B. V. MATIOC where we have employe the multiplier inequality 4. Combining 16 an 1 we obtain the esire conclusion. 4. Error estimates For each ω { 1, 1} an n N \ {0}, we consier the solution u of equation 1 whose initial ata is given by the approximate solution u evaluate at time zero, i.e. u satisfies the equations { t u = u x u + 14u x u + x Λ 2 Ru t > 0, 18 u 0 = u 0. Note that u 0 is boune in H s for any s R. Inee, since u 0 = u 0 an recalling the efinition 11 we fin that u 0 H s which yiels 1 H s ωn 1 H s + n s cosnx H s π 2π lim inf u 0 H s lim sup u 0 H s n 28 n for ω { 1, 1}. Furthermore, we obtain in view of 12 that 2π lim u 0 H 1 = n 14 for ω { 1, 1}. Therefore, if s > 3/2 we may infer from the Theorem 2.1 that there exists an integer n0 1 an positive constants C an Tu T0 u 0, such that 19 u t H s C for all t [0, Tu ], n n0 an ω { 1, 1}. In the following lemma, we fin that the exact solutions u have very nice regularity properties, which allow us to estimate the ifference to the approximate solution u as follows: Lemma 4.1 Estimating the error u u H k. Define k := s + 2. Then, for each ω { 1, 1} an n n0 we have that u C[0, Tu ], H k+1 C 1 [0, Tu ], H k. Moreover, there is a constant CTu > 0 such that max u t u t H k CTu n2 t [0,Tu ] for all ω { 1, 1}, n n0. 20 Proof. Let ω { 1, 1} an let n n0 be arbitrary. For simplicity we set u := u. Because u0 is smooth, we know in view of Theorem 2.1 that the Cauchy problem 18 has a unique maximal solution u C[0, T, H k+1 C 1 [0, T, H k with existence time T := T ω, n. In orer to erive a boun on the absolute error in H k we have to prove first that this aitional regularity hols up to an incluing the time Tu. That is, we have to show that T > Tu for all ω { 1, 1} an n n0. To this en we procee as in the proof of Theorem 2.1 an compute that 2 k 2 u 2H k C u W x Ru L2 in [0, T. 1 u k + u H k Λ H t

7 NON-UNIFORM CONTINUITY OF THE FLOW MAP We stuy the last term more carefully an obtain in view of the commutator estimate 2 that Λk 2 x u2x L2 2 [Λk 2, ux ]uxx L2 + 2 ux Λk 2 uxx L2 C u W 2 u H k 1 + u W 1 u H k, whereas Λk 2 x u L2 C u H k 1, an Λk 2 x up L2 p [Λk 2, up 1 ]ux L2 + p up 1 Λk 2 ux L2 p 1 p 1 C u p 1 u H s + u W 1 u s + u H L u H k 1 W1 for 2 p 4. In view of the embeing H s, C 1 an 19 we fin that u 2H k C u 2H k + u W 2 u H k 1 u H k t Next we employ the well-known interpolation inequality r r/r2 r1 u H r u H2r1 in [0, min{t, Tu }. r r /r2 r1 u H r for r = k 1, r1 = s an r2 = k an obtain that u 2H k 1 u H s u H k for all u H k. Recalling that H k 1, C 2, we arrive at u 2H k C u 2H k in [0, min{t, Tu } t which we may integrate with respect to time to obtain u H k ec Tu u0 H k in [0, min{t, Tu }. 22 This inequality shows that T > Tu for ω { 1, 1} an for all n n0. Inee, assuming to the contrary that T < Tu, then ut H k as t approaches the maximal existence time T of u H k. This is a contraiction to the fact that u is boune in H k in view of 22. Finally, the error estimate 20 is a simple consequence of 22 an of the estimate for all n n0, cf. 13. u0 H k = u 0 H k Cnk s It turns out that estimate 20 can be improve when we choose k = 1 an s 2. The argument relies on the regularity properties erive in the previous Lemma 4.1. Lemma 4.2 Estimating the error u u H 1. Assume that s 2. Then, for all n n0 an ω { 1, 1} we have that max u t u t H 1 CTu n s. t [0,Tu ] 23 Proof. Denoting the ifference between the approximate solution an the exact solution by v := u u, we see that v is a solution of the initial value problem { 2 vt = vx 14vvx + 14u vx + 14u for t > 0, x v + E + x Λ F 24 v0 = 0,

8 8 N. DURUK MUTLUBAŞ, A. GEYER, AND B. V. MATIOC whereby E is the error term efine by 14 an F := Ru Ru = 14u x v x v 2 x 2v 20u v + 10v 2 + 6u 2 v 6u v 2 + 2v 3 12u 3 v + 18u 2 v 2 12u v 3 + 3v 4. In view of the regularity property erive for u in Lemma 4.1, we may apply Λ 2 on both sies of 24 an fin that Λ 2 v t = Λ 2 v x 14Λ 2 vv x + 14Λ 2 u v x + 14Λ 2 u x v + Λ 2 E + x F, an therewith 1 2 t v 2 H = vλ 2 v 1 t x = vλ 2 v x x vλ 2 u x v x + for all t [0, T u ]. Taking into account that vλ 2 v x x = an noting that we fin t v 2 H = 1 = 14 vλ 2 vv x x + 14 vλ 2 u v x x vλ 2 E x + v x F x vv x + v x v xx x = 0 14 vλ 2 vv x x = 14 v 2 v x x + vx 3 x = vx 3 x vx 3 x + 14 vλ 2 u v x x + 14 vλ 2 u x v x + vλ 2 E x v x 14u x v x vx 2 2v 20u v + 10v 2 + 6u 2 v 6u v 2 x v x 2v 3 12u 3 v + 18u 2 v 2 12u v 3 + 3v 4 x vλ 2 u v x x + 14 vλ 2 u x v x + vλ 2 E x v x 14u x v x 20u v + 6u 2 v 6u v 2 x v x 12u 3 v + 18u 2 v 2 12u v 3 x. This leas us to the following inequality t vt 2 H C u 1 x W 1 v 2 H + E 1 H 1 v H u L 2 u x L v 2 H 1 + u L u x L v 3 H + u 1 x L v 4 H. 1 Observing that the relation 13 implies sup [0,Tu] u t H 2 C, we fin together with 19 that t v 2 H C u 1 x W 1 v 2 H + E 1 H 1 v H 1 + u x L v 2 H. 1

9 NON-UNIFORM CONTINUITY OF THE FLOW MAP 9 Taking now into account the estimates 1 s u x L Cn 2 s u 1 Cn x W an for n n0 an ω { 1, 1}, we obtain in view of the error estimate 15 in Lemma 3.1 that v 2H 1 C v 2H 1 + n s v H 1. t The latter estimate leas us to v H 1 C v H 1 + n s in [0, Tu ], t the esire estimate 23 following in view of Gronwall s inequality an since v0 = 0. Before proving the main result, we show the analog of Lemma 4.2 in the situation when 3/2 < s < 2. The regularity properties erive in Lemma 4.1 are once again essential. Lemma 4.3 Estimating the error u u H σ. Let 3/2 < s < 2. For all n n0, ω { 1, 1} an 1/2 < σ s 1 we have that max u t u t H σ CTu n s. 25 t [0,Tu ] Proof. In this case, we interpret the function v = u u as a solution of the initial value problem { vt = vx + u + u vx + E + x Λ 2 G for t > 0, 26 v0 = 0, with E given by 14 an with G := Ru Ru. It is useful to bring G in the following form G = u + u x vx 3v u 3 + u 2 u + u u 2 + u 3 + 2v u 2 + u u + u 2 10v u + u 2v. In view of 26, we have 1 v 2H σ = Λσ vλσ vt x = Λσ vλσ vx x + Λσ vλσ E x 2 t σ σ + Λ vλ u + u vx x + Λσ vλσ x Λ 2 G x. The first term in the previous equation vanishes Λσ vλσ vx x = 0, while applying the Cauchy-chwarz inequality for the secon an fourth term we obtain the estimates Λσ vλσ E L1 v H σ E H σ Λσ vλσ x Λ 2 G L1 v H σ G H σ 1. To erive a boun for the thir term, we use the Caleron-Coifman-Meyer type estimate 3. We first commute the operator Λσ x with the function u + u an obtain Λσ vλσ u + u vx x = Λσ vu + u Λσ x v x + Λσ v[λσ x, u + u ]v x.

10 10 N. DURUK MUTLUBAŞ, A. GEYER, AND B. V. MATIOC After integrating by parts, we estimate the first integral as follows Λ σ vu + u Λ σ x v x x u + u L v 2 H σ. To estimate the secon integral, we apply the Cauchy-chwarz inequality an then use the estimate 3 to fin Λ σ v[λ σ x, u + u ]v x Λ σ v L 2 [Λ σ x, u + u ]v L2 C u + u H s v 2 H σ. In view of the bouneness of the family {max [0,Tu] u + u H s : n n 0, ω = ±1} we may combine the preceing estimates an obtain that Λ σ vλ σ u + u v x L1 C v 2 H σ. The latter argument an the multiplier inequality 4 show that G H σ 1 C v H σ, an together with the error boun 15 obtaine in Lemma 3.1 we conclue that Whence, t v 2 H σ C v 2 H σ + n 2s+1+σ v H σ. t v H σ C v H σ + n 2s+1+σ in [0, T u ], an the conclusion follows, as in Lemma 4.2, by taking into account that 2s σ s for all σ 1/2, s 1]. 5. Proof of the main result In the remaining part we prove that the functions u n := u 1,n+n0 an ũ n := u 1,n+n0, n N, satisfy all the properties require in Theorem 1.1. Recalling the estimate 19, which ensures that the strong solutions u ±1,n, n n 0, are boune in H s, proves the first claim sup max u 1,nt H s + u 1,n t H s C, t [0,T u] n n 0 where T u is the constant introuce right before Lemma 4.1. The secon assertion follows by taking into account the efinition of the approximate solutions 11, which yiels u 1,n 0 u 1,n 0 H s = 2n H s n 0. To show that the thir claim of Theorem 1.1 hols, we have to erive a ecay estimate for the ifference between the two unknown exact solutions. The trick is to work with inequalities involving the estimates for the absolute errors euce in the preceing lemmas. We assume first that s 2, an observe that u 1,n t u 1,n t H s u 1,n t u 1,n t H s u 1,n t u 1,n t H s u 1,n t u 1,n t H s 2

11 NON-UNIFORM CONTINUITY OF THE FLOW MAP 11 for all t [0, Tu ] an n n0. Now we fin lower bouns for each of these three terms. A simple calculation yiels that 1 u1,n t u 1,n t H s = 2n 1 n s cosnx + t cosnx t H s 14 n s 2πn 1 s sint sinnx H 1 π sint 2πn 28 for all n n0 an t [0, Tu ], where we have use 12 in the last inequality. To estimate the secon an thir term in 2, we apply the interpolation inequality 21 with r = s, r1 = 1 an r2 = k, an fin 2 s 1 k 1 k 1 ±1,n t u±1,n t H u±1,n t u±1,n t H s u±1,n t u±1,n t H 1 u k 2 29 CTu n k 1 for all n n0, in view of the estimates 23 an 25 obtaine in Lemma 4.1 an 4.2, respectively. Gathering 28 an 29, we obtain that 2 π sint 2πn 1 u1,n t u 1,n t H s CTu n k 1, for all t [0, Tu ] an n n0. Finally, we let n to complete the proof in the case s 2. For 3/2 < s < 2 we note that the estimate 28 is still vali, whereas the analog of 29 hols in view of Lemma 4.3. Inee, we fin that 2 s σ k σ k σ ±1,n t u±1,n t H u±1,n t u±1,n t H s u±1,n t u±1,n t H σ u k 2σ CTu n k σ, where σ 1/2, s 1] is fixe an n n0. The final argument of the proof is analogous to the one presente in the case when s 2. Acknowlegements. A. Geyer was supporte by the FWF project J 3452 Dynamical ystems Methos in Hyroynamics of the Austrian cience Fun. References [1] A. Boutet e Monvel, A. Kostenko, D. hepelsky, an G. Teschl. Long-time asymptotics for the Camassa-Holm equation. IAM J. Math. Anal., 414: , [2] A. Bressan an A. Constantin. Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal., 1832: , 200. [3] R. Camassa an D. D. Holm. An integrable shallow water equation with peake solitons. Phys. Rev. Lett., 111: , [4] A. Constantin. On the scattering problem for the Camassa-Holm equation. R. oc. Lon. Proc. er. A Math. Phys. Eng. ci., :953 90, [5] A. Constantin. Nonlinear Water Waves with Applications to Wave-Current Interactions an Tsunamis, volume 81 of CBM-NF Conference eries in Applie Mathematics. IAM, Philaelphia, [6] A. Constantin an J. Escher. Well-poseness, global existence, an blowup phenomena for a perioic quasi-linear hyperbolic equation. Comm. Pure Appl. Math., 515:45 504, [] A. Constantin, V.. Gerjikov, an R. I. Ivanov. Inverse scattering transform for the Camassa-Holm equation. Inverse Problems, 226: , [8] A. Constantin an D. Lannes. The hyroynamical relevance of the Camassa-Holm an DegasperisProcesi equations. Arch. Ration. Mech. Anal., 1921: , 2009.

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