Luc Molinet
email: luc.molinet@univ-tours.fr

ASYMPTOTIC STABILITY FOR SOME NON POSITIVE PERTURBATIONS OF THE CAMASSA-HOLM PEAKON WITH APPLICATION TO THE ANTIPEAKON-PEAKON PROFILE

Keywords: Mathematics Subject Classification. 35Q35, 35Q51, 35B40 Camassa-Holm equation, asymptotic stability, peakon, antipeakonpeakon

We continue our investigation on the asymptotic stability of the peakon . In a first step we extend our asymptotic stability result [34] in the class of functions whose negative part of the momentum density is supported in ] -∞, x 0 ] and the positive part in [x 0 , +∞[ for some x 0 ∈ R. In a second step this enables us to prove the asymptotic stability of well-ordered train of antipeakons-peakons and, in particular, of the antipeakon-peakon profile. Finally, in the appendix we prove that in the case of a non negative momentum density the energy at the left of any given point decays to zero as time goes to +∞. This leads to an improvement of the asymptotic stability result stated in [34].

Introduction

In this paper we continue our investigation of the asymptotic stability of the peakon for the Camassa-Holm equation (C-H) by studying a particular case of solutions with a non signed momentum density. This leads to an asymptotic stability result for the antipeakon-peakon profile with respect to some perturbations.

Recall that the Camassa-Holm equation reads

(1.1)

u t -u txx = -3uu x + 2u x u xx + uu xxx , (t, x) ∈ R 2 ,
and can be derived as a model for the propagation of unidirectional shalow water waves over a flat bottom ( [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF], [START_REF] Johnson | Camassa-Holm, Korteweg-de Vries and related models for water waves[END_REF]). A rigorous derivation of the Camassa-Holm equation from the full water waves problem is obtained in [START_REF] Alvarez-Samaniego | Large time existence for 3D water-waves and asymptotics[END_REF] and [13].

(C-H) is completely integrable (see [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF],[8], [10] and [START_REF] Constantin | Inverse scattering transform for the Camassa-Holm equation[END_REF]) and enjoys also a geometrical derivation (cf. [START_REF] Kolev | Lie groups and mechanics: an introduction[END_REF], [30]). It possesses among others the following invariants

E E (1.2) M (v) = R (v-v xx ) dx, E(v) = R v 2 (x)+v 2 x (x) dx and F (v) = R v 3 (x)+v(x)v 2 x (x) dx
and can be written in Hamiltonian form as

(1.3) ∂ t E ′ (u) = -∂ x F ′ (u) .
It is also worth noticing that (1.6) can be rewritted as

(1.4) y t + uy x + 2u x y = 0 that is a transport equation for the momentum density y = u -u xx .

Camassa and Holm [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] exhibited peaked solitary waves solutions to (C-H) that are given by u(t, x) = ϕ c (x -ct) = cϕ(x -ct) = ce -|x-ct| , c ∈ R.

They are called peakon whenever c > 0 and antipeakon whenever c < 0, the profile ϕ c being the unique H 1 -weak solution of the differential equation eqpeakon eqpeakon (1.5) 2 . Note that the initial value problem associated with (C-H) has to be rewriten as CH CH (1.6)

-cϕ c + cϕ ′′ c + 3 2 ϕ 2 c = ϕ c ϕ ′′ c + 1 2 (ϕ ′ c )
u t + uu x + (1 -∂ 2 x ) -1 ∂ x (u 2 + u 2 x /2) = 0 u(0) = u 0 ,
to give a meaning to these solutions.

Their stability seems not to enter the general framework developed for instance in [START_REF] Benjamin | The stability of solitary waves[END_REF], [22], especially because of the non smoothness of the peakon. However, Constantin and Strauss [START_REF] Constantin | Stability of peakons[END_REF] succeeded in proving their orbital stability by a direct approach.

In [34], making use of the finite speed propagation of the momentum density, we proved a rigidity result for solutions with a non negative momentum density. Following the framework developed by Martel and Merle (see for instance [32], [33]) this leads to an asymptotic stability result for the peakon with respect to non negative perturbations of the momentum density.

In this paper we continue our study of the asymptotic stability of the peakons of the Camassa-Holm equation. In a first part we extend our result in [34] on the asymptotic stability result for non negative initial momentum density y 0 ∈ M + by considering the case where supp y - 0 ⊂] -∞, x 0 ] and supp y + 0 ⊂ [x 0 , +∞[ for some x 0 ∈ R. Here y - 0 and y + 0 denote respectively the negative and positive part of y 0 . In a second part we use this last result to prove an asymptotic stability result for well-ordered train of antipeakons and peakons.

It is well-known (cf. [START_REF] Wahlén | Global existence of weak solutions to the Camassa-Holm equation[END_REF]) that under the above hypothesis on y 0 , the solution u exists for all positive times in Y and that there exists a C 1 -function t → x 0 (t) such that for all non negative times, supp y -(t) ⊂] -∞, x 0 (t)] and supp y + (t) ⊂ [x 0 (t), +∞[. On the other hand, in contrast to the case of a non negative momentum density, no uniform in time bound on the total variation of y is known in this case. Actually, only an exponential estimate on y(t) M has been derived (see (2.14)). The main novelty of this work is the proof that, also in this case, the constructed asymptotic object has a non negative momentum density and is Y -almost localized. To prove our result we separate two possible behaviors of x 0 (t).The first case corresponds to the case where the point x 0 (t) travels far to the left of the bump (at least for large times). Then locally around the bump, the momentum density is non negative and we can argue as in the case of a non negative momentum density. The boundedness of the total variation of y around the bump being proved by using an almost decay result of E(•) + M (•) at the right of a point that travels to the right between x 0 (•) and the bump but far away from these last ones. The second case is more involved and corresponds to the case where x 0 (t) stays always close to the bump. Then we prove that the total variation of y has to decay exponentially fast in time in a small interval at the left of x 0 (•). This enables us to prove an uniform in time estimate on the total variation of y + (t) and thus of y(t) by using the conservation of M (u) along the trajectory. We can thus pass to the limit on a sequence of times. The fact that the asymptotic object has a non negative momentum density follows by extending the decay of the total variation of y(t) to an interval that grows with time.

At this stage we would like to emphasize that we are not able to consider antipeakon-peakon collisions. Actually, our hypothesis forces the antipeakons to be initially at the left of the peakons and this property is preserved for positive times. Recently, Bressan and Constantin ([5], [START_REF] Bressan | Global dissipative solutions of the Camassa-Holm equation[END_REF]) succeed to construct global conservative and dissipative solutions of the (1.6) for initial data in H 1 (R) by using scalar conservation laws techniques (see also [START_REF] Holden | Global conservative solutions of the Camassa-Holm equation -a Lagrangian point of view[END_REF], [START_REF] Holden | Dissipative solutions for the Camassa-Holm equation[END_REF] for a slightly different point of view). These class of solutions take into account the antipeakon-peakon collisions which are studied in more details in [START_REF] Holden | Global conservative multipeakon solutions of the Camassa-Holm equation[END_REF] and [START_REF] Holden | Global dissipative multipeakon solutions for the Camassa-Holm equation Com[END_REF] (see also [START_REF] Grunert | The general peakon-antipeakon solution for the Camassa-Holm equation[END_REF] and references therein).

1.1. Statement of the results. Before stating our results let us introduce the function space where our initial data will take place. Following [START_REF] Constantin | Global weak solutions for a shallow water equation[END_REF], we introduce the following space of functions

(1.7) Y = {u ∈ H 1 (R) such that u -u xx ∈ M(R)} .
We denote by Y + the closed subset of Y defined by

Y + = {u ∈ Y / u -u xx ∈ M + }
where M + is the set of non negative finite Radon measures on R.

Let C b (R) be the set of bounded continuous functions on R, C 0 (R) be the set of continuous functions on R that tends to 0 at infinity and let I ⊂ R be an interval. A sequence {ν n } ⊂ M is said to converge tightly (resp. weakly) towards ν ∈ M if for any φ ∈ C b (R) (resp. C 0 (R)), ν n , φ → ν, φ . We will then write ν n ⇀ * ν tightly in M (resp. ν n ⇀ * ν in M).

Throughout this paper, y ∈ C ti (I; M) (resp. y ∈ C w (I; M)) will signify that for any φ ∈ C b (R) (resp. φ ∈ C 0 (R)) , t → y(t), φ is continuous on I and y n ⇀ * y in C ti (I; M) (resp. y n ⇀ * y in C w (I; M)) will signify that for any

φ ∈ C b (R) (resp. C 0 (R)), y n (•), φ → y(•), φ in C(I).
As explained above, the aim of this paper is to extend the results in [34] under the following hypothesis. hyp Hypothesis 1. We will say that u 0 ∈ Y satisfies Hypothesis 1 if there exists x 0 ∈ R such that its momentum density y 0 = u 0 -u 0,xx satisfies hyp1 hyp1

(1.8) supp y - 0 ⊂] -∞, x 0 ] and supp y + 0 ⊂ [x 0 , +∞[ . As in [34],
the key tool to prove our results is the following rigidity property for Y -almost localized solutions of (1.6) with non negative density momentum : (1.6) that is not identically vanishing. Then there exists c * > 0 and x 0 ∈ R such that

liouville Theorem 1.1 ([34]). Let u ∈ C(R; H 1 (R)), with u -u xx ∈ C w (R; M + ), be a Y - almost localized solution of
u(t) = c * ϕ(• -x 0 -c * t), ∀t ∈ R .
Recall that a Y -almost localized solution is defined in the following way :

defYlocalized Definition 1.1. We say that a solution u ∈ C(R; H 1 (R)) with u-u xx ∈ C w (R; M + ) of (1.6) is Y -almost localized if there exist c > 0 and a C 1 -function x(•), with
x t ≥ c > 0, for which for any ε > 0, there exists R ε > 0 such that for all t ∈ R and all

Φ ∈ C(R) with 0 ≤ Φ ≤ 1 and supp Φ ⊂ [-R ε , R ε ] c . defloc defloc (1.9) R (u 2 (t) + u 2 x (t))Φ(• -x(t)) dx + Φ(• -x(t)), u(t) -u xx (t) ≤ ε .
Our first new result is the asymptotic stability of the peakon with respect to perturbations that satisfy Hypothesis 1 : asympstab Theorem 1.2. Let c > 0 be fixed. There exists an universal constant 0 < η ≤ 2 -10 such that for any 0 < θ < c and any u 0 ∈ Y , satisfying Hypothesis 1., such that difini difini

(1.10) u 0 -ϕ c H 1 ≤ η θ c 8 , there exists c * > 0 with |c -c * | ≪ c and a C 1 -function x : R → R with lim t→∞ ẋ(t) = c * such that (1.11) u(t, • + x(t)) ⇀ t→+∞ ϕ c * in H 1 (R) ,
where u ∈ C(R; H 1 ) is the solution emanating from u 0 . Moreover, cvforte cvforte

(1.12) lim t→+∞ u(t) -ϕ c * (• -x(t)) H 1 (|x|>θt) = 0 .
Remark 1.1. We emphasize that (1.12) gives a strong H 1 -convergence result at the left of the line x = -θt. This is due to the fact that, contrary to KdV type equation, there is no small linear waves travelling to the left for the Camassa-Holm equation.

In the appendix we even show that in the case of a non negative momentum density, all the energy is traveling to the right.

Combining this asymptotic stability result with the one obtained in [34] and the orbital stability of well ordered trains of antipeakons-peakons proven in [21], we are able to prove the asymptotic stability of such trains that contains, as a particular case, the asymptotic stability of the antipeakon-peakon profile.

asympt-mult-peaks Theorem 1.3. Let be given

N -negative velocities c -N-< .. < c -2 < c -1 < 0, N + positive velocities 0 < c 1 < c 2 < .. < c N+ and 0 < θ 0 < min(|c -1 |, c 1 )/4. There exist L 0 > 0 and ε 0 > 0 such that if the initial data u 0 ∈ Y satisfies Hypothesis 1 with inini inini (1.13) u 0 - N- j=1 ϕ c-j (• -z 0 -j ) - N+ j=1 ϕ cj (• -z 0 j ) H 1 ≤ ε 2 0 ,
for some

z 0 N-< •• < z 0 -1 < z 0 1 < •• < z 0 N+ with |z 0 j -z 0 q | ≥ L 0 for j = q , then there exist c * -N-< .. < c * -2 < c * -1 < 0 < c * 1 < .. < c * N+ with |c * j -c j | ≪ |c j | and C 1 -functions t → x j (t), with ẋj (t) → c * j as t → +∞, j ∈ [[-N -, N + ]]/{0}, such that the solution u ∈ C(R + ; H 1 (R)) of (1.6) emanating from u 0 satisfies mul1 mul1 (1.14) u(• + x j (t)) ⇀ t→+∞ ϕ c * j in H 1 (R), ∀j ∈ [[-N -, N + ]]/{0} . Moreover, mul2 mul2 (1.15) u - N- j=1 ϕ c * -j (• -x -j (t)) - N+ j=1 ϕ c * j (• -x j (t)) -→ t→+∞ 0 in H 1 |x| > θ 0 t .
This paper is organized as follows : in the next section we recall the wellposedness results for the class of solutions we will work with. In Section 3, we derive an almost monotonicity result that is a straightforward adaptation of the one proven in [34]. Section 4 is devoted to the proof of the Y -almost localization of the elements of the ω-limit set of the orbits associated with initial data satisfying Hypothesis 1 and being H 1 -close enough to a peakon. This is the main new contribution of this paper. In Section 5 we deduce the asymptotic stability results. Finally in the appendix we present an improvement of the asymptotic stability result given in [34] by noticing that all the energy of any solution to (1.6) with a non negative density momentum is traveling to the right. Moreover, for such a solution, the energy at the the left of any given point decays to zero as t → +∞.

Global well-posedness results

We first recall some obvious estimates that will be useful in the sequel of this paper. Noticing that p(x) = 1 2 e -|x| satisfies p * y = (1 -∂ 2 x ) -1 y for any y ∈ H -1 (R) we easily get

u W 1,1 = p * (u -u xx ) W 1,1 u -u xx M and u xx M ≤ u L 1 + u -u xx M which ensures that bv bv (2.1) Y ֒→ {u ∈ W 1,1 (R) with u x ∈ BV (R)} .
It is also worth noticing that for v ∈ C ∞ 0 (R), satisfying Hypothesis 1, formulev formulev

(2.2) v(x) = 1 2 x -∞ e x ′ -x (v -v xx )(x ′ )dx ′ + 1 2 +∞ x e x-x ′ (v -v xx )(x ′ )dx ′ and v x (x) = - 1 2 x -∞ e x ′ -x (v -v xx )(x ′ )dx ′ + 1 2 +∞ x e x-x ′ (v -v xx )(x ′ )dx ′ ,
so that for x ≤ x 0 we get

v x (x) = v(x) -e -x x -∞ e x ′ y(x ′ ) dx ′ ≥ v(x)
whereas for x ≥ x 0 we get

v x (x) = -v(x) + e x +∞ x e -x ′ y(x ′ ) dx ′ ≥ -v(x)
Throughout this paper, we will denote {ρ n } n≥1 the mollifiers defined by rho rho

(2.3) ρ n = R ρ(ξ) dξ -1 nρ(n•) with ρ(x) = e 1/(x 2 -1) for |x| < 1 0 for |x| ≥ 1
Following [START_REF] Wahlén | Global existence of weak solutions to the Camassa-Holm equation[END_REF] we approximate v ∈ Y satisfying Hypothesis 1 by the sequence of functions app app

(2.4) v n = p * y n with y n = -(ρ n * y -)(• + 1 n ) + (ρ n * y + )(• - 1 n ) and y = v -v xx
that belong to Y ∩ H ∞ (R) and satisfy Hypothesis 1 with the same x 0 . It is not too hard to check that estyn estyn

(2.5)

y n L 1 ≤ y M Moreover, noticing that v n = -ρ n * (p * y -) (• + 1 n ) + ρ n * (p * y + ) (• - 1 n ) with p * y ∓ ∈ H 1 (R) ∩ W 1,1 (R), we infer that mm mm (2.6) v n → v ∈ H 1 (R) ∩ W 1,1 (R) .
that ensures that for any v ∈ Y satisfying Hypothesis 1 it holds dodo dodo

(2.7)

v x ≥ v on ] -∞, x 0 [ and v x ≥ -v on ]x 0 , +∞[ . Proposition 2.1.
(Global weak solution [START_REF] Wahlén | Global existence of weak solutions to the Camassa-Holm equation[END_REF])

prop1
Let u 0 ∈ Y satisfying Hypothesis 1 for some x 0 ∈ R.

1. Uniqueness and global existence : (1.6) has a unique solution

u ∈ C(R + ; H 1 (R)) ∩ C 1 (R + ; L 2 (R)) such that y = (1 -∂ 2 x )u ∈ C ti (R + ; M). E(u), F (
u) and M (u) = y, 1 are conservation laws . Moreover, for any t ∈ R + , the density momentum y(t) satisfies supp y -(t) ⊂] -∞, q(t, x 0 )] and supp y + (t) ⊂ [q(t, x 0 ), +∞[ where q(•, •) is defined by

defq defq (2.8) q t (t, x) = u(t, q(t, x)) , (t, x) ∈ R 2 q(0, x) = x , x ∈ R .
2. Continuity with respect to the H 1 -norm: For any sequence {u 0,n } bounded in Y that satisfy Hypothesis 1 and such that u 0,n → u 0 in H 1 (R), the emanating sequence of solutions {u n } ⊂ C 1 (R + ; L 2 (R))∩C(R + ; H 1 (R)) satisfies for any T > 0 cont1 cont1

(2.9)

u n → u in C([0, T ]; H 1 (R)) .
Moreover, if {u 0,n } is the sequence defined by (2.4)

then cont2 cont2 (2.10) (1 -∂ 2 x )u n ⇀ * y in C ti ([0, T ], M) .
3. Continuity with respect to initial data in Y equipped with its weak topology: Let {u 0,n } be a bounded sequence of Y such that u 0,n ⇀ u 0 in H 1 (R) and such that the emanating sequence of solution

{u n } is bounded in C([-T -, T + ]; H 1 )∩ L ∞ (-T -, T + ; Y ) for some (T -, T + ) ∈ (R + ) 2 . Then the solution u of (1.6) ema- nating from u 0 belongs to C([-T -, T + ]; H 1 ) with y = u -u xx ∈ C w ([-T -, T + ]; M). Moreover, weakcont weakcont (2.11) u n ⇀ n→∞ u in C w ([-T -, T + ]; H 1 (R)) ,
and cont22 cont22 (2.12) (1 -∂ 2 x )u n ⇀ * y in C w ([-T -, T + ], M) .
Proof. For sake of completeness let us recall that the global existence result follows from the following estimate on the density momentum y of smooth solutions with initial data u 0 ∈ Y that satisfies Hypothesis 1 :

d dt R y + (t, x) dx = d dt +∞ q(t,x0) y(t, x) dx = - +∞ q(t,x0) u x (t, x)y(t, x) dx ≤ sup x∈R (-u x (t, x)) +∞ q(t,x0) y(t, x) dx ≤ sup x∈R (|u(t, x|)) R y + (t, x) dx ≤ E(u 0 ) R y + (t, x) dx ,
where one used (1.4), (2.7), the conservation of the energy and the classical Sobolev inequality :

sobo sobo (2.13) v L ∞ ≤ 1 √ 2 v H 1 , ∀v ∈ H 1 (R) .
Indeed, then by Gronwall estimate and the conservation of M one gets yL1 yL1

(2.14)

y L 1 ≤ 2 exp( E(u 0 )t) y 0 L 1
which ensures that the associated solution u can be extended for all positive times. Finally the global existence result for u 0 ∈ Y follows by approximating u 0 as in (2.4) and proceeding as in [START_REF] Constantin | Global weak solutions for a shallow water equation[END_REF]. In this way, the uniqueness and global existence results are obtained in [START_REF] Wahlén | Global existence of weak solutions to the Camassa-Holm equation[END_REF] except the conservation of M (u) and the fact that y belongs to C ti (R+; M). In [START_REF] Wahlén | Global existence of weak solutions to the Camassa-Holm equation[END_REF], only the fact that y ∈ L ∞ loc (R + , M) is stated. But these properties will follow directly from (2.10) since M (u) is a conservation law for any smooth solution

u ∈ C(R + ; H 3 ) with u -u xx ∈ L ∞ loc (R + ; L 1 (R)).
To prove (2.9), it suffices to notice that, according to the conservation of the H 1norm and (2.14), the sequence of emanating solution

{u n } is bounded in C(R + ; H 1 (R))∩ L ∞ (]0, T [; W 1,1 (R)) with {u n,x } bounded in L ∞ (]0, T [; BV (R)), for any T > 0. Therefore, there exists v ∈ L ∞ (R + ; H 1 (R)) with (1 -∂ 2 x )v ∈ L ∞ loc (R + ; M(R)) such that, for any T > 0, u n ⇀ n→∞ v ∈ L ∞ (]0, T [; H 1 (R)) and (1-∂ 2 x )u n ⇀ n→∞ * (1-∂ 2 x )v in L ∞ (]-T, T [; M(R)) . Moreover, in view of (1.6), {∂ t u n } is bounded in L ∞ (]0, T [; L 2 (R) ∩ L 1 (R)) and
Helly's, Aubin-Lions compactness and Arzela-Ascoli theorems ensure that v is a solution to (1.6) that belongs to

C w ([0, T ]; H 1 (R)) with v(0) = u 0 and that (2.10) holds. In particular, v t ∈ L ∞ (]0, T [; L 2 (R)) and thus v ∈ C([0, T ]; L 2 (R)). Since v ∈ L ∞ (]0, T [; H 3 2 -(R)), this actually implies that v ∈ C([0, T ]; H 3 2 -(R))
. Therefore, v belongs to the uniqueness class which ensures that v = u. The conservation of E(•) and the above weak convergence results then lead to (2.9).

To prove (2.10) we use the following W 1,1 -Lipschitz bound that is proven in [START_REF] Constantin | Global weak solutions for a shallow water equation[END_REF] and [START_REF] Wahlén | Global existence of weak solutions to the Camassa-Holm equation[END_REF] :

Let u i 0 ∈ Y for i = 1, 2 and let u i ∈ C([0, T ]; H 1 ) ∩ L ∞ (0, T ; Y ) be the associated solution of (1.6). Setting M = 2 i=1 u i -u i xx L ∞ (0,T ;M) , it holds lip lip (2.15) u 1 -u 2 L ∞ (0,T ;W 1,1 ) e 6MT u 1 0 -u 2 0 W 1,1 .
Let {u 0,n } the sequence defined by (2.4). The sequence of emanating solutions

{u n } is included in C(R + ; H ∞ ) ∩ C(R + ; W 1,1
). Moreover in view of (2.5), (2.6) and (2.15), for any T > 0, {u n } is a Cauchy sequence in C([0, T ]; W 1,1 ) and thus convW convW

(2.16) u n → u ∈ C(R + ; W 1,1
) . Now, we notice that for any v ∈ BV (R) and any φ ∈ C 1 (R), it holds

v ′ , φ = -vφ ′ .
Therefore, setting

y n = u n -u n,xx , (2.16) ensures that, for any t ∈ R + , R y n φ = R (u n φ + u n,x φ ′ ) → R (uφ + u x φ ′ ) = y, φ
and thus y n (t) ⇀ * y(t) tightly in M. Using again that {∂ t u n } is bounded in L ∞ (0, T ; L 1 ), Arzela-Ascoli theorem leads then to (2.10). Finally (2.11)-(2.12) can be proven exactly in the same way, since u 0 ∈ Y and

{u n } is bounded in L ∞ (] - T -, T + [; Y ) by hypotheses.

Monotonicity results

We have to prove a monotonicity result for our solutions. The novelty with respect to the monotonicity result proven in [34] is that we only require the momentum density to be non negative at the right of some curve. This is possible since, the differential equation satisfied by y being local and of order 1, we may test y with a test function Φ that vanishes on R -. Note that this is not possible to use such test function for the energy since the non local term contained in the time derivative of the energy density imposes to require a condition similar to (3.4) on the test function.

As in [33], we introduce the C ∞ -function Ψ defined on R by defPsi defPsi

(3.1) Ψ(x) = 2 π arctan exp(x/6)
It is easy to check that Ψ(-•) = 1 -Ψ on R, Ψ ′ is a positive even function and that there exists C > 0 such that ∀x ≤ 0, psipsi psipsi

(3.2) |Ψ(x)| + |Ψ ′ (x)| ≤ C exp(x/6) .
Moreover, by direct calculations, it is easy to check that psi3 psi3

(3.3) |Ψ ′′′ | ≤ 1 2 Ψ ′ on R
and that po po

(3.4) Ψ ′ (x) ≥ Ψ ′ (2) = 1 3π e 1/3 1 + e 2/3 , ∀x ∈ [0, 2] .
We also introduce the function Φ defined by defPhi defPhi (1.6), emanating from an initial datum u 0 ∈ Y that satisfies Hypothesis 1, such that there exist x : R

(3.5) Φ(x) =    0 for x ≤ 0 x/2 for x ∈ [0, 2] 1 for x ≥ 1 Lemma 3.1. Let 0 < α < 1 and let u ∈ C(R; H 1 ) with y = (1-∂ 2 x )u ∈ C w (R; M+) be a solution of
+ → R of class C 1 with inf R ẋ ≥ c 0 > 0 and R 0 > 0 with loc loc (3.6) u(t) L ∞ (|x-x(t)|>R0) ≤ (1 -α)c 0 2 6 , ∀t ∈ R + . For 0 < β ≤ α, 0 ≤ γ ≤ 1 3π(1+e 2/3 ) (1 -α)c 0 , R > 0, t 0 ∈ R + and any C 1 -function z : R + → R such that condz condz (3.7) (1 -α) ẋ(t) ≤ ż(t) ≤ (1 -β) ẋ(t), ∀t ∈ R + ,
we set

defI defI (3.8) I ∓R t0 (t) = u 2 (t) + u 2 x (t), Ψ • -z ∓R t0 (t) + γ y(t), Φ • -z ∓R t0 (t)
, 6 , ∀t ≥ t 0 , for some constant K 0 > 0 that only depends on E(u), c 0 , R 0 and β.

where condz2 condz2 (3.9) z ∓R t0 (t) = x(t 0 ) ∓ R + z(t) -z(t 0 ) Let also q(•, •) be defined as in (2.8). Then if zo1 zo1 (3.10) z +R t0 (t) ≥ q(t, x 0 ), for 0 ≤ t ≤ t 0 , it holds mono mono (3.11) I +R t0 (t 0 ) -I +R t0 (t) ≤ K 0 e -R/6 , ∀0 ≤ t ≤ t 0 whereas if zo2 zo2 (3.12) z -R t0 (t) ≥ q(t, x 0 ), for t ≥ t 0 it holds mono2 mono2 (3.13) I -R t0 (t) -I -R t0 (t 0 ) ≤ K 0 e -R/
Proof. The proof is the same as in [34]. The only difference is that we replace Ψ by Φ to test the density momentum y.

We first approximate u 0 by the sequence {u 0,n } ∈ Y ∩ C ∞ (R) as in (2.4). Then, by Proposition 2.1, the emanating solutions u n exist for all positive time and satisfy (2.14). In particular, for any T 0 > 0 fixed,

u n,x L ∞ (]0,T0[×R) ≤ sup t∈]0,T0[ y n (t) L 1 (R) ≤ 2 exp(2 E(u 0 )T 0 ) y 0 M
and thus for any ε > 0 there exists

θ = θ ε,T0 > 0 such that if u n -u L ∞ (]0,T0[×R) < θ ε,T0 then qqq qqq (3.14) |q n (t, x 0 ) -q(t, x 0 )| < ε , ∀t ∈ 0, T 0 ] .
Now, for any fixed t 0 , T > 0, (2.9) ensures that there exists n 0 = n 0 (t 0 + T ) ≥ 0 such that for any n ≥ n 0 ,

u n -u L ∞ (]0,t0+T [×R) < max αc 0 2 6 , θ 1 2 ,t0+T
. which together with (3.6) and (3.14) force dif dif (3.15) sup

t∈]0,t0+T [ u n L ∞ (|x-x(t)|>R0) < (1 -α)c 0 2 5 and difq difq (3.16) sup t∈]0,t0+T [ |q n (t, x 0 -q(t, x 0 )| ≤ 1 2 .
We first prove that (3.11) holds on [0, t 0 ] with u replaced by u n for n ≥ n 0 . The following computations hold for u n with n ≥ n 0 but , to simplify the notation, we drop the index n. According to [34] we have go go

(3.17) d dt R (u 2 + u 2 x )g = R uu 2 x g ′ + 2 R uhg ′ ,
where

h := (1 -∂ 2 x ) -1 (u 2 + u 2 x /2) and d dt R yg dx = R yug ′ + 1 2 R (u 2 -u 2 x )g ′ . go2 (3.18) Applying (3.17) with g(t, x) = Ψ(x -z R t0 (t)) and (3.18) with g(t, x) = Φ(x -z R t0 (t)) we get d dt I +R t0 (t) = -ż(t) R Ψ ′ (u 2 + u 2 x ) + γΦ ′ y + γ 2 R (u 2 -u 2 x )Φ ′ + R Ψ ′ uu 2 x + γΦ ′ yu + 2 R uhΨ ′ ≤ -ż(t) R Ψ ′ (u 2 + u 2 x ) + γΦ ′ y + γ 2 R (u 2 -u 2 x )Φ ′ + J 1 + J 2 . go3 (3.19)
Now, in view of (3.4) and the conditions on γ, we have

ż(t)Ψ ′ - γ 2 Φ ′ ≥ (1 -α) 4 c 0 Ψ ′ on R that leads to -ż(t) R Ψ ′ (u 2 +u 2 x )+γΦ ′ y + γ 2 R (u 2 -u 2 x )Φ ′ ≤ - (1 -α)c 0 4 R Ψ ′ (u 2 +u 2 x )+γΦ ′ y
where we used that, according to (3.9) and (3.16), y ≥ 0 on the support of Φ ′ . Finally the terms J 1 and J 2 are treated as in [21]. For instance, to estimate J 1 we divide R into two regions relating to the size of |u| as follows

J 1 (t) = |x-x(t)|<R0 Ψ ′ uu 2 x + γΦ ′ yu + |x-x(t)|>R0 Ψ ′ uu 2 x + γΦ ′ yu = J 11 + J 12 . J0 (3.20)
Observe that (3.7) ensures that ẋ(t) -ż(t) ≥ βc 0 for all t ∈ R and thus, for |x -

x(t)| < R 0 , to1 to1 (3.21) x-z R t0 (t) = x-x(t)-R+(x(t)-z(t))-(x(t 0 )-z(t 0 )) ≤ R 0 -R-βc 0 (t 0 -t)
and thus the decay properties of Ψ ′ and the compact support of Φ ′ ensure that

J 11 (t) u(t) L ∞ ( u x (t) 2 L 2 + c 0 y(t) L 1 ) e R0/6 e -R/6 e -β 6 c0(t0-t) u 0 H 1 ( u 0 2 H 1 + c 0 y 0 L 1 )e R0/6 e -R/6 e -β 6 c0(t0-t) . J11 (3.22)
On the other hand, (3.15) ensures that for all t ∈ [t 0 -T, t 0 ] it holds

J 12 ≤ 4 u L ∞ (|x-x(t)|>R0) |x-x(t)|>R0 Ψ ′ u 2 x + γΦ ′ y ≤ (1 -α)c 0 8 |x-x(t)|>R0 Ψ ′ u 2 x + γΦ ′ y . J12 (
d dt I +R t0 (t) ≤ - (1 -α)c 0 8 R Ψ ′ (u 2 + u 2 x ) + γΦ ′ y + Ce -R/6 e -β 6 (t0-t) .
Integrating between t and t 0 we obtain (3.11) for any t ∈ [0, t 0 ] and u replaced by u n with n ≥ n 0 . Note that the constant appearing in front of the exponential now also depends on β. The convergence results (2.9)-(2.10) then ensure that (3.11) holds also for u and t ∈ [0, t 0 ]. Finally, (3.13) can be proven in exactly the same way by noticing that for |x -

x(t)| < R 0 it holds to2 to2 (3.25) x -z -R t0 (t) = x -x(t) + R + (x(t) -z(t)) -(x(t 0 ) -z(t 0 )) ≥ -R 0 + R + βc 0 (t -t 0 ) . Remark 3.

Properties of the asymptotic object

The aim of this section is to prove that for u 0 ∈ Y satisfying Hypothesis 1 and H 1 -close enough to a peakon, one can extract from the orbit of u 0 an asymptotic object that has a non negative density momentum and gives rise to a Y -almost localized solution.

Let u 0 ∈ Y satisfying Hypothesis 1, such that stab stab

(4.1) u 0 -cϕ H 1 < ε 2 3c 2 4
, 0 < ε < c, then, according to [START_REF] Constantin | Stability of peakons[END_REF] and [START_REF] Wahlén | Global existence of weak solutions to the Camassa-Holm equation[END_REF],

stabo stabo

(4.2) sup t∈R+ u(t) -cϕ(• -ξ(t)) H 1 < ε 2 c , where u ∈ C(R + ; H 1 )
(4.5) R u(t)(ρ n0 * ϕ ′ )(• -x(t)) = 0, ∀t ∈ R + ,
where

{ρ n } is defined in (2.
3) and where n 0 satisfies :

unic unic (4.6) ∀y ∈ [-1/2, 1/2], R ϕ(• -y)(ρ n0 * ϕ ′ ) = 0 ⇔ y = 0 . Moreover, x(•) ∈ C 1 (R) with estc estc (4.7) sup t∈R+ | ẋ(t) -c| ≤ c 8 and if gf gf (4.8) sup t∈R+ u(t) -cϕ(• -z(t)) H 1 < ε 2 c = c ε c 2 for 0 < ε < cε 0 then fg fg (4.9) sup t∈R+ u(t) -cϕ(• -x(t)) H 1 ≤ Kε .
At this stage, we fix 0 < θ < c and we take

(4.14) u(t n k , • + x(t n k )) -→ n k →+∞ ũ0 in H 1 loc (R)
where x(•) is a C 1 -function satisfying (4.5), (4.7) and (4.9). Moreover, the solution of (1.6) emanating from ũ0 is Y -almost localized.

Proof. In the sequel we set x 0 (t) = q(t, x 0 ), t ≥ 0, so that yyyy yyyy (4.15) supp y -(t) ⊂] -∞, x 0 (t)] and supp y + (t) ⊂ [x 0 (t), +∞[, ∀t ≥ 0 .

We separate two possible behaviors of x 0 (•) which has to be treated in different ways. In the first case we can prove that x(t) -x 0 (t) → +∞ so that for t > 0 large enough we are very close to the case of a non negative density momentum. On the other hand, in the second case x(t) -x 0 (t) is uniformly bounded and we have to use other arguments. In this case we will first prove that the total variation of y stays bounded for all positive times which enables us to pass to the limit. The non negativity of the limit follows from a decay estimate on the total variation of y(t) on a growing interval at the left of x 0 (t).

Case

, • + x(t n k )) ⇀ ũ0 in H 1 (R) ȗ(t n k , • + x(t n k )) → ũ0 in H 1 loc (R) (ȗ -ȗxx )(t n k , • + x(t n k )) ⇀ * ỹ0 = ũ0 -ũ0,xx in M(R) .
Since for t ≥ t * , according to (4.18) and the support property of Φ and Φ ′ , (ȗȗxx )(t,

• + x(t)) = y(t, •) on ] -c 8 (t -t * ) + 2,
y(t n k + t, • + x(t n k ) + x(t)) -y(t n k + t, • + x(t n k + t)), φ = y(t n k + t), φ(• -x(t n k ) -x(t)) -φ(• -x(t n k + t)) → 0
In view of (4.23) we infer that (ũ, x(•)) satisfies (4.5) and (4.9) with the same ε than (u, x(•)). Therefore, (4.10) forces (ũ, x(•)) to satisfy (4.3) and the uniqueness result in Lemma 4.1 ensures that x(•) is a C 1 -function and satisfies (4.7).

Let us now prove that ũ is Y -almost localized. In the sequel, for u ∈ Y and γ ≥ 0, we defined the quantity G(u) by

G γ (u) = E(u) + c 4 u -u xx , 1 .
We will also make use of the following functionals that measure the quantity G(u) at the right and at the left of u. For 0 ≤ γ ≤ c2 -4 , v ∈ Y and R > 0 we set defJr defJr

(4.25) J R γ,r (v) = v 2 + v 2 x , Ψ(• -R) + γ v -v xx , Φ(• -R) and defJl defJl (4.26) J R γ,l (v) = v 2 + v 2 x , (1 -Ψ(• + R)) + γ v -v xx , 1 -Φ(• + R) .
We separate G(v) into two parts :

G R o,γ (v) = v 2 + v 2 x , 1 -Ψ(• + R) + Ψ(• -R) + γ v -v xx ), 1 -Φ(• + R) + Φ(• -R) = J R γ,r (v) + J R γ,l (
v) , which almost "localizes" outside the ball of radius R and

G R i,γ (v) = v 2 + v 2 x , Ψ(• + R) -Ψ(• -R) + γ v -v xx , Φ(• + R) -Φ(• -R) = G(v) -G R o (v)
, which almost "localizes" inside this ball. It is however worth mentioning that, in view of (3.5), the function Φ(

• + R) -Φ(• -R) is indeed supported in [-R, R + 1].
We first notice that the almost monotonicity result (3.11) ensures that for any ε > 0 there exists R ′ ε > 0 such that decR decR

(4.27) J R ′ ε γ,r (ũ(t, • + x(t)
)) < ε, ∀t ≥ 0 . Indeed, let t 0 > 0 be fixed. Fixing α = β = 1/4 and taking z(•) = (1 -α)x(•), z(•) clearly satisfies (3.7) and (4.9)-(4.18) ensure that it also satisfies (3.10) for R ≥ 2. Moreover, we have J R γ,r (u(t 0 , • + x(t 0 )) = I +R t0 (t 0 ) where

I +R t0 is defined in (3.8). Since obviously, J R γ,r u(t, • + x(t)) ≥ I +R t0 (t) , ∀0 ≤ t ≤ t 0 ,
we deduce from (3.11) that monoJr monoJr

(4.28) J R γ,r u(t 0 , • + x(t 0 )) ≤ J R γ,r u(t, • + x(t)) + K 0 e -R/6 , ∀0 ≤ t ≤ t 0 ,
where K 0 is the constant appearing in (3.11). Therefore, taking R ′ ε ≥ 2 such that K 0 e -Rε/6 < ε/2 and J R ′ ε r (u(0, • + x(0)) < ε/2 we get that for all t ≥ 0 dede dede

(4.29) J R ′ ε γ,r u(t, • + x(t)) ≤ J R ′ ε γ,r u(0, • + x(0)) + ε/2 ≤ ε
Passing to the weak limit, this leads to (4.27). Moreover, (4.18) ensures that for any R > 0 there exists t(R) > 0 such that

x 0 (t) ≤ x(t R ) -R + 9 10 (x(t) -x(t R ), ∀t ≥ t(R) .
Therefore, the hypotheses (3.7) and (3.12) of Lemma 3.1 are fulfilled for z(t) = x(t R ) -R + 9 10 (x(t) -x(t R )) and t ≥ t R and, proceeding as above, (3.13) leads to monoJl monoJl

(4.30) J R γ,l u(t, • + x(t)) ≥ J R γ,l u(t(R), • + x(t(R))) -K 0 e -R/6 , ∀t ≥ t(R)
Now we notice that to prove Y -almost localization of ũ, it suffices to prove that for all ε > 0, there exists R ε > 0 such that po2 po2

(4.31) G Rε o,γ ũ(t, • + x(t)) < ε , ∀t ∈ R .
Indeed if (4.31) is true for some (ε, R ε ) then (ũ, x) satisfies (1.9) with (ε/2, 2R ε ).

As indicated above, we prove (4.31) by contradiction. Assuming that (4.31) is not true, there exists ε 0 > 0 such that for any R > 0 there exists t R ∈ R satisfying

(4.32) G R o,γ ũ(t R , • + x(t R )) ≥ ε 0 Let R 0 > R ′ ε 10 such that (4.33) G R0 o,γ ũ(0) ≤ ε 0 10 and K 0 e -R0/6 < ε 0 10 .
The conservation of G then forces

G R0 i,γ (ũ(t R0 , • + x(t R0 )) ≤ G R0 i,γ (ũ(0)) - 9 10 ε 0 . Recalling that Ψ(• + R) -Ψ(• -R) and Φ(• + R) -Φ(• -R) belong to C 0 (R),
G R0 i,γ (u(t n k + t R0 ), • + x(t n k + t R0 )) ≤ G R0 i,γ (u(t n k , • + x(t n k ))) - 4 5 ε 0 .
We first assume that t R0 > 0. By (4.27) and the conservation of G this ensures that bb bb (4.34)

J R0 γ,l (u(t n k + t R0 ), • + x(t n k + t R0 ) ≥ J R0 γ,l (u(t n k , • + x(t n k )) + 7 10 ε 0 . Now we take a subsequence {t n ′ k } of {t n k } such that t n ′ 0 ≥ t(R 0 ), t n ′ k+1 -t n ′ k ≥ t R0 and n ′ k ≥ n k0 .
From (4.34) and again (4.30), we get that for any k ≥ 0,

J R0 γ,l (u(t n ′ k , • + x(t n ′ k )) ≥ J R0 γ,l (u(t n ′ 0 , • + x(t n ′ 0 )) + 3 5 k ε 0 -→ k→+∞ + ∞
that contradicts the conservation of G in view of (4.29) and To prove this claim we approximate u 0 by {u 0,n } as in (2.4) and work with the global solutions u n emanating from u 0,n that satisfy Hypothesis 1 with the same x 0 . We define q n as the flow associated with u n and we set

G R0 i,γ (u(t n ′ k , •+x(t n ′ k ))) → G R0 i,γ (ũ(0, • + x0 ))). Finally, if t R0 < 0, then for k ≥ k 0 such that t n k > |t R0 | we get in the same way 0 ≤ J R0 γ,r u(t n k , • + x(t n k ) ≤ J R0 γ,r u(t n k -|t R0 |, • + x(t n k -|t R0 |)) - 7
x 0,n (t) = q n (t, x 0 ) Let us recall that, in [9], it is shown that for any (t, x) ∈ R + × R, yy yy (4.39) y n (0, x) = y n (t, q n (t, x))q n,x (t, x) 2 with formula1 formula1 (4.40) q n,x (t, x) = exp t 0 u x (s, q(s, x)) ds .
Let t 0 > 0 be fixed. According to (2.9) and (4.36) there exists n 0 ≥ 0 such that for all n ≥ n 0 , fc2 fc2

(4.41) u n (t) ≥ c 4 on [x 0,n (t) -ln 2, x 0,n (t)], ∀t ∈ [0, t 0 ]
We proceed in three steps.

Step 1. In this step, we prove that for all t ∈ [0, t 0 ] and n large enough, claimx0 claimx0

(4.42) x0,n(t) x0,n(t)-ln 2 y n (t, s) ds ≤ exp(- c 4 t) y 0,n L 1 .
Let t ∈ [0, t 0 ] and x ∈ [x 0,n (t) -ln 2, x 0,n (t)], then one has, ∀0 ≤ τ ≤ t,

q n (τ, q -1 n (t, x)) ≤ q n (τ, q -1 n (t, x 0,n (t)) = x 0,n (τ )
where for all t ≥ 0, q -1 n (t, •) is the inverse mapping of q n (t, •). Since, according to (2.7) and (4.41),

u n,x ≥ u n ≥ c 4 on [x 0,n (τ ) -ln 2, x 0,n (τ )] it holds u n (τ, x 0,n (τ )) ≥ u n τ, q n (τ, q -1 n (t, x)) ≥ 0 and thus d dτ x 0,n (τ ) -q n (τ, q -1 n (t, x) ≥ 0 on [0, t] .
Therefore, for all τ ∈ [0, t] one has

q n (τ, q -1 n (t, x)) ∈ [x 0,n (τ ) -ln 2, x 0,n (τ )]
which ensures that u n,x τ, q n (τ, q -1 n (t, x)) ≥ c 4 and (4.40) leads to

q n,x (t, q -1 n (t, x)) = exp t 0 u n,x τ, q n (τ, q -1 n (t, x)) dτ ≥ exp( c 4 t) .
Therefore for all t ∈ [0, t 0 ], (4.39) leads to

- x0,n(t) x0,n(t)-ln 2 y n (t, x) dx = - x0 q -1 n (t,x0,n(t)-ln 2)
y n (t, q n (t, θ))q n,x (t, θ) dθ

≤ -e -c 4 t x0 q -1 n (t,x0(t)-ln 2)
y n (t, q n (t, θ))q 2 n,x (t, θ) dθ

≤ -e -c 4 t x0 x0-ln 2
y n (0, θ) dθ which proves (4.42).

Step 2. In this step we prove that for t ∈ [0, t 0 ] and n ≥ 0 large enough, step2 step2

(4.43) y n (t) L 1 ≤ 2 2 + 9 c E(u 0,n ) y 0,n M .
Let Φ : R → R + defined by Φ ≡ 0 on ] -∞, -ln 2], Φ = 1 on R + and Φ(x) = (x + ln 2)/ ln 2 for x ∈ [-ln 2, 0]. Then

d dt R y n (t)Φ(• -x 0,n (t)) = -ẋ0,n (t) R yΦ ′ (• -x 0,n (t)) + R u n y n Φ ′ (• -x 0,n (t)) + R (u 2 n -u 2 n,x )Φ ′ (• -x 0,n (t)) gv (4.44)
Since supp Φ ′ (• -x 0,n (t)) ⊂ [x 0,n (t) -ln 2, x 0,n (t)], (2.7) ensures that the last term of the right-hand side of (4.44) is non positive and (4.42) ensures that

R (u n (t) -ẋ0,n (t))y n (t)Φ ′ (• -x 0,n (t)) ≤ 2E(u 0,n ) R y n (t)Φ ′ (• -x 0,n (t)) ≤ √ 2 ln 2 E(u 0,n ) y 0,n L 1 exp(- c 4 t) . (4.45) Therefore (4.44) leads to R y n (t)Φ(• -x 0,n (t)) ≤ R y 0,n Φ(• -x 0,n ) + 8 c E(u 0,n ) y 0,n L 1 ≤ 1 + 4 √ 2 c ln 2 E(u 0,n ) y 0,n L 1 (4.46)
Gathering this last estimate with (4.42) we get that

+∞ x0,n(t) y n (t, s) ds ≤ 2 + 4 √ 2 c ln 2 E(u 0,n ) y 0,n M
that leads to (4.43) by making use of the conservation of M (•). (4.37) then follows by passing to the limit in n and using (2.10).

Step 3. Finally we prove that for all t ∈ [0, t 0 ] and n ≥ 0 large enough,

- n (t) M]x0(t)-c 8 t,+∞[ ≤ exp(- c 4 t) y 0,n L 1 . step3 step3 (4.47) y
For this we first notice that since (2.7) leads to u x (t, •) ≥ u(t, •) on ] -∞, x 0 (t)[, it holds u(t, x) ≤ e x-x0(t) u(t, x 0 (t)), ∀x ≤ x 0 (t) .

Therefore, (4.11) and (2.13) force

u(t, x) ≤ 1 2 u(t, x 0 (t)) ≤ 1 2 17 16 c ≤ 17 32 c, ∀(t, x] ∈ R + ×] -∞, x 0 (t) -ln 2] .
On the other hand, u x (t) ≥ u(t) on ] -∞, x 0 (t)] and (4.35) forces

u(t, x 0 (t)) ≥ 2c 3 - c 16 = 29c 48 , ∀t ≥ 0.
Moreover, (4.11) ensures that u ≥ -2 -4 c on R + × R. Taking n large enough we can thus assume that pl1 pl1

(4.48) u n (t, x) ≤ 9c 16 , ∀(t, x) ∈ [0, t 0 ]×] -∞, x 0,n (t) -ln 2], pl2 pl2 (4.49) u n (t, x 0,n (t)) ≥ 7c 12 , ∀t ∈ [0, t 0 ] . and pl3 pl3 (4.50) u n (t, x) ≥ - c 8 on [0, t 0 ] × R .
For any t 1 ≥ 0 we define the function q n,t1 on R + × R by defqn defqn (4.51)

∂ t q n,t1 (t, x) = u n (t, q n,t1 (t, x)) , (t, x) ∈ R + × R q n,t1 (t 1 , x) = x , x ∈ R .
q n,t1 is the flow associated with u n that satisfies q n,t1 (t 1 ) = Id. Obviously q n = q n,0 and one can easily check that for q n,t1 (4.40) becomes

∂ x q n,t1 (t, x) = exp t t1
u n,x (s, q n,t1 (s, x)) ds , ∀t ≥ 0, so that (4.50) and u n,x ≥ u n on ] -∞, x 0,n (t)] ensure that dw dw

(4.52) ∂ x q n,t/2 (t, x) ≥ exp(- c 16 t), ∀(t, x) ∈ [0, t 0 ] × R .
For any t ≥ 0, we denote by q -1 n,t1 (t, •) the inverse mapping of q n,t1 (t). By a continuity argument, (4.48)-(4.49) and (2.8) lead to

x 0,n (t) -q n,t/2 (t, x 0,n (t/2) -ln 2) ≥ ln 2 + c 96 t .
Therefore, for any x ∈ [x 0,n (t)-ln 2-c 96 t, x 0,n (t)], we have q -1 n,t/2 (t, x) ∈ [x 0,n (t/2)ln 2, x 0,n ] and thus for all t ∈ [0, t 0 ], a change of variables along the flow, (4.39), (4.52) and (4.42) lead to

- x0,n(t) x0,n(t)-ln 2-c 96 t, x0,n(t)
y n (t, x) dx = - x0,n(t/2) q -1 n,t/2 (t,x0,n(t)-ln 2-c 96 t) y n (t, q n,t/2 (t, θ))∂ x q n,t/2 (t, θ) dθ ≤ -e c 16 t x0,n(t/2) x0,n(t/2)-ln 2 y n (t, t/2, q n,t/2 (t, θ))[∂ x q n,t/2 (t, θ)] 2 dθ ≤ -e c 16 t x0,n(t/2) x0,n(t/2)-ln 2 y n (t/2, θ) dθ ≤ e c 16 t e -c 8 t y 0,n L 1 ≤ e -c 16 t y 0,n L 1 .
This proves that

y - n (t) L 1 (]x0,n(t)-ln 2-c 96 t,+∞[) ≤ e -c 16 t y 0,n L 1 .
Now we notice that (2.10) ensures that y - n (t) ⇀ * z in M with z ∈ M + (R) and 0 ≤ y -(t) ≤ z. Therefore, passing to the limit in n → ∞ and then in t 0 → +∞, we obtain that

y -(t) M(]x0(t)-ln(3/2)-c 96 t,+∞[) ≤ e -c 16 t y 0 M , ∀t ≥ 0.
Finally, (4.11), (2.13) and

u x ≥ u on ] -∞, x 0 (t)[ ensure that fl4 fl4 (4.53) x 0 (t) ≤ x(t) + ln(3/2), ∀t ≥ 0,
which completes the proof of (4.38). Now, in view of (4.37), for any sequence t n ր +∞, there exists ũ0 ∈ Y and a subsequence of {t n } (that we still denote by t n to simplify the notation) such that To prove the Y -almost localization of the solution ũ emanating from ũ0 , we proceed as in the Case 1. We first obtain as in Case 1 that there exists a C 1 -function x such that (4.22), (4.23) hold for some subsequence {t n k } of {t n }. Moreover, according to the boundedness of { y n M } we may also require that for any φ ∈ C 0 (R) and any t ∈ R, Let u 0 ∈ Y satisfying hypothesis 1 and (4.1) with ε defined as in (4.10) and let {t n } be an increasing sequence that tends to +∞. Combining Theorem 1.1 and Proposition 4.2 we infer that there exists a subsequence {t n k } ⊂ {t n }, x 0 ∈ R and c 0 > 0 close to c such that zz zz (5.1)

u(t n , • + x(t n )) → ũ0 in H 1 loc (R) 442 (4.54) y(t n , • + x(t n )) ⇀ * ỹ0 = ũ0 -ũ0,xx in M(R) .
u(t n k , • + x(t n k )) -→ n k →+∞ ũ0 = ϕ c0 (• -x 0 ) in H 1 loc (R)
where x(•) is a C 1 -function satisfying (4.5), (4.7) and (4.9). Moreover, (4.9), (4.10) and (5.1) ensure that |x 0 | ≪ 1/2. Since by (5.1), ũ0 satisfies the orthogonality condition (4.5), (4.6) forces x 0 = 0. On the other hand, (4.14) ensures that c 0 = lim

n→+∞ max R u(t ′ n ) and thus u(t ′ n , • + x(t ′ n )) -λ(t ′ n )ϕ -→ n→+∞ 0 in H 1 loc (R)
where we set λ(t) := max R u(t), ∀t ∈ R. Since this is the only possible limit, it follows that pp3 pp3

(5.2)

u(t, • + x(t)) -λ(t)ϕ -→ t→0 0 in H 1 loc (R)
The convergence of the scaling parameter λ(t) and of the derivative function ẋ towards c 0 at +∞, as well as the strong H 1 -convergence on ]θt, +∞[ follow exactly as in ([34], Section 5) and will thus be omitted.

It remains to prove the H 1 -convergence in ] -∞, -θt[. Note that such convergence at the left was not established in [34]. In the appendix we complete the result in [34] by proving that for u 0 ∈ Y + the energy at the left of any given point decays to zero as time goes to +∞. In the present case the desired H 1 -convergence is a direct consequence of the following monotony result : Let Ψ be the function defined in (3.1), then for any z ∈ R, the functional

Λ z : t → R Ψ(• + θ 2 t -z)(u 2 (t) + u 2 x (t))
is not increasing on R + . Indeed, one has clearly lim z→-∞ Λ z (0) = 0 and for any given z ∈ R, -θt < -θ 2 t + z for t ≥ 0 large enough. Therefore the monotony of Λ z ensures that u(t) H 1 (x<θt) → 0 as t → +∞ and the result follows since (4.12) obviously forces as well that ϕ(• -x(t)) H 1 (x<θt) → 0 as t → +∞.

By continuity with respect to initial data in H 1 , it suffices prove this monotony result for smooth solutions. Then according to (3.17) it holds

d dt Λ z (t) = θ (u 2 + u 2 x )Ψ ′ (• + θ 2 t -z)
≤ 0 which yields the desired result.

(5.3) u 0 - N+ j=-N - j =0 ϕ cj (• -z 0 j ) H 1 ≤ ε 2 0 for some 0 < ε < ε 0 and z 0 N-< z 0 N-+1 < • • • < z N+ such that z 0 j -z 0 j-1 ≥ L ≥ L 0 , ∀j ∈ [[-N -+ 1, N + ]] ,
1 Actually, in the statement given in [21], ∂xϕc i appears instead of ρn 0 * ∂xϕc i in the orthogonality condition (5.5). But it was noticed in [34] that then there is a gap in the proof of the C 1 -regularity of the functions x i , i = 1, .., N . The proof of this version uses exactly the same arguments as developed in the appendix of [34].

then there exist N -+ N + C 1 -functions t → x -N-(t), .., x -1 (t), x 1 (t), .., t → x N+ (t) uniquely determined such that ini2 ini2

(5.4) sup t∈R+ u(t, •) -

N+ j=-N - j =0 ϕ cj (• -x j (t)) H 1 ≤ A √ ε + L -1/8
and mod2 mod2

(5.5)

R u(t, •) - N+ j=-N - j =0 ϕ cj (• -x j (t)) (ρ n0 * ∂ x ϕ ci )(• -x i (t)) dx = 0 , i ∈ {1, .., N }. Moreover, for i ∈ [[-N -, N + ]]/{0}, difdif difdif (5.6) | ẋi -c i | ≤ A √ ε + L -1/8 , ∀t ∈ R + .
Combining this result with the asymptotic stability of a peakon established in the preceding section, we are able to extend the asymptotic result to a train of well ordered antipeakons-peakons by following the strategy developped in [31] (see also [START_REF] Dika | Stability of N solitary waves for the generalized BBM equations[END_REF]).

Proof of Theorem 1.3. We first notice that it suffices to prove the result for the part of the train travelling at the right of the line x = θ 0 t, i.e. there exist 0

< c * 1 < .. < c * N+ and C 1 -functions t → x j (t),, with ẋj (t) → c * j as t → +∞, j ∈ [[1, N + ]], such that mul22 mul22 (5.7) u - N+ j=1 ϕ c * j (• -x j (t)) -→ t→+∞ 0 in H 1 x > θ 0 t .
Indeed, then the result at the left of x = -θ 0 t follows by simply using the symmetry u(t, x) → -u(t, -x) of the C-H equation.

As in [31] to prove the asymptotic result for the part of the train that travels to the right we proceed by induction starting from the fastest bump. More precisely, setting z 1 (t) = θ 0 t and

z i (t) = x i-1 (t) + x i (t) 2
, ∀i ∈ {2, .., N } , we will prove by induction from i = N to i = 1 that x i (t) → c * i as t → +∞ and that It remains to tackle the slowest bump travelling to the right. By separating the two same cases as in the preceding section and using the monotonicity result at the right of x 1 (t) that is proven in [[34], Lemma 6.2] (Note that we can directly apply this lemma here since supp y -(t) ⊂] -∞, x 1 (t) + 2] ), we obtain in the same way an asymptotic object that gives rise to an Y + almost localized solution of (1.6). The rest of the proof follows exactly the same line as in [34] since it does not used the non negativity of the momentum density. 5.3. Asymptotic stability of a not well-ordered train of antipeakons-peakons. Finally we notice that we can drop the hypothesis that the set of peakons and the set of anti-peakons are each one well-ordered. Note anyway that it is crucial for us that the set of antipeakons are at the left of the set of peakons since otherwise our initial datum cannot satisfy Hypothesis 1. To do so, we make use of the fact that Camassa-Holm equation possesses special solutions called multipeakons given by

u(t) - N j=i ϕ c * I (• -x i (t)) → 0 in H 1 (]z i (t),
u(t, x) = N i=1 p i (t)e -|x-qi(t)|
where (p i (t), q i (t)), i = 1, .., N , satisfy a differential hamiltonian system (cf. [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF]). In [START_REF] Beals | Multi-peakons and the classical moment problem[END_REF] (see also [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF]), the limit as t → ∓∞ of p i (t) and qi (t), i = 1, .., N , are determined. Combining the orbital stability of well ordered train of antipeakonspeakons, the continuity with respect to initial data in H 1 (R) and the asymptotics of multipeakons, the H 1 -stability of the variety defined for (N

-, N + ) ∈ (N * ) 2 by N N-,N+ : = v = N+ j=-N - j =0 p j e -|•-qj| , (p -N-, .., p -1 , p 1 , .., p N+ ) ∈ (R * -) N-× (R * + ) N + , q N-< q -1 < q 1 < .. < q N .
is proved in ([21], Corollary 1.1). Gathering this last result with the asymptotics of the multipeakons and Theorem 1.3, the following asymptotic stability result for not well ordered train of peakons can be deduced quite directly.

cor-mult-peaks Corollary 5.1. Let be given

N -negative real numbers c -N-< .. < c -2 < c -1 < 0, N + positive real numbers 0 < c 1 < c 2 < .. < c N+ , N -+ N + real numbers q 0 -N-< .. < q 0 -1 < q 0 1 < .. < q 0 N and let λ -N-< •• < λ -1 < 0 < λ 1 < •• < λ N+ be the N -+ N + distinct eigenvalues of the matrix (p 0 j e -|q 0 i -q 0 j |/2 ) (i,j)∈([[-N-,N+]]/{0}) 2 .
For any B > 0 there exists a positive function ε with ε(y) → 0 as y → 0 and α 0 > 0 such that if u 0 ∈ Y satisfies the Hypothesis 1 with ini3 ini3

(5.8)

m 0 M ≤ B and u 0 - N+ j=-N - j =0 p 0 j exp(• -q 0 j ) H 1 ≤ α for some 0 < α < α 0 , then there exists c * -N-< •• < c * -1 < 0 < c * 1 < •• < c * N+ and C 1 -functions (x -N-, .., x -1 , x 1 , .., x N+ ) with |c * i -λ i | ≤ ε(α) and lim t→+∞ ẋi (t) = c * i , ∀i ∈ [[-N -, N + ]]/{0}, such that coromul2 coromul2
(5.9) u -

N+ j=-N - j =0 ϕ c * i (• -x i (t)) -→ t→+∞ 0 in H 1 (|x| > min(-λ -1 , λ 1 )/4) .
6. Appendix: An improvement of the asymptotic stability result in the class of solutions with non negative momentum density

In this subsection we make a simple observation that enables to improve the asymptotic stability result given in [34]. This observation is that all the energy of the solutions of the C-H equation, that have a non negative density momentum, is traveling to the right. Moreover, as stated in the following lemma, the energy at the left of any fixed point tends to zero as t → +∞. This differs from KdV like equations since for these last equations the linear waves are travelling to the left. This causes a small loss of energy to the left and the so called dispersive tail. Note that for CH-type equation, there is no linear part and if moreover the momentum density is non negative, there is also no (even small) antipeakon and thus all the energy is travelling to the right that formally leads to (6.1). Before proving Lemma 6.1, let us state the improved asymptotic stability result : asympstab+ Theorem 6.2. Let c > 0 be fixed. There exists a universal constant 0 < η 0 ≪ 1 such that for any 0 < θ < c and any u 0 ∈ Y + satisfying difini+ difini+ To prove (6.6) we first claim that for any t ∈ R and any ∆ > 0 it holds td4 td4 (6. Let us prove this claim. First we notice that by continuity with respect to initial data, it suffices to prove (6.7) for u ∈ C ∞ (R; H ∞ ) ∩ L ∞ (R; Y + ). Then a simple application of the implicit function theorem ensures that t → x γ (t) is of class C 1 . Indeed,

ψ : (z, v) → R (v 2 + v 2 x )Φ(• -z)
is of class C 1 from R × H 1 (R) into R and for any (z 0 , v) ∈ R × Y + /{0}, ∂ z ψ(z 0 , v) = R (v 2 + v 2 x )Φ ′ (• -z 0 ) > 0. It then follows from (3.17) that ẋγ (t)

R (u 2 + u 2 x )Φ ′ (• -x γ (t)) = R uu 2 x Φ ′ (• -x γ (t)) + 2 R uhΦ ′ (• -x γ (t))
where h = p * (u 2 + u 2 x /2). Now, it is worth noticing that according to [9], the following convolution estimate holds : For any v ∈ H 1 (R), Combining (6.7)-(6.8) and the fact that |v x | ≤ v on R for any v ∈ Y + , we eventually get

2 ẋγ (t) R u 2 Φ ′ (• -x γ (t)) ≥ R u(u 2 + u 2 x )Φ ′ (• -x γ (t)) ≥ R u 3 Φ ′ (• -x γ (t))
and Hölder inequality together with the fact that Φ ′ is a non negative function of total mass 1 lead to td6 td6 (6.9) ẋγ (t) ≥

1 2 R u 2 Φ ′ (• -x γ (t)) 1/2
Integrating this inequality between t and t + ∆ yields (6.7) that obviously implies that x γ (•) is an increasing function. In particular there exists x ∞ γ ∈ R ∩ {+∞} such that x γ (t) ր x ∞ γ as t → +∞ and it remains to prove that x ∞ γ = +∞. Assuming the contrary, we first notice that since |u x | ≤ u ≤ u 0 H 1 on R 2 , (6.5) then yields x ∞ γ u 2 (τ, s) ds dτ = 0 .

In particular there exists a sequence (t n , x n ) n≥1 ⊂ R × [x ∞ γ , x ∞ γ + 2] with t n ր +∞ such that u(t n , x n ) → 0 as n → ∞. Therefore, making use of the fact that |u x | ≤ u on R 2 forces, for any (t, x 0 ) ∈ R 2 , that (6.11) u(t, x) ≤ e |x0-x| u(t, x 0 ), ∀x ∈ R , we infer that for any A > 0, td7 td7 (6.12) lim

n→∞ sup x∈[x ∞ γ -A,x ∞ γ +A]
u(t n , x) = 0 .

  cvy

  [START_REF] Wahlén | Global existence of weak solutions to the Camassa-Holm equation[END_REF]) and (4.38) ensure that ỹ0 is a non negative bounded measure. Finally, according to part 3. of Proposition 2.1 and (4.37), the solution ũ emanating from ũ0 belongs to C b (R; Y + ).

5 .

 5 Asymptotic stability results 5.1. Asymptotic stability of a peakon. With Proposition 4.2 in hands, the proof of the asymptotic stability of a single peakon follows very closely the proof in [34] since the rest of the proof only uses Theorem 1.1 and the monotonicity results for γ = 0 (i.e. only E(•) is involved).

5. 2 .

 2 Asymptotic stability of well ordered trains of antipeakons-peakons. In [21] the orbital stability in H 1 (R) of well ordered trains of antipeakons-peakons is established. More precisely, the following theorem 1 is proved :mult-peaks Theorem 5.1 ([21]). Let be given N -negative velocities c -N-< .. < c -2 < c -1 < 0, N + positive velocities 0 < c 1 < c 2 < .. < c N+ There exist n 0 ∈ N satisfying (4.6), A > 0, L 0 > 0 and ε 0 > 0 such that if u ∈ C(R + ; H 1 ) is the solution of (C-H) emanating from u 0 ∈ Y ,satisfying Hypothesis 1 with huhu huhu

apen Lemma 6 . 1 .

 61 For any u 0 ∈ Y + and any z ∈ R, denoting by u ∈ C(R; H 1 ) solution of (1.6) emanating from u 0 it holds tdt tdt (6.1) lim t→+∞ u(t) H 1 (]-∞,z[) = 0 .

( 6 . 2 ) u 0 -ϕ c H 1 ≤ η 0 θ c 8 , 2 H 1 (u 2 2 H 1 [

 62821221 there exists c * > 0 with |c -c * | ≪ c and a C 1 -function x : R → R with lim t→∞ ẋ = c * such that(6.3) u(t, • + x(t)) ⇀ t→+∞ ϕ c * in H 1 (R) ,where u ∈ C(R; H 1 ) is the solution emanating from u 0 . Moreover, for any z > 0,cvforte+ cvforte+ (6.4) lim t→+∞ u(t) -ϕ c * (• -x(t)) H 1 (R/[z,θt]) = 0 .Proof of Lemma 6.1. Let 0 < γ < u 0 and let x γ : R → R be defined by+ u 2 x )(t)Φ(• -x γ (t)) = γ with Φ defined in (3.5). Note that x γ (•) is well-defined since u 0 ∈ Y + forces u > 0 on R 2 and thus for any fixed t ∈ R, z → R (u 2 + u 2 x )(t)Φ(• -z) is a decreasing continuous bijection from R to ]0, u 0 . Moreover, u ∈ C(R; H 1 ) ensures that x γ (•)is a continuous function. (6.1) is clearly a direct consequence of the fact that

7 )

 7 x γ (t + ∆) -x γ (t)

  p * (v 2 + v 2 x /2)(x) ≥ 1 2 v 2 on R.

(u 2 u 2

 22 + u 2 x )Φ ′ (• -x γ (t)) = lim t→+∞ R (u 2 + u 2 x )Φ ′ (• -x ∞ γ ) = γ .Now, taking ∆ = 1, (6.(τ, s) ds dτ = 0 which, recalling that x γ (t) → x ∞ γ

  3.23) Gathering(3.20),(3.22),(3.23) and the estimates on J 2 that we omit, since it is exactly the same terms as in [21], we conclude that there exists C > 0 only depending on R 0 and E(u) such that for R ≥ R 0 and t ∈ [0, t 0 ] it holds

	nini nini	(3.24)

  is the solution emanating from u 0 and ξ(t) ∈ R is any point where the function u(t, •) attains its maximum. According to[34], by the implicit function theorem, we have the following lemma.

	distxz distxz	(4.4)	sup t∈R+	|x(t) -z(t)| < κ 0
		and
	ort ort
	gff gff	(4.3)	sup

modulation Lemma 4.1. There exists 0

< ε 0 < 1, κ 0 > 0, n 0 ∈ N and K > 1 such that if a solution u ∈ C(R; Y ) to (1.6) satisfies t∈R+ u(t) -cϕ(• -z(t)) H 1 < cε 0 ,

for some function z : R + → R, then there exists a unique function x : R + → R such that

  ≤ γ ≤ (1 -α)c. In particular, u satisfies the hypotheses of Lemma 3.1 for α = 1/3. Note that the hypothesis (1.10) with

	defep defep	(4.10)	ε =	1 2 4 K	min	θ 2 5 , c ε 0
	Kep Kep	For u 0 ∈ Y satisfying Hypothesis 1 and (4.1) with this ε, (4.2) ensures that (4.3) 2 9 , cε0 2 4 so that and thus (4.7) hold. Moreover, (4.9) is satisfies with Kε ≤ min θ (4.11) sup cε 0 2 4 ≤ c 2 4 .
		It follows that
	cc cc	(4.12)	ẋ(t) ≥	7 8	c , ∀t ≥ 0.
		and that u satisfies the hypotheses of Lemma 3.1 for any 0 < α < 1 such that
	okok okok	(4.13)		(1 -α) ≥	θ 4c
		and any 0 η 0 =	1 K 8 min	1 2 10 ,	2 4 ε 0	8
		implies that (4.1) holds with ε given by (4.10).

t∈R+ u(t) -cϕ(• -x(t)) H 1 ≤ propasym Proposition 4.2.

Let u 0 ∈ Y satisfying Hypotheses 1 and (4.1) with ε defined as in (4.10) and let u ∈ C(R; H 1 (R)) the emanating solution of

(1.6)

. For any sequence t n ր +∞ there exists a subsequence {t n k } ⊂ {t n } and ũ0 ∈ Y + such that pp2 pp2

  Since, by (4.7), {x(t n + •) -x(t n )} is uniformly equi-continuous, Arzela-Ascoli theorem ensures that there exists a subsequence {t n k } ⊂ {t n } and x ∈ C(R) such that for all T > 0,

	+∞[ and thus is a non negative measure 8 (t -t * ) + 2, +∞[, we infer that ũ0 ∈ Y + . Moreover, the support properties on ] -c of Φ and Φ ′ imply that (4.20) u(t n vcc vcc vc vc (4.21) (4.22) x(t n k + •) -x(t n k ) -→ t→+∞ x in C([0, T ]) . Therefore, on account of (4.20), (4.22) and part 3. of Proposition 2.1 for any t ∈ R, u(t n k + t, • + x(t n k + t)) → ũ(t, • + x(t)) in H 1 loc (R) strongcv (4.23) where ũ ∈ C(R + ; H 1 (R)) is the solution of (1.6) emanating from ũ0 ∈ Y + . More-over, for all A > 0 and φ ∈ C 0 (R) with supp φ ⊂ [-A, +∞[, it holds y(t n cvx cvx weakcvy weakcvy (4.24) y(t n

k , • + x(t n k )) → ũ0 in H 1 loc (R) and, for all A > 0 and φ ∈ C 0 (R) with supp φ ⊂ [-A, +∞[, k , • + x(t n k )), φ → ỹ0 , φ . k + t, • + x(t n k + t)), φ → ỹ(t, • + x(t)), φ ,

where ỹ = ũ -ũxx . Indeed, on one hand, it follows from part 3. of Proposition 2.1 that

y(t n k + t, • + x(t n k ) + x(t)), φ → ỹ(t, • + x(t)), φ

and on the other hand, the uniform continuity of φ together with (4.22) ensure that

  is almost non increasing which ensures that (4.27) holds. Then, proceeding exactly as in the case 1, but with γ = 0, we obtain that ũ is H 1 -almost localized, i.e. for any ε > 0 there exists R ε > 0 such thatG Rε o,0 (ũ(t, • + x(t))) < ε, ∀t ∈ R .This last estimate together with (4.27) and (4.57) prove the Y -localization of ũ.

	weakcvy2 weakcvy2	(4.56)	y(t n k + t, • + x(t n k + t)), φ → ỹ(t, • + x(t)), φ .
		But then (4.35), (4.38) force
	sup sup	(4.57) Now, according to (4.53), supp y -(t) ⊂]-∞, x(t)+ ln(3/2)] for all t ≥ 0. Therefore, supp ỹ(t) ⊂ [x(t) -ln 2, +∞[, ∀t ∈ R . for R > 0 big enough, we can thus apply Lemma 3.1 to obtain that J R r,γ (u(t, •+x(t)))

Finally, taking A > 0 such that x ∞ γ -A < x γ ′ (0) with γ < γ ′ < u 0 2 H 1 , we infer from (6.12) and the monotony of t → x γ ′ (t) that

This contradicts (6.10) and concludes the proof of the lemma.

Conflict of Interest : The author declares that he has no conflict of interest.