André Kabakouala
email: andre.kabakouala@lmpt.univ-tours.fr

Stability in the Energy Space of the Sum of N Peakons for the Degasperis-Procesi Equation

Introduction

The Degasperis-Procesi (DP) equation

u t -u txx + 4uu x = 3u x u xx + uu xxx , (t, x) ∈ R * + × R (1.1)
is completely integrable (see [START_REF] Degasperis | A new integrable equation with peakon solutions[END_REF]) and possesses, among others, the following invariants

E(u) = R yv and F (u) = R u 3 , (1.2)
where y = (1 -∂ 2 x )u and v = (4 -∂ 2 x ) -1 u. Substituting u by 4v -v xx in (1.2) and using integration by parts (we suppose that u(±∞) = v(±∞) = v x (±∞) = 0), the conservation laws can be rewritten as

E(u) = R 4v 2 + 5v 2 x + v 2 xx and F (u) = R -v 3 xx + 12vv 2 xx -48v 2 v xx + 64v 3 . (1.3)
One can see that the conservation law E(•) is equivalent to • 2 L 2 (R) . Indeed, using integration by parts

u 2 L 2 (R) = R u 2 = R (4v -v xx ) 2 = R 16v 2 + 8v 2 x + v 2 xx ≤ 4E(u), (1.4)
and applying Plancherel-Parseval identity

E(u) = R yv = R 1 + ω 2 4 + ω 2 | u(ω)| 2 ≤ R | u(ω)| 2 = u 2 L 2 (R) , (1.5)
where u denotes the Fourier transform of u. In the sequel we will denote u H = E(u).

(1.6)

Note that, by reversing the operator (1 -∂ 2 x )(•) in (1.1), the DP equation can be rewritten in conservation form as

u t + 1 2 ∂ x u 2 + 3 2 (1 -∂ 2 x ) -1 ∂ x u 2 = 0, (t, x) ∈ R * + × R. (1.7)
The DP equation possesses solitary waves called peakons (see Fig. 1a) and defined by

u(t, x) = ϕ c (x -ct) = cϕ(x -ct) = ce -|x-ct| , c ∈ R * , (t, x) ∈ R * + × R, (1.8)
but they are not smooth since ϕ c / ∈ C1 (R) (see Fig. 1b). The peakons are only global weak solutions of (1.7). It means, for any smooth test function φ ∈ C ∞ (R + × R), it holds

+∞ 0 R ϕ c (x -ct)φ t (t, x)dtdx + 1 2 +∞ 0 R ϕ 2 c (x -ct)φ x (t, x)dtdx + 3 2 +∞ 0 R (1 -∂ 2 x ) -1 ϕ 2 c (x -ct)φ x (t, x)dtdx + R ϕ c (x)φ(0, x)dx = 0.
The goal of our work is to prove that ordered trains of peakons are stable under small perturbations in the energy space H (equivalent to L 2 ).

Definition 1.1 (Stability). Let c > 0 be given. The peakon ϕ c is said stable in H, if for all ε > 0, there exists δ > 0 such that if

u 0 -ϕ c H ≤ δ, (1.9)
then for all t ≥ 0, there exists ξ(t) such that

u(t, •) -ϕ c (• -ξ(t)) H ≤ ε, (1.10)
where u(t) is the solution to (1.1) emanating from u 0 .

Lin and Liu proved in [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF] the stability of a single peakon under the additional condition that (1 -∂ 2

x )u 0 ∈ M + (R). Using this result and the general strategy introduced by Martel, Merle and Tsai in [START_REF] Martel | Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations[END_REF] for the generalized Korteweg-de Vries (gKdV) equation and adapted by El Dika and Molinet in [START_REF] El | Stability of multipeakons[END_REF] and [START_REF] El | Stability of multi antipeakon-peakons profile[END_REF] for the Camassa-Holm (CH) equation, we prove here the stability of the sum of N peakons for the DP equation.

Before stating the main result we introduce the function space where will live our class of solutions to the equation. For I a finite or infinite time interval of R + , we denote by X (I) the function space 1 X (I) = u ∈ C I; H 1 (R) ∩ L ∞ I; W 1,1 (R) , u x ∈ L ∞ (I; BV (R)) .

(1.11)

The main result of the present paper is the following theorem.

Theorem 1.1 (Stability of the Sum of N Peakons). Let be given N velocities c 1 , . . . , c N such that 0 < c 1 < . . . < c N . Let u ∈ X ([0, T [), with 0 < T ≤ +∞, be a solution of the DP equation. There exist C > 0, L 0 > 0 and ε 0 > 0 only depending on the speeds (c i ) N i=1 , such that if

y 0 = (1 -∂ 2 x )u 0 ∈ M + (R) (1.12) and u 0 - N i=1 ϕ ci (• -z 0 i ) H ≤ ε 2 , with 0 < ε < ε 0 , (1.13)
for some z 0 1 , . . . , z 0 N satisfying z 0 1 < . . . < z 0 N and z 0 i -z 0 i-1 ≥ L, with L > L 0 > 0, i = 2, . . . , N, (1.14) then there exist ξ 1 1 (t), . . . , ξ N 1 (t) such that

u(t) - N i=1 ϕ ci (• -ξ i 1 (t)) H ≤ C( √ ε + L -1/8 ), ∀t ∈ [0, T [ (1.15) and ξ i 1 (t) -ξ i-1 1 (t) > L 2 , ∀t ∈ [0, T [, i = 2, . . . , N, (1.16)
where ξ 1 1 (t), . . . , ξ N 1 (t) are defined in Subsection 4.1.

Preliminaries

In this section, we briefly recall the global well-posedness results for the DP equation and its consequences (see [START_REF] Escher | Global weak solutions and blow-up structure for the Degasperis-Procesi equation[END_REF] and [START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF] for details).

Theorem 2.1 (Global Weak Solution; See [START_REF] Escher | Global weak solutions and blow-up structure for the Degasperis-Procesi equation[END_REF] and [START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF]). Assume that u 0 ∈ L 2 (R) with y 0 = (1 -∂ 2 x )u 0 ∈ M + (R). Then the DP equation has a unique global weak solution u ∈ X (R + ) such that (2.2)

y(t, •) = (1 -∂ 2 x )u(t, •) ∈ M + (R), ∀t ∈ R + . ( 2
Then, using the Sobolev embedding of H 1 (R) into L ∞ (R) and (2.2), we infer that there exists a constant

C S > 0 such that u L ∞ (R) ≤ C S u H 1 (R) ≤ 2C S u L 2 (R) . (2.3)
Lemma 2.1 (Positivity; See [START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF]). Let u ∈ H 1 (R) with y = (1 -∂ 2 x )u ∈ M + (R). If k 1 ≥ 1, then we have

(k 1 ± ∂ x )u(x) ≥ 0, ∀x ∈ R. (2.4)
Lemma 2.2 (Positivity; See [START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF]). Let w(x) = (k 1 ±∂ x )u(x). Assume that u ∈ H 1 (R) with y = (1-∂ 2 x )u ∈ M + (R). If k 1 ≥ 1 and k 2 ≥ 2, then we have

(k 2 ± ∂ x )(4 -∂ 2
x ) -1 w(x) ≥ 0, ∀x ∈ R.

(2.5)

Stability of a single peakon

The proof of Lin and Liu in [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF] is not entirely suitable for our work, because it involves all local extrema of the function v = (4 -∂ 2

x ) -1 u on R, and thus is not local. For our work, we have to localize the estimates. Therefore, we need to modify a little the proof of Lin and Liu. We do this first for a single peakon. Theorem 3.1 (Stability of Peakons). Let u ∈ X ([0, T [), with 0 < T ≤ +∞, be a solution of the DP equation and ϕ c be the peakon defined in (1.8), traveling to the right at the speed c > 0. There exist C > 0 and ε 0 > 0 only depending on the speed c > 0, such that if

y 0 = (1 -∂ 2 x )u 0 ∈ M + (R) (3.1)
and u 0 -ϕ c H ≤ ε 2 , with 0 < ε < ε 0 , (3.2) then u(t, •) -ϕ c (• -ξ 1 (t)) H ≤ C √ ε, ∀t ∈ [0, T [, (3.3
)

where ξ 1 (t) ∈ R is any point where the function v(t, •) = (4 -∂ 2 x ) -1 u(t,
•) attains its maximum. To prove this theorem we first need the following lemma that enables to control the distance of E(u) and F (u) to respectively E(ϕ c ) and F (ϕ c ).

Lemma 3.1 (Control of Distances Between Energies). Let u ∈ H 1 (R) with y = (1 -∂ 2 x )u ∈ M + (R). If u -ϕ c H ≤ ε 2 , then |E(u) -E(ϕ c )| ≤ O(ε 2 ) (3.4) and |F (u) -F (ϕ c )| ≤ O(ε 2 ), (3.5)
where O(•) only depends on the speed c.

Proof. For the first estimate, applying triangular inequality, and using that u -ϕ c H ≤ ε 2 and

ϕ c H = c/ √ 3, we have |E(u) -E(ϕ c )| = | u H -ϕ c H | ( u H + ϕ c H ) ≤ u -ϕ c H ( u -ϕ c H + 2 ϕ c H ) ≤ ε 2 ε 2 + 2c √ 3 ≤ O(ε 2 ).
For the second estimate, applying the Hölder inequality, and using that u -ϕ c H ≤ ε 2 and (2.3), we have

|F (u) -F (ϕ c )| ≤ R u 3 -ϕ 3 c ≤ R |u -ϕ c |(u 2 + uϕ c + ϕ 2 c ) ≤ u -ϕ c L 2 (R) R (u 2 + uϕ c + ϕ 2 c ) 2 1/2 = u -ϕ c L 2 (R) R (u 4 + 2u 3 ϕ c + 3u 2 ϕ 2 c + 2uϕ 3 c + ϕ 4 c ) 1/2 ≤ u -ϕ c L 2 (R) • 4C 2 S u 4 L 2 (R) + 4cC S u 2 L 2 (R) + 3c 2 u 2 L 2 (R) + 8 3 c 3 C S u L 2 (R) + 1 2 c 4 1/2 ≤ O(ε 2 ),
where we also use that the L 2 norm of u is bounded and the following measures of peakon:

ϕ c L ∞ (R) = c, ϕ c L 3 (R) = 3 2 3 c and ϕ c L 4 (R) = 1 4 √ 2 c.
This proves the lemma. Now, to prove Theorem 3.1, by the conservation of E(•), F (•) and the continuity of the map t → u(t)

from [0, T [ to H 1 (R) ֒→ H (since H ≃ L 2 and u L 2 (R) ≤ u H 1 (R) ), it suffices to prove that for any function u ∈ H 1 (R) satisfying y = (1 -∂ 2 x )u ∈ M + (R), (3.4) and (3.5), if inf ξ∈R u -ϕ c (• -ξ) H ≤ ε 1/4 , (3.6) then u -ϕ c (• -ξ 1 ) H ≤ C √ ε, (3.7)
where ξ 1 ∈ R is any point where the function v = (4 -∂ 2 x ) -1 u attains its maximum. We divide the proof of Theorem 3.1 into a sequence of lemmas. In the sequel, we will need to introduce the following smooth-peakons defined for all x ∈ R by:

ρ c (x) = cρ(x) = (4 -∂ 2 x ) -1 ϕ c (x) = c 3 e -|x| - c 6 e -2|x| . (3.8) One can check that ρ c ∈ H 3 (R) ֒→ C 2 (R) (by the Sobolev embedding) since ϕ c ∈ H 1 (R)
. Indeed, we have

ρ c 2 H 3 (R) = R (1 + ω 2 ) 3 (4 + ω 2 ) 2 | ϕ c (ω)| 2 ≤ R (1 + ω 2 )| ϕ c (ω)| 2 ≤ ϕ c 2 H 1 (R) = 2c 2 . (3.9)
Moreover, ρ c is a positive even function which decays to 0 at infinity, and admits a single maximum c/6 at point 0 (see Fig. 1a-1c).

Lemma 3.2 (Uniform Estimates). Let u ∈ H 1 (R) with y = (1 -∂ 2 x )u ∈ M + (R), and ξ ∈ R. If u -ϕ c (• -ξ) H ≤ ε 1/4 , then u -ϕ c (• -ξ) L ∞ (R) ≤ O(ε 1/8 ) (3.10) and v -ρ c (• -ξ) L ∞ (R) ≤ O(ε 1/4 ), (3.11
)

where v = (4 -∂ 2 x ) -1 u and ρ c is defined in (3.8).
Proof. For the second estimate, applying the Hölder inequality and using assumption, we get for all x ∈ R,

|v(x) -ρ c (x -ξ)| ≤ 1 4 R e -2|x ′ | |u(x -x ′ ) -ϕ c [(x -x ′ ) -ξ]| dx ′ ≤ 1 4 R e -4|x ′ | dx ′ 1/2 R |u(x ′ ) -ϕ c (x ′ -ξ)| 2 dx ′ 1/2 ≤ 1 2 √ 2 u -ϕ c (• -ξ) H ≤ O(ε 1/4 ).
For the first estimate, note that the assumption y = (1

-∂ 2 x )u ≥ 0 implies that u = (1 -∂ 2 x
) -1 y ≥ 0 and satisfies (2.2). Then, applying triangular inequality, and using that |ϕ ′ c | = ϕ c on R and (2.3), we have

u -ϕ c (• -ξ) H 1 (R) ≤ u H 1 (R) + ϕ c H 1 (R) ≤ 2 u L 2 (R) + 2 ϕ c L 2 (R) ≤ 2 u -ϕ c (• -ξ) L 2 (R) + 4 ϕ c L 2 (R) ≤ O(ε 1/4 ) + O(1).
Now, applying the Gagliardo-Nirenberg inequality and using assumption, we obtain

u -ϕ c (• -ξ) L ∞ (R) ≤ C G u -ϕ c (• -ξ) 1/2 L 2 (R) u -ϕ c (• -ξ) 1/2 H 1 (R) ≤ O(ε 1/8 ) O(ε 1/8 ) + O(1) ≤ O(ε 1/8 ).
This proves the lemma. Lemma 3.3 (Quadratic Identity; See [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF]). For any u ∈ L 2 (R) and ξ ∈ R, it holds

(c) (1 -∂ 2 x )ϕ(x) = 2δ 0 and (1 -∂ 2 x )ρ(x) = (1/2)e -2|x| profiles
E(u) -E(ϕ c ) = u -ϕ c (• -ξ) 2 H + 4c v(ξ) - c 6 , (3.12
)

where v = (4 -∂ 2 x ) -1 u and c/6 = ρ c (0) = max x∈R ρ c (x -ξ).
Sketch of proof. The proof follows by direct computation, with the aid of two integration by parts, and using that (1 -∂2

x )ϕ c (• -ξ) = 2cδ ξ , where δ ξ denotes the Dirac mass applied at point ξ.

Let u ∈ H 1 (R) with y = (1 -∂ 2 x )u ∈ M + (R)
, and assume that there exists ξ ∈ R such that (3.6) holds for some ξ ∈ R. We consider now the interval in which the peakon (respectively the smooth-peakon) is concentrated, and we will decompose this interval according to the variation of v = (4 -∂ 2

x ) -1 u in the following way: we set with local minimal values. We rename α = η 0 and β = η k+1 so that it holds

α = sup x < ξ, v(x) = c 2400 and β = inf x > ξ, v(x) = c 2400 . ( 3
η 0 < ξ 1 < η 1 < . . . < ξ j < η j < ξ j+1 < η j+1 < . . . < η k < ξ k+1 < η k+1 .
(3.14) Let M j = v(ξ j ), j = 1, . . . , k + 1, and m j = v(η j ), j = 1, . . . , k.

(3.15)

By construction v x (x) ≥ 0, ∀x ∈ [η j-1 , ξ j ], j = 1, . . . , k (3.16)
and

v x (x) ≤ 0, ∀x ∈ [ξ j , η j ], j = 1, . . . , k + 1. (3.17) We claim that v(x) ≤ c 300 , ∀x ∈ R \ [η 0 , η k+1 ], (3.18) u(x) ≤ c 300 , ∀x ∈ R \ [η 0 , η k+1 ], (3.19)
and there exits

C 0 > 0 such that [η 0 , η k+1 ] ⊂ [ξ -C 0 , ξ + C 0 ]. (3.20)
Indeed, for some 0 < ε ≪ 1 fixed, using (3.11) we have

ρ c (η 0 -ξ) = v(η 0 ) + O(ε 1/4 ) = c 2400 + O(ε 1/4 ) ≤ c 350 .
Please note that, we abuse notation by writing that the difference between v and ρ c (• -ξ) is equal to O(ε 1/4 ). Therefore, using that

ρ c (• -ξ) is increasing on ] -∞, ξ], it holds for all x ∈] -∞, η 0 [, v(x) = ρ c (x -ξ) + O(ε 1/4 ) ≤ c 350 + O(ε 1/4 ) ≤ c 300 .
Proceeding in the same way for x ∈]η k+1 , +∞[, we obtain (3.18). One can remark that for all x ∈ R,

ϕ c (x) -6ρ c (x) = ce -|x| -6 c 3 e -|x| - c 6 e -2|x| = -ce -|x| + ce -2|x| ≤ 0. ( 3
ρ c (η 0 -ξ) = v(η 0 ) + O(ε 1/4 ) = c 2400 + O(ε 1/4 ) ≥ c 3000 .
Therefore, since ρ c = (c/3)e -|•| -(c/6)e -2|•| and that x → (1/3)e -|x| -(1/6)e -2|x| is a positive even function decreasing to 0 on R + (see Fig. 1a), there exists a universal constant C 0 > 0 such that (3.20) holds.

We now are ready to establish the connection between the conservation laws. Please note that, we will change the order of the extrema of v = (4 -∂ 2

x ) -1 u while keeping the same notations as in (3.15).

Lemma 3.4 (Connection Between E(•) and the Local Extrema of v). Let u ∈ H 1 (R) and v = (4 - ∂ 2 x ) -1 u ∈ H 3 (R). Define the function g by g(x) =          2v + v xx -3v x , x < ξ 1 , 2v + v xx + 3v x , ξ j < x < η j , 2v + v xx -3v x , η j < x < ξ j+1 , 2v + v xx + 3v x , x > ξ k+1 , j = 1, . . . , k. (3.22) Then it holds R g 2 (x)dx = E(u) -12   k j=0 M 2 j+1 - k j=1 m 2 j   . (3.23) Proof. We have R g 2 (x)dx = ξ1 -∞ g 2 (x)dx + k j=1 ξj+1 ξj g 2 (x)dx + +∞ ξ k+1 g 2 (x)dx. (3.24) For j = 1, . . . , k, ξj+1 ξj g 2 (x)(x)dx = ηj ξj (2v + v xx + 3v x ) 2 + ξj+1 ηj (2v + v xx -3v x ) 2 = J + I.
Let us compute I,

I = ξj+1 ηj 4v 2 + v 2 xx + 9v 2 x + 4vv xx -12vv x -6v x v xx = ξj+1 ηj 4v 2 + v 2 xx + 9v 2 x + 4 ξj+1 ηj vv xx -12 ξj+1 ηj vv x -6 ξj+1 ηj v x v xx = ξj+1 ηj 4v 2 + v 2 xx + 9v 2 x + I 1 + I 2 + I 3 .
Applying integration by parts and using that v x (ξ j ) = v x (η j ) = 0, we get

I 1 = -4 ξj+1 ηj v 2 x , I 2 = -6 ξj+1 ηj ∂ x (v 2 ) = -6v 2 (ξ j+1 ) + 6v 2 (η j ) and I 3 = -3 ξj+1 ηj ∂ x (v 2 x ) = 0. Therefore I = ξj+1 ηj 4v 2 + 5v 2 x + v 2 xx -6v 2 (ξ j+1 ) + 6v 2 (η j ). (3.25)
Similar computations lead to

J = ηj ξj 4v 2 + 5v 2 x + v 2 xx -6v 2 (ξ j )+6v 2 (η j ), ξ1 -∞ g 2 (x) dx = ξ1 -∞ 4v 2 + 5v 2 x + v 2 xx -6v 2 (ξ 1 ) (3.26) and +∞ ξ k+1 g 2 (x) dx = +∞ ξ k+1 4v 2 + 5v 2 x + v 2 xx -6v 2 (ξ k+1 ). (3.27)
Adding I and J, and summing over j ∈ {1, . . . , k}, we obtain

ξ k+1 ξ1 g 2 (x)dx = ξ k+1 ξ1 4v 2 + 5v 2 x + v 2 xx -6 k j=1 v 2 (ξ j+1 ) -6 k j=1 v 2 (ξ j ) + 12 k j=1 v 2 (η j ). ( 3
v). Let u ∈ H 1 (R) and v = (4 - ∂ 2 x ) -1 u ∈ H 3 (R). Define the function h by h(x) =          -v xx -6v x + 16v, x < ξ 1 , -v xx + 6v x + 16v, ξ j < x < η j , -v xx -6v x + 16v, η j < x < ξ j+1 , -v xx + 6v x + 16v, x > ξ k+1 , j = 1, . . . , k. (3.29) Then it holds R h(x)g 2 (x)dx = F (u) -144   k j=0 M 3 j+1 - k j=1 m 3 j   . (3.30)
Proof. We have

R h(x)g 2 (x)dx = ξ1 -∞ h(x)g 2 (x)dx + k j=1 ξj+1 ξj h(x)g 2 (x)dx + +∞ ξ k+1 h(x)g 2 (x)dx. (3.31)
For j = 1, . . . , k,

ξj+1 ξj h(x)g 2 (x)dx = ηj ξj (-v xx -6v x + 16v) (2v + v xx -3v x ) 2 + ξj+1 ηj (-v xx + 6v x + 16v) (2v + v xx + 3v x ) 2 = J + I.
Let us compute I,

I = ξj+1 ηj -v 3 xx + 12vv 2 xx + 64v 3 + 60v 2 v xx -54 ξj+1 ηj v 3 x + 27 ξj+1 ηj v 2 x v xx -108 ξj+1 ηj vv x v xx -216 ξj+1 ηj v 2 v x + 216 ξj+1 ηj vv 2 x = ξj+1 ηj -v 3 xx + 12vv 2 xx + 64v 3 + 60v 2 v xx -54 ξj+1 ηj v 3 x + I 1 + I 2 + I 3 + I 4 .
Applying integration by parts and using that v x (ξ j ) = v x (η j ) = 0, we get

I 1 = 9 ξj+1 ηj ∂ x (v 3 x ) = 0, I 2 = 54 ξj+1 ηj v 3 x , I 3 = -72 ξj+1 ηj ∂ x (v 3 ) = -72v 3 (ξ j+1 ) + 72v 3 (η j )
and

I 4 = 108 ξj+1 ηj ∂ x (v 2 )v x = -108 ξj+1 ηj v 2 v xx .
Therefore

I = ξj+1 ηj -v 3 xx + 12vv 2 xx + 64v 3 + 60v 2 v xx -72v 3 (ξ j+1 ) + 72v 3 (η j ). (3.32)
Similar computations lead to

J = ηj ξj -v 3 x + 12vv 2 xx + 64v 3 + 60v 2 v xx -72v 3 (ξ j ) + 72v 3 (η j ), (3.33) ξ1 -∞ h(x)g 2 (x)dx = ξ1 -∞ -v 3 x + 12vv 2 xx + 64v 3 + 60v 2 v xx -72v 3 (ξ 1 ) (3.34) and +∞ ξ k+1 h(x)g 2 (x) dx = +∞ ξ k+1 -v 3 x + 12vv 2 xx + 64v 3 + 60v 2 v xx -72v 3 (ξ k+1 ). ( 3
ξ k+1 ξ1 h(x)g 2 (x)dx = ξ k+1 ξ1 -v 3 x + 12vv 2 xx + 64v 3 + 60v 2 v xx -72 k j=1 v 3 (ξ j+1 ) -72 k j=1 v 3 (ξ j ) + 72 k j=1 v 3 (η j ). (3.36)
The lemma follows by combining (3.31) and (3.34)-(3.36).

Lemma 3.6 (Connection Between E(•) and

F (•)). Let u ∈ H 1 (R), with y = (1 -∂ 2 x )u ∈ M + (R), that satisfies (3.6) for some ξ ∈ R. Assume that v = (4 -∂ 2
x ) -1 u satisfies (3.13)-(3.20), with local extrema on [η 0 , η k+1 ] arranged in decreasing order in the following way:

M 1 ≥ M 2 ≥ . . . ≥ M k+1 ≥ 0, m 1 ≥ m 2 ≥ . . . ≥ m k ≥ 0, M j+1 ≥ m j , j = 1, . . . , k.
(

3.37)

There exists ε 0 > 0 only depending on the speed c, such that if 0 < ε < ε 0 , then it holds

M 3 1 - 1 4 E(u)M 1 + 1 72 F (u) ≤ 0. (3.38)
Proof. The key is to show that h ≤ 18M 1 on R. Note that by (3.6) we know that 18M 1 ≥ c/4. We rewrite the function h as

h(x) =                      -v xx -6v x + 16v, x < η 0 , -∂ 2 x + 3∂ x + 2 v -3v x + 18v, η 0 < x < ξ 1 , -∂ 2 x -3∂ x + 2 v + 3v x + 18v, ξ j < x < η j , -∂ 2 x + 3∂ x + 2 v -3v x + 18v, η j < x < ξ j+1 , -∂ 2 x -3∂ x + 2 v + 3v x + 18v, ξ k+1 < x < η k+1 , -v xx + 6v x + 16v, x > η k+1 , j = 1, . . . , k.
First, one can remark that for all x ∈ R,

v(x) = e -2x 4 x -∞ e 2x ′ u(x ′ )dx ′ + e 2x 4 +∞ x e -2x ′ u(x ′ )dx ′ and v x (x) = - e -2x 2 x -∞ e 2x ′ u(x ′ )dx ′ + e 2x 2 +∞ x e -2x ′ u(x ′ )dx ′ .
Then using that u = (1 -∂ 2 x ) -1 y ≥ 0 on R, we get (3.19) and (3.39), it holds

|v x (x)| ≤ 2v(x), ∀x ∈ R. (3.39) Next, if x ∈ R \ [η 0 , η k+1 ], using that v xx = 4v -u, (3.18),
h ≤ |v xx | + 6|v x | + 16v ≤ u + 32v ≤ c 9 . If η 0 < x < ξ 1 , then v x ≥ 0, and using that y = (1 -∂ 2 x )u ≥ 0, it follows from Lemma 2.2 that h = -(∂ 2 x + 3∂ x + 2)v -3v x + 18v = -(2 + ∂ x )(4 -∂ 2 x ) -1 (1 + ∂ x )u -3v x + 18v ≤ 18v. If ξ j < x < η j , then v x ≤ 0, and similarly using that y = (1 -∂ 2 x )u ≥ 0, it follows from Lemma 2.2 that h = -(∂ 2 x -3∂ x + 2)v + 3v x + 18v = -(2 -∂ x )(4 -∂ 2 x ) -1 (1 -∂ x )u + 3v x + 18v ≤ 18v. Therefore, it holds h(x) ≤ 18 max x∈R v(x) = 18M 1 , ∀x ∈ R. ( 3
F (u) -144   k j=0 M 3 j+1 - k j=1 m 3 j   = R h(x)g 2 (x)dx ≤ h L ∞ (R) R g 2 (x)dx ≤ 18M 1   E(u) -12   k j=0 M 2 j+1 - k j=1 m 2 j     .
For j = 1, . . . , k, we set

A j = M 3 j+1 -m 3 j and B j = M 2 j+1 -m 2 j
, and our inequality becomes

M 3 1 - 1 4 E(u)M 1 + 1 72 F (u) ≤ 2 k j=1 A j - 3 2 M 1 B j . (3.41)
On the other hand, using that M j+1 ≥ m j , we have

A j - 3 2 M 1 B j = - 1 2 (M j+1 -m j ) 3M 1 m j + 3M 1 M j+1 -2M j+1 m j -2M 2 j+1 -2m 2 j ≤ 0. (3.42)
Finally, combining (3.41) and (3.42), we obtain the lemma.

Proof of Theorem 3.1. We argue as El Dika and Molinet in [START_REF] El | Stability of multipeakons[END_REF]. As noticed after the statement of the theorem, it suffices to prove (3.7) assuming that u ∈ H 1 (R) satisfies (3.1), (3.2) and (3.4)- (3.6). We set M 1 = v(ξ 1 ) = max x∈R v(x) and δ = c/6 -M 1 . We first remark that if δ ≤ 0, combining (3.4) and (3.12), it holds

u -ϕ c (• -ξ 1 ) H ≤ |E(u 0 ) -E(ϕ c )| 1/2 ≤ O(ε),
that yields the desired result. Now suppose that δ > 0, that is the maximum of the function v is less than the maximum of ρ c . Combining (3.4), (3.5) and (3.38), we get

M 3 1 - 1 4 E(ϕ c )M 1 + 1 72 F (ϕ c ) ≤ O(ε 2 ).
Using that E(ϕ c ) = c 2 /3 and F (ϕ c ) = 2c 3 /3, our inequality becomes

M 1 - c 6 2 M 1 + c 3 ≤ O(ε 2 ).
Substituting M 1 by c/6 -δ and using that (M 1 + c/3) -1 < 3/c, it holds

δ 2 ≤ O(ε 2 ) ⇒ δ ≤ O(ε). (3.43)
Finally, combining (3.4), (3.12) and (3.43), we obtain

u -ϕ c (• -ξ 1 ) H ≤ C √ ε,
where C > 0 only depends on the speed c. This completes the proof of the stability of a single peakon.

Stability of the trains of peakons

For γ > 0 and L > 0, we define the following neighborhood of all the sums of N peakons of speed c 1 , ..., c N with spatial shifts z i that satisfied

z i -z i-1 ≥ L, U (γ, L) = u ∈ H 1 (R); inf zi-zi-1>L u - N i=1 ϕ ci (• -z i ) H ≤ γ . (4.1)
By the continuity of the map t → u(t) from [0, T [ into H 1 (R) ֒→ H, to prove Theorem 1.1 it suffices to prove that there exist A > 0, ε 0 > 0 and L 0 > 0 such that for all L > L 0 and 0 < ε < ε 0 , if u 0 satisfies (1.12)-(1.14), and if for some 0 < t 0 < T ,

u(t) ∈ U A( √ ε + L -1/8 ), L 2 , ∀t ∈ [0, t 0 ], (4.2)
then

u(t 0 ) ∈ U A 2 ( √ ε + L -1/8 ), 2L 3 . (4.3)
Therefore, in the sequel of this section we will assume (4.2) for some 0 < ε < ε 0 and L > L 0 , with A, ε 0 and L 0 to be specified later, and we will prove (4.3).

that u H is equivalent to v H 2 (R) , where v = (4 -∂ 2 x ) -1 u. Let t 1 ∈ [0, t 0 ] fixed, if u(t 1 ) ∈ U (γ, L/2), then there exists Z = (z i ) N i=1 with zi -zi-1 ≥ L/2, such that v(t 1 ) ∈ H 3 (R) ֒→ C 2 (R) stays close to N i=1 ρ ci (• -zi ) in the H 2 norm, where ρ ci is defined in (3.8).

Control of the distance between the peakons

In this subsection, we want to prove that the different bumps of u (respectively of v) that are individually close to a peakon (respectively a smooth-peakon) get away from each others as time is increasing. This is crucial in our analysis since we do not know how to manage strong interactions. Lemma 4.1 (Decomposition of the Solution Around ϕ c ). Let u 0 satisfying (1.12)-(1.14). There exist γ 0 > 0, L 0 > 0 and C 0 > 0 such that for all 0 < γ < γ 0 and 0

< L 0 < L, if u(t) ∈ U (γ, L/2) on [0, t 0 ] for some 0 < t 0 < T , then there exist N C 1 functions x1 , . . . , xN defined on [0, t 0 ] such that u(t) - N i=1 ϕ ci (• -xi (t)) H ≤ O(γ), (4.4
)

v(t) - N i=1 ρ ci (• -xi (t)) C 1 (R) ≤ O(γ), (4.5) ẋi (t) -c i ≤ c -2 1 O(γ) + O(e -L/4 ) , i = 1, . . . , N, (4.6
)

and xi (t) -xi-1 (t) ≥ 3L 4 + (c i -c i-1 )t 2 , i = 2, . . . , N. (4.7)
Moreover, for i = 1, . . . , N , setting

J i = [y i (t), y i+1 (t)], with          y 1 = -∞, y i (t) = xi-1 (t) + xi (t) 2 , y N +1 = +∞, i = 2, . . . , N, (4.8
)

it holds ξ i 1 (t) -xi (t) ≤ L 12 , i = 1, . . . , N, (4.9)
where ξ 1 1 (t), . . . , ξ N 1 (t) are any point such that

v t, ξ i 1 (t) = max x∈Ji v(t, x), i = 1, . . . , N, (4.10)
and where v = (4 -∂ 2 x ) -1 u and O(•) only depends on the speeds (c i ) N i=1 .

Proof. We will slightly modify the construction done by El Dika and Molinet in [START_REF] El | Stability of multipeakons[END_REF]. One can remark that the peakons ϕ ci (• -c i t) and the smooth-peakons ρ ci (• -c i t) travel at the same speed c i , thanks to this, we will do our construction with v = (4 -∂ 2

x ) -1 u instead of u. We do that because the H (equivalent to L 2 ) approximation (4.2) does not permit us to construct a C 1 function, which is crucial for application of the Implicit Function Theorem. We note that the same approach can also be used for the CH equation.

For

Z = (z 1 , . . . , z N ) ∈ R N fixed such that |z i -z i-1 | > L/2, we set R Z (•) = N i=1 ρ ci (• -z i ) and S Z (•) = N i=1 ϕ ci (• -z i ). (4.11)
For 0 < γ < γ 0 , we define the function

Y : (-γ, γ) N × B H 2 (R Z , γ) → R N , (y 1 , . . . , y N , v) → Y 1 (y 1 , . . . , y N , v), . . . , Y N (y 1 , . . . , y N , v) with Y i (y 1 , . . . , y N , v) = R   v - N j=1 ρ cj (• -z j -y j )   ∂ x ρ ci (• -z i -y i ).
Y is clearly of class C 1 . For i = 1, . . . , N ,

∂Y i ∂y i (y 1 , . . . , y N , v) = - R     v - 1≤j≤N j =i ρ cj (• -z j -y j )     ∂ 2 x ρ ci (• -z i -y i ) and for j = i, ∂Y i ∂y j (y 1 , . . . , y N , v) = R ∂ x ρ cj (• -z j -y j )∂ x ρ ci (• -z i -y i ).
Hence

∂Y i ∂y i (0, . . . , 0, R Z ) = ∂ x ρ ci 2 L 2 (R) = c 2 i 54 ≥ c 2 1 54
and for j = i, using the exponential decay of ϕ ci and that

|z i -z i-1 | > L/2, for L > L 0 > 0 with L 0 ≫ 1, it holds ∂Y i ∂y j (0, . . . , 0, R Z ) = R ∂ x ρ cj ,α (• -z j )∂ x ρ ci,α (• -z i ) = R ρ cj ,α (• -z j )∂ 2 x ρ ci,α (• -z i ) ≤ 1 9 (c i -α)(c j -α) R e -|x-zj|-|x-zi| dx + 2 R e -|x-zj|-2|x-zi| dx + 1 2 R e -2|x-zj|-|x-zi| dx + R e -2|x-zj|-2|x-zi| dx ≤ O(e -L/4 ).
We deduce that, for L > 0 large enough, D (y1,...,yN ) Y(0, . . . , 0, R Z ) = D + P where D is an invertible diagonal matrix with D -1 ≤ (c 1 /3 √ 6) -2 and P ≤ O(e -L/4 ). Hence there exists L 0 > 0 such that for L > L 0 , D (y1,...,yN ) Y(0, . . . , 0, R Z ) is invertible with an inverse matrix of norm smaller than 2(c 1 /3 √ 6) -2 . From the Implicit Function Theorem we deduce that there exists β 0 > 0 and C 1 functions (y 1 , . . . , y N ) from B H 2 (R Z , β 0 ) to a neighborhood of (0, . . . , 0) which are uniquely determined such that

Y(y 1 (v), . . . , y N (v), v) = 0, ∀v ∈ B H 2 (R Z , β 0 ).
In particular, there exists

C 0 > 0 such that if v ∈ B H 2 (R Z , β), with 0 < β ≤ β 0 , then N i=1 |y i (v)| ≤ C 0 β. (4.12)
Note that β 0 and C 0 only depend on c 1 and L 0 and not on the point (z

1 , . . . , z N ). For v ∈ B H 2 (R Z , β 0 ) we set xi (v) = z i + y i (v). Assuming that β 0 ≤ L 0 /8C 0 , (x 1 (v), . . . , xN (v)) are thus C 1 functions on B H 2 (R Z , β) satisfying xi (v) -xi-1 (v) = z i -z i-1 + y i (v) -y i-1 (v) > L 2 -2C 0 β > L 4 . (4.13)
For L > L 0 and 0 < γ < γ 0 < β 0 /2 to be chosen later, we define the modulation of v in the following way: we cover the trajectory of v by a finite number of open balls in the following way:

{v(t), t ∈ [0, t 0 ]} ⊂ k=1,...,M B H 2 (R Z k , 2γ).
This is possible thanks to Remark 4.1. It is worth noticing that, since 0 < γ < γ 0 < β 0 /2, the functions xi (v) are uniquely determined for

v ∈ B H 2 (R Z k , 2γ) ∩ B H 2 (R Z k ′ , 2γ
). We can thus define the functions

t → xi (t) on [0, t 0 ] by setting xi (t) = xi (v(t)). By construction R   v(t, •) - N j=1 ρ cj (• -xj (t))   ∂ x ρ ci (• -xi (t)) = 0. (4.14)
For 0 < γ < γ 0 , with γ 0 ≪ 1, using that u ∈ U (γ, L/2) and (4.12), we have

u(t) -S X(t) H ≤ u(t) -S Z (t) H + N i=1 ϕ ci (• -z i ) -ϕ ci (• -z i -y i (v(t))) L 2 (R) ≤ γ + √ 2 N i=1 R ϕ 2 ci (x)dx - R ϕ ci (x -z i )ϕ ci (x -z i -y i (v(t)))dx 1/2 = γ + √ 2 N i=1 c i 1 -e -|yi(v(t))| -|y i (v(t))|e -|yi(v(t))| 1/2 ≤ γ + N i=1 O(|y i (v(t))|) ≤ O(γ),
where we apply two time the mean value theorem with the function ϕ on [0, |y i (v(t))|] for substituting (1 -e -|yi(v(t))| ) by |y i (v(t))|e -θ , with θ ∈]0, |y i (v(t))|[, and this proves (4.4) (see Fig. 2a-2b).

The estimate (4.5) follows directly by using (4.4), Remark 4.1 and the Sobolev embedding of

H 2 (R) into C 1 (R).
To prove that the speed of xi (•) stays close to c i , we set

S j (t) = ϕ cj (• -xj (t)), ε 1 (t) = u(t) - N j=1 S j (t)
and

R j (t) = ρ cj (• -xj (t)), ε 2 (t) = v(t) - N j=1 R j (t).
One can notice that

∂ 2 x R i = 4R i -S i , (4.15)
and using the Fourier transformation Differentiating (4.14) with respect to time and using (4.15), we get

(1 -∂ 2 x ) -1 (4 -∂ 2 x ) -1 (•) = F -1 1 (1 + ω 2 )(4 + ω 2 ) (•) = F -1 1 3(1 + ω 2 ) - 1 3(4 + ω 2 ) (•) = 1 3 (1 -∂ 2 x ) -1 (•) - 1 3 (4 -∂ 2 x ) -1 (•). ( 4
R ∂ t ε 2 ∂ x R i = ẋi (t) 4 R ε 2 R i - R ε 2 S i and thus R ∂ t ε 2 ∂ x R i ≤ | ẋi (t)| 4 ε 2 L ∞ (R) R i L 1 (R) + ε 2 L ∞ (R) S i L 1 (R) ≤ | ẋi (t) -c i |O(γ) + O(γ), (4.17)
we point out that S i L 1 (R) = c and R i L 1 (R) = c/2. Substituting u by ε 1 + N j=1 S j in (1.7) and using that S j satisfies

∂ t S j = -( ẋj (t) -c j )∂ x S j - 1 2 ∂ x S 2 j - 3 2 (1 -∂ 2 x ) -1 ∂ x S 2 j ,
we infer that ε 1 satisfies on [0, t 0 ],

∂ t ε 1 - N j=1 ( ẋj (t) -c j )∂ x S j = - 1 2 ∂ x      ε 1 + N j=1 S j   2 - N j=1 S 2 j    - 3 2 ∂ x (1 -∂ 2 x ) -1      ε 1 + N j=1 S j   2 - N j=1 S 2 j    .
Multiplying by (4 -∂ 2 x ) -1 (•) and using (4.16), we get

∂ t ε 2 - N j=1 ( ẋj (t) -c j )∂ x R j = - 1 2 ∂ x (1 -∂ 2 x ) -1      ε 1 + N j=1 S j   2 - N j=1 S 2 j    .
Taking the L 2 scalar product with ∂ x R i , integrating by parts, we find

-( ẋi (t) -c i ) R (∂ x R i ) 2 = - R ∂ t ε 2 ∂ x R i + 1≤j≤N j =i ( ẋj (t) -c j ) R (∂ x R i )(∂ x R j ) + 1 2 R (1 -∂ 2 x ) -1      ε 1 + N j=1 S j   2 - N j=1 S 2 j    ∂ 2 x R i . (4.18)
We set

Q =   ε 1 + N j=1 S j   2 - N j=1 S 2 j = ε 2 1 + 2ε 1   N j=1 S j   + 1≤i,j≤N j =i S i S j , then (1 -∂ 2 x ) -1 Q = (1 -∂ 2 x ) -1 ε 2 1 + 2 N j=1 (1 -∂ 2 x ) -1 (ε 1 S j ) + 1≤i,j≤N j =i (1 -∂ 2 x ) -1 (S i S j ) = I + J + K.
We have the following estimates

I = 1 2 R e -|x-x ′ | ε 2 1 (x ′ ) ≤ 1 2 e -|•| L ∞ (R) ε 1 2 L 2 (R) = 1 2 ε 1 2 L 2 (R) , J = N j=1 R e -|x-x ′ | ε 1 (x ′ )S j (x ′ ) ≤ e -|•| L ∞ (R) ε 1 L 2 (R) N j=1 S j L 2 (R) =   N j=1 c j   ε 1 L 2 (R) and K = 1 2 1≤i,j≤N j =i R e -|x-x ′ | S j (x ′ )S i (x ′ ) ≤ 1 2 1≤i,j≤N j =i R S j (x ′ )S i (x ′ ).
Thus, using (4.4) and the exponential decay of S j , it holds

(1 -∂ 2 x ) -1 Q L ∞ (R) ≤ O(γ) + O(e -L/4 )
and then

1 2 R [(1 -∂ 2 x ) -1 Q]∂ 2 x R i ≤ 1 2 (1 -∂ 2 x ) -1 Q L ∞ (R) ∂ 2 x R i L 1 (R) ≤ O(γ) + O(e -L/4 ), (4.19)
where ∂ 2

x R i L 1 (R) = c/3. Now, combining (4.17), (4.19), and using the exponential decay of

∂ x R i , it holds ẋi (t) -c i ∂ x R i 2 L 2 (R) ≤ ẋi (t) -c i O(γ) + O(γ) + O(e -L/4 ), then ẋi (t) -c i c 2 i 54 -O(γ) ≤ O(γ) + O(e -L/4 ),
which yields (4.6). Taking 0 < γ < γ 0 and L > L 0 > 0 with γ 0 ≪ 1 and L 0 ≫ 1, combining (1.12)-(1.14), (4.6) and (4.13), we deduce that

xi (t) -xi-1 (t) = xi (0) -xi-1 (0) + (c i -c i-1 )t ≥ L -2C 0 γ 0 + (c i -c i-1 )t 2 ≥ 3L 4 + (c i -c i-1 )t 2 ,
this proves (4.7). From (4.5), we infer that

v(x) = N j=1 ρ cj (x -xj ) + O(γ), ∀x ∈ R,
please note that we abuse notation by writing ε 2 (x) = O(γ). Applying this formula with x = ξ i 1 and v(ξ i 1 ) = max x∈Ji v(x), and using (4.7), it holds

v(ξ i 1 ) = max x∈Ji    N j=1 ρ cj (x -xj )    + O(γ) = c i 6 + O(e -L/4 ) + O(γ) ≥ c i 7 .
On the other hand, for x ∈ J i \ [x i (t) -L/12, xi (t) + L/12], we get

v(x) ≤ c i 3 e -L/12 + O(e -L/4 ) + O(γ) ≤ c i 8 .
This ensures that ξ i 1 ∈ [x i (t) -L/12, xi (t) + L/12], and this concluded the proof of the lemma.

Monotonicity property

Thanks to the preceding lemma, for ε 0 > 0 small enough and L 0 > 0 large enough, one can construct N C 1 functions x1 , . . . , xN defined on [0, t 0 ] such that (4.4)-(4.8) are satisfied. In this subsection, we state the almost monotonicity of functionals that are very close to the energy at the right of ith bump, i = 1, . . . , N -1 of u (respectively of v). Let ψ be a C ∞ test-function (see Fig. 3) such that Setting ψ K = ψ(•/K), we introduce for i = 2, . . . , N , Figure 3: ψ (q) (x), q = 0, 1, 2, 3, 4, 5, profiles. Note that ψ (5) will not be used.

0 < ψ(x) < 1, ψ ′ (x) > 0, x ∈ R, |ψ (q) (x)| ≤ 10ψ ′ (x), q = 2,
J i,K (t) = R 4v 2 + 5v 2 x + v 2 xx ψ i,K (t), (4.22)
where

ψ i,K (t, x) = ψ K (x -y i (t)) with y i 's as in (4.8). Note that J i,K (t) is close to u(t) 2 H(x>yi(t)) (respectively to v(t) 2 H 2 (x>yi(t))
) and thus measures the energy at the right of the (i -1)th bump of u (respectively of v). Finally, we set

σ 0 = 1 4 min{c 1 , c 2 -c 1 , . . . , c N -c N -1 }. (4.23)
We have the following monotonicity result.

d dt R 4v 2 + 5v 2 x + v 2 xx g = 2 3 R u 3 g ′ -4 R u 2 vg ′ - 1 2 R u 2 vg ′′′ + 1 2 R u 2 v x g ′′ + R uhg ′ + 1 2 R uh x g ′′ - 5 2 R vh x g ′′ -2 R v x hg ′′ + 1 2 R vh x g (4) (4.25) where y = (1 -∂ 2 x )u, v = (4 -∂ 2 x ) -1 u and h = (1 -∂ 2 x ) -1 u 2 .
The full proof of Lemma 4.2 is given in the Appendix 4.4. Proof of Proposition 4.1. We first note that, combining (4.6) and (4.8), it holds for i = 2, . . . , N ,

ẏi (t) = ẋi-1 (t) + ẋi (t) 2 = c i-1 + c i 2 + O(γ) ≥ c i-1 + O(γ) ≥ c 1 2 . ( 4

.26)

Recall that the assumption (1.12) ensures that u ≥ 0 and v ≥ 0 on R. Now, applying the Virial type identity (4.25) with g = ψ i,K and using (4.26), we get

d dt J i,K (t) = -ẏi R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + 2 3 R u 3 ψ ′ i,K -4 R u 2 vψ ′ i,K - 1 2 R u 2 vψ ′′′ i,K + 1 2 R u 2 v x ψ ′′ i,K + R uhψ ′ i,K + 1 2 R uh x ψ ′′ i,K - 5 2 R vh x ψ ′′ i,K -2 R v x hψ ′′ i,K + 1 2 R vh x ψ (4) i,K ≤ -ẏi R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + 2 3 R u 3 ψ ′ i,K - 1 2 R u 2 vψ ′′′ i,K + 1 2 R u 2 v x ψ ′′ i,K + R uhψ ′ i,K + 1 2 R uh x ψ ′′ i,K - 5 2 R vh x ψ ′′ i,K -2 R v x hψ ′′ i,K + 1 2 R vh x ψ (4) i,K ≤ - c 1 2 R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + 8 k=1 J k . (4.27)
We claim that for k = 1, . . . , 8, it holds

J k ≤ c 1 20 R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + C K u 0 3 H e -1 K (σ0t+L/8) . (4.28)
We divide R into two regions D i and D c i with

D i = xi-1 (t) + L 4 , xi (t) - L 4 , i = 2, . . . , N.
Combining (4.7) and (4.8), one can check that for x ∈ D c i ,

|x -y i (t)| ≥ xi (t) -xi-1 (t) 2 - L 4
≥ c i -c i-1 4 t + L 8 ≥ σ 0 t + L 8 . ( 4

.29)

Let us begin by an estimate of J 1 . Using (2.3), (4.29) and the exponential decay of ψ ′ i,K on D c i , we get

2 3 R u 3 ψ ′ i,K = 2 3 Di u 3 ψ ′ i,K + 2 3 D c i u 3 ψ ′ i,K ≤ 2 3 u L ∞ (Di) R u 2 ψ ′ i,K + 2 3 ψ ′ i,K L ∞ (D c i ) u L ∞ (R) u 2 L 2 (R) ≤ 2 3 u L ∞ (Di) R u 2 ψ ′ i,K + C K u 3 L 2 (R) e -1 K (σ0t+L/8) . (4.30)
Note that, using the exponential decay of

|ψ ′′′ j,K | on D c i , |ψ ′′′ i,K | ≤ (10/K 2 )ψ ′ i,K on D i , with K ≥ 4, and that u H = u 0 H , we have R u 2 ψ ′ i,K = R (4v -v xx ) 2 ψ ′ i,K = 16 R v 2 ψ ′ i,K + R v 2 xx ψ ′ i,K -8 R vv xx ψ ′ i,K = 16 R v 2 ψ ′ i,K + R v 2 xx ψ ′ i,K + 8 R v 2 x ψ ′ i,K + 4 R ∂ x (v 2 )ψ ′′ i,K = 16 R v 2 ψ ′ i,K + R v 2 xx ψ ′ i,K + 8 R v 2 x ψ ′ i,K -4 R v 2 ψ ′′′ i,K ≤ R 16v 2 + 8v 2 x + v 2 xx ψ ′ i,K + 4 Di v 2 |ψ ′′′ j,K | + 4 D c i v 2 |ψ ′′′ j,K | ≤ R 16v 2 + 8v 2 x + v 2 xx ψ ′ i,K + 40 K 2 R v 2 ψ ′ i,K + C K 3 u 0 2 H e -1 K (σ0t+L/8) ≤ 5 R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + C K 3 u 0 2 H e -1 K (σ0t+L/8) . (4.31)
Now, using the exponential decay of ϕ ci on D i , (4.4), and proceeding as for the estimate (3.10) (see Lemma 3.2), it holds

u L ∞ (Di) ≤ u - N j=1 ϕ cj (• -xj (t)) L ∞ (Di) + N j=1 ϕ cj (• -xj (t)) L ∞ (Di) ≤ O(γ 1/2 ) + O(e -L/8 ). (4.32)
Therefore, for 0 < γ < γ 0 and L > L 0 > 0, with γ 0 ≪ 1 and L 0 ≫ 1, combining (4.30)-(4.31), we obtain

J 1 ≤ c 1 20 R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + C K u 3 L 2 (R) e -1 K (σ0t+L/8) .
Next, the estimate of J 2 on D c i gives us

1 2 D c i u 2 vψ ′′′ i,K ≤ 1 2 ψ ′′′ i,K L ∞ (D c i ) v L ∞ (R) u 2 L 2 (R) .
Note that, applying the Hölder inequality, we have for all x ∈ R,

v(x) = 1 4 R e -2|x-x ′ | u(x ′ )dx ′ ≤ 1 4 R e -4|x-x ′ | dx ′ 1/2 R |u(x ′ )| 2 dx ′ 1/2 = 1 4 √ 2 u L 2 (R) (4.33)
and thus, using (4.33) and the exponential decay of

|ψ ′′′ i,K | on D c i , it holds 1 2 D c i u 2 vψ ′′′ i,K ≤ 1 8 √ 2K 3 u 3 L 2 (R) e -1 K (σ0t+L/8) . (4.34) Using that |ψ ′′′ i,K | ≤ (10/K 2 )ψ ′ i,K on D i , the estimate of J 2 on D i leads to 1 2 Di u 2 vψ ′′′ i,K ≤ 5 K 2 u L ∞ (Di) R uvψ ′ i,K . (4.35)
Also, one can notice that, using the exponential decay of |ψ

′′′ j,K | on D c i , |ψ ′′′ i,K | ≤ (10/K 2 )ψ ′ i,K on R, with K ≥ 4, and that u H = u 0 H , we have R uvψ ′ i,K = R (4v -v xx )vψ ′ i,K = 4 R v 2 ψ ′ i,K + R ∂ x (vψ ′ i,K )v x = 4 R v 2 ψ ′ j,K + R v 2 x ψ ′ i,K + R vv x ψ ′′ i,K = R 4v 2 + v 2 x ψ ′ i,K - 1 2 R v 2 ψ ′′′ i,K ≤ R 4v 2 + v 2 x ψ ′ i,K + 1 2 Di v 2 |ψ ′′′ i,K | + 1 2 D c i v 2 |ψ ′′′ i,K | ≤ R 4v 2 + v 2 x ψ ′ i,K + 5 K 2 R v 2 ψ ′ i,K + C K 3 u 0 2 H e -1 K (σ0t+L/8) ≤ 2 R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + C K 3 u 0 2 H e -1 K (σ0t+L/8) . (4.36)
Therefore, for 0 < γ < γ 0 and L > L 0 > 0, with γ 0 ≪ 1 and L 0 ≫ 1, combining (4.32), (4.34)-( 4.36), it holds

J 2 ≤ c 1 20 R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + C K 3 u 3 L 2 (R) e -1 K (σ0t+L/8) .
In the same way, using that |v x | ≤ 2v on R (see (3.39)), and the definition of ψ i,K (see (4.20) and (4.21)), we deduce the estimate of J 3 . Let us tackle now the estimate of J 4 . On D c i we have

D c i uhψ ′ i,K ≤ ψ ′ i,K L ∞ (D c i ) R uh = ψ ′ i,K L ∞ (D c i ) R u[(1 -∂ 2 x ) -1 u 2 ] = ψ ′ i,K L ∞ (D c i ) R u 2 [(1 -∂ 2 x ) -1 u] ≤ ψ ′ i,K L ∞ (D c j ) (1 -∂ 2 x ) -1 u L ∞ (R) u 2 L 2 (R) .
Remark that, applying the Hölder inequality, we have for all x ∈ R,

(1 -∂ 2 x ) -1 u(x) ≤ 1 2 R e -|x-x ′ | u(x ′ )dx ′ ≤ 1 2 R e -2|x-x ′ | dx ′ 1/2 R |u(x ′ )| 2 dx ′ 1/2 = 1 2 u L 2 (R) (4.37)
and thus, using (4.37) and the exponential decay of ψ ′ i,K on D c i , it holds

D c i uhψ ′ i,K ≤ 1 2K u 3 L 2 (R) e -1 K (σ0t+L/8) . (4.38)
The estimate of J 4 on D i leads to

Di uhψ ′ i,K ≤ u L ∞ (Di) R ψ ′ i,K [(1 -∂ 2 x ) -1 u 2 ] = u L ∞ (Di) R u 2 [(1 -∂ 2 x ) -1 ψ ′ i,K ]. (4.39)
On the other hand, using that

|ψ ′′′ i,K | ≤ (10/K 2 )ψ ′ i,K on R, we have (1 -∂ 2 x )ψ ′ i,K (x) = ψ ′ i,K (x) -ψ ′′′ i,K (x) ≥ 1 - 10 K 2 ψ ′ i,K (x), ∀x ∈ R,
and since K ≥ 4, it holds

(1 -∂ 2 x ) -1 ψ ′ i,K (x) ≤ 1 - 10 K 2 -1 ψ ′ i,K (x), ∀x ∈ R. (4.40)
Therefore, for 0 < γ < γ 0 and L > L 0 > 0, with γ 0 ≪ 1 and L 0 ≫ 1, combining (4.32), (4.38)-( 4.40), it holds

J 4 ≤ c 1 20 R 4v 2 + 5v 2 x + v 2 xx ψ ′ i,K + C K u 3 L 2 (R) e -1 K (σ0t+L/8) .
Noticing that for all x ∈ R,

h(x) = e -x 2 x -∞ e x ′ u 2 (x ′ )dx ′ + e x 2 +∞ x e -x ′ u 2 (x ′ )dx ′ and h x (x) = - e -x 2 x -∞ e x ′ u 2 (x ′ )dx ′ + e x 2 +∞ x e -x ′ u 2 (x ′ )dx ′ , we infer that |h x (x)| ≤ h(x), ∀x ∈ R. (4.41)
Then, combining (4.20), (4.41), and proceeding as for the estimate of J 4 , we deduce the estimate of J 5 . Now, combining (4.33) and (4.37), we have for all x ∈ R,

(1 -∂ 2 x ) -1 v(x) = 1 3 (1 -∂ 2 x ) -1 u(x) - 1 3 v(x) ≤ 1 3 (1 -∂ 2 x ) -1 u L ∞ (R) + 1 3 v L ∞ (R) ≤ 4 + √ 2 24 u L 2 (R) , (4.42)
and using the exponential decay of ρ ci on D i and (4.5), it holds

v L ∞ (Di) ≤ v - N j=1 ρ cj (• -xj (t)) L ∞ (Di) + N j=1 ρ cj (• -xj (t)) L ∞ (Di) ≤ O(γ) + O(e -L/
d dt J i,K (t) ≤ C K u 0 3 H e -1 K (σ0t+L/8) .
Integrating between 0 and t, we obtain

J i,K (t) -J i,K (0) ≤ C K u 0 3 H - K σ 0 e -1 K (σ0t+L/8) + K σ 0 e -L 8K ≤ C σ 0 u 0 3 H e -L 8K ,
and this proves the proposition for smooth initial solutions. For u ∈ X ([0, T [), we will use that for any T 0 > 0 and any sequence (u 0,n

) n≥1 ⊂ L 2 (R) such that (u 0,n -∂ 2 x u 0,n ) n≥1 ⊂ M + (R) and u 0,n → u 0 in L 2 (R)
, the sequence of emanating global weak solutions (u n ) n≥1 to the DP equation satisfies

u n -→ n→+∞ u in C [0, T 0 ]; L 2 (R) , (4.44)
where u is the global weak solution emanating from u 0 . This fact can be easily deduced from the proof of the existence of the global weak solutions in [START_REF] Escher | Global weak solutions and blow-up structure for the Degasperis-Procesi equation[END_REF]. Indeed, by the same arguments developed in this proof, we obtain that, up to a subsequence, (

u n ) n≥1 converges in C [0, T 0 ]; L 2 (R)
where v = (4 -∂ 2 x ) -1 u. For all t ∈ [0, T [, we set J n i,K (t) = J i,K (u n (t)) = R 4v 2 n + 5v 2 n,x + v 2 n,xx ψ i,K (t), (4.46)
and we claim that lim

n→+∞ sup 0≤t<T J n i,K (t) -J i,K (t) = 0. (4.47)
Let t ∈ [0, T [ be fixed, we compute

J n i,K (t) -J i,K (t) = 4 R (v 2 n -v 2 )ψ i,K (t) + 5 R (v 2 n,x -v 2 x )ψ i,K (t) + R (v 2 n,xx -v 2 xx )ψ i,K (t) = K n 1 (t) + K n 2 (t) + K n 3 (t). (4.48)
Then it is easy to check that

|K n 1 (t)| ≤ 4 R |v n -v|(v n + v)ψ i,K (t) = 4 v n -v L 2 (R) v n + v L 2 (R) ψ i,K L ∞ (R) ≤ O( v n -v L 2 (R) ) → 0 as n → +∞, (4.49)
and

|K n 2 (t)| ≤ 5 R |v n,x -v x | |v n,x + v x | ψ i,K (t) = 4 v n,x -v x L 2 (R) v n,x + v x L 2 (R) ψ i,K L ∞ (R) ≤ O( v n,x -v x L 2 (R) )
→ 0 as n → +∞. (4.50)

Recalling that v xx = 4v -u and thus v 2 xx = 16v 2 + u 2 -8uv, we also get

|K n 3 (t)| ≤ 16 R |v n -v|(v n + v)ψ i,K (t) + R |u n -u|(u n + u)ψ i,K (t) + 8 R u n |v n -v|ψ i,K (t) + 8 R v|u n -u|ψ i,K (t) ≤ 16 v n -v L 2 (R) v n + v L 2 (R) ψ i,K L ∞ (R) + u n -u L 2 (R) u n + u L 2 (R) ψ i,K L ∞ (R) + 8 u n L 2 (R) v n -v L 2 (R) ψ i,K L ∞ (R) + 8 v L 2 (R u n -u L 2 (R) ψ i,K L ∞ (R) ≤ O u n -u L 2 (R) ) + O( v n -v L 2 (R) → 0 as n → +∞. ( 4

A localized and a global estimate

Let K = √ L/8 and define the function

φ i = φ i (t, x) (see Fig. 4) by      φ 1 = 1 -ψ 2,K = 1 -ψ K (• -y 2 (t)), φ i = ψ i,K -ψ i+1,K = ψ K (• -y i (t)) -ψ K (• -y i+1 (t)), φ N = ψ N,K = ψ K (• -y N (t)), i = 2, . . . , N -1, (4.52)
where ψ i,K 's and y i 's are defined in Subsection 4.2. One can see that the φ i 's are positive functions and that N i=1 φ i = 1. We take L/K > 4 so that φ i satisfies for i = 1, . . . , N ,

|1 -φ i | ≤ 2e -L 8K on y i + L 8 , y i+1 - L 8 (4.53)
and

|φ i | ≤ 2e -L 8K on R \ y i - L 8 , y i+1 + L 8 . ( 4

.54)

We will use the following localized version of the conservation laws defined for i = 1, . . . , N by

E i (t) = R 4v 2 + 5v 2 x + v 2 xx φ i (t) and F i (t) = R -v 3 xx + 12vv 2 xx -48v 2 v xx + 64v 3 φ i (t). (4.55)
One can remark that the functional E i (•) and F i (•) do not depend on time in the statement below since we fix -∞ = y 1 < y 2 < . . . < y N < y N +1 = +∞. For i = 1, . . . , N , we set

Ω i =]y i -L/8, y i+1 + L/8[. First, one can notice that N j=1 ρ cj (x -xj ) = ρ ci (x -xi ) + O(e -L/4 ), ∀x ∈ Ω i , (4.56)
we abuse notation by writing ρ ci (x -xi ) = O(e -L/4 ) for all x ∈ R \ Ω i . We will now decompose this interval according to the variation of v = (4 -∂ 2 x ) -1 u in the same way as in Section 3. We set

α i = sup x < xi , v(x) = c i 2400
and

β i = inf x > xi , v(x) = c i 2400 . ( 4
N i=1 ρ c (• -xi ) in L ∞ norm with ρ ci (0) = c i /6. Therefore v must have at least one local maximum on [α i , β i ]. Assume that on [α i , β i ] the function v admits k i + 1 points (ξ i j ) ki+1 j=1
with local maximal values for some integer k i ≥ 0, where ξ i 1 is the first local maximum point and ξ i ki+1 the last local maximum point3 . Then between ξ i 1 and ξ i ki+1 , the function v admits k i points (η i j ) ki j=1 with local minimal values. We rename α i = η i 0 and β i = η i k+1 so that it holds

η i 0 < ξ i 1 < η i 1 < . . . < ξ i j < η i j < ξ i j+1 < η i j+1 < . . . < η i ki < ξ i ki+1 < η i ki+1 . (4.58)
Let M i j = v(ξ i j ), j = 1, . . . , k i + 1, and m i j = v(η i j ), j = 1, . . . , k i . (4.59)

By construction v x (x) ≥ 0, ∀x ∈ [η i j-1 , ξ i j ], j = 1, . . . , k i (4.60) and v x (x) ≤ 0, ∀x ∈ [ξ i j , η i j ], j = 1, . . . , k i + 1. (4.61)
Proceeding as for (3.18)-(3.20), we also have

v(x) ≤ c i 300 , ∀x ∈ Ω i \ [η i 0 , η i ki+1 ], (4.62)
u(x) ≤ c i 300 , ∀x ∈ Ω i \ [η i 0 , η i ki+1 ], (4.63)
and taking L > L 0 > 8C 0 , it holds

[η i 0 , η i ki+1 ] ⊂ [x i -C 0 , xi + C 0 ] ⊂ y i + L 8 , y i+1 - L 8 , (4.64)
where C 0 > 0 is the universal constant appearing in (3.20). We now derive versions of Lemma 3.4, Lemma 3.5 and Lemma 3.6 where the global functional E(•) and F (•) are replaced by their localized versions E i (•) and F i (•). Please note that, we will change the order of the extrema of v = (4 -∂ 2

x ) -1 u while keeping the same notations as in (4.59).

Lemma 4.3 (Connection Between

E i (•) and the Local Extrema of v). Let u ∈ H 1 (R) and v = (4 - ∂ 2 x ) -1 u ∈ H 3 (R)
. For i = 1, . . . , N , define the function g i by

g i (x) =            2v + v xx -3v x , x < ξ i 1 , 2v + v xx + 3v x , ξ i j < x < η i j , 2v + v xx -3v x , η i j < x < ξ i j+1 , 2v + v xx + 3v x , x > ξ i ki+1 , j = 1, . . . , k i . (4.65) Then it holds R g 2 i (x)φ i (x) = E i (u) -12   ki j=0 (M i j+1 ) 2 φ i (ξ i j+1 ) - ki j=1 (m i j ) 2 φ i (η j )   + u 2 H O(L -1/2 ). (4.66) Proof. We have R g 2 i (x)φ i (x)dx = ξ i 1 -∞ g 2 i (x)φ i (x)dx + ki j=1 ξ i j+1 ξ i j g 2 i (x)φ i (x)dx + +∞ ξ i k i +1 g 2 i (x)φ i (x)dx. (4.67) For j = 1, . . . , k i , ξ i j+1 ξ i j g 2 i (x)φ i (x)dx = η i j ξ i j (2v + v xx + 3v x ) 2 φ i (x)dx + ξ i j+1 η i j (2v + v xx -3v x ) 2 φ i (x)dx = J + I.
Computing I , we obtain

I = ξ i j+1 η i j 4v 2 + v 2 xx + 9v 2 x + 4vv xx -12vv x -6v x v xx φ i = ξ i j+1 η i j 4v 2 + v 2 xx + 9v 2 x φ i + 4 ξ i j+1 η i j vv xx φ i -12 ξ i j+1 η i j vv x φ i -6 ξ i j+1 η i j v x v xx φ i = ξ i j+1 η i j 4v 2 + v 2 xx + 9v 2 x φ i + I 1 + I 2 + I 3 (4.68)
with

I 1 = -4 ξ i j+1 η i j ∂ x (vφ i )v x = -4 ξ i j+1 η i j v 2 x φ i -4 ξ i j+1 η i j vv x φ ′ i = -4 ξ i j+1 η i j v 2 x φ i -2 ξ i j+1 η i j (v 2 ) x φ ′ i = -2v 2 (ξ i j+1 )φ ′ i (ξ i j+1 ) + 2v 2 (η i j )φ ′ i (η i j ) -4 ξ i j+1 η i j v 2 x φ i + 2 ξ i j+1 η i j v 2 φ ′′ i (4.69) I 2 = -6 ξ i j+1 η i j ∂ x (v 2 )φ i = -6v 2 (ξ i j+1 )φ i (ξ i j+1 ) + 6v 2 (η i j )φ i (η i j ) + 6 ξ i j+1 η i j v 2 φ ′ i (4.70)
and

I 3 = -3 ξ i j+1 η i j ∂ x (v 2 x )φ i = 3 ξ i j+1 η i j v 2 x φ ′ i . (4.71)
Adding (4.68)-(4.71), we get

I = ξ i j+1 η i j 4v 2 + 5v 2 x + v 2 xx φ i -6v 2 (ξ i j+1 )φ i (ξ i j+1 ) + 6v 2 (η i j )φ i (η i j ) -2v 2 (ξ i j+1 )φ ′ i (ξ i j+1 ) + 2v 2 (η i j )φ ′ i (η i j ) + R 1 , (4.72)
where using that K = √ L/8, we have

|R 1 | ≤ 6( φ ′ i L ∞ (R) + φ ′′ i L ∞ (R) ) ξ i j+1 η i j (v 2 + v 2 x ) ≤ O(L -1/2 ) ξ i j+1 η i j (v 2 + v 2 x ).
Similar computations lead to

J = η i j ξ i j 4v 2 + 5v 2 x + v 2 xx φ i -6v 2 (ξ i j )φ i (ξ i j ) + 6v 2 (η i j )φ i (η i j ) + 2v 2 (ξ i j )φ ′ i (ξ i j ) -2v 2 (η i j )φ ′ i (η i j ) + R 2 , (4.73)
ξ i 1 -∞ g 2 (x)φ i (x) dx = ξ i 1 -∞ 4v 2 + 5v 2 x + v 2 xx φ i -6v 2 (ξ i 1 )φ i (ξ i 1 ) -2v 2 (ξ i 1 )φ ′ i (ξ i 1 ) + R 3 (4.74) and +∞ ξ i k i +1 g 2 (x)φ i (x) dx = +∞ ξ i k i +1 4v 2 + 5v 2 x + v 2 xx φ i -6v 2 (ξ i 1 )φ i (ξ i ki+1 ) + 2v 2 (ξ i ki+1 )φ ′ i (ξ i ki+1 ) + R 4 , (4.75)
with

|R 2 | ≤ O(L -1/2 ) η i j ξ i j (v 2 + v 2 x ), |R 3 | ≤ O(L -1/2 ) ξ i 1 -∞ (v 2 + v 2 x ) and |R 4 | ≤ O(L -1/2 ) +∞ ξ i k i +1 (v 2 + v 2 x ).
Then, adding (4.72) and (4.73), and summing over j ∈ {1, . . . , k i }, we infer that

ξ i k i +1 ξ i 1 g 2 i (x)φ i (x)dx = ξ i k i +1 ξ i 1 4v 2 + 5v 2 x + v 2 xx φ i -6 ki j=1 v 2 (ξ i j+1 )φ i (ξ i j+1 ) -6 ki j=1 v 2 (ξ i j )φ i (ξ i j ) + 12 ki j=1 v 2 (η i j )φ i (η i j ) -2 ki j=1 v 2 (ξ i j+1 )φ ′ i (ξ i j+1 ) + 2 ki j=1 v 2 (ξ i j )φ ′ i (ξ i j ) + R, (4.76)
with

|R| ≤ O(L -1/2 ) ξ i k i +1 ξ i 1 (v 2 + v 2 x ).
Finally, adding (4.74)-(4.76), and recalling that v H 1 ≤ u H , we obtain the lemma.

Lemma 4.4 (Connection Between F i (•) and the Local Extrema of v). Let u ∈ H 1 (R) and v = (4 - ∂ 2 x ) -1 u ∈ H 3 (R). For i = 1, . . . , N , define the function h i by h i (x) =            -v xx -6v x + 16v, x < ξ i 1 , -v xx + 6v x + 16v, ξ i j < x < η i j , -v xx -6v x + 16v, η i j < x < ξ i j+1 , -v xx + 6v x + 16v, x > ξ i ki+1 , j = 1, . . . , k i . (4.77) Then it holds R h i (x)g 2 i (x)φ i (x) = F i (u) -144   ki j=0 (M i j+1 ) 3 φ i (ξ i j+1 ) - ki j=1 (m i j ) 3 φ i (η i j )   + u 3 H O(L -1/2
). (4.78)

Proof. We have

R h i (x)g 2 i (x)φ i (x)dx = ξ i 1 -∞ h i (x)g 2 i (x)φ i (x)dx + ki j=1 ξ i j+1 ξ i j h i (x)g 2 i (x)φ i (x)dx + +∞ ξ i k i +1 h i (x)g 2 i (x)φ i (x)dx. ( 4

.79)

For j = 1, . . . , k i ,

ξ i j+1 ξ i j h i (x)g 2 i (x)φ i (x)dx = η i j ξ i j (-v xx -6v x + 16v) (2v + v xx -3v x ) 2 φ i + ξ i j+1 η i j (-v xx + 6v x + 16v) (2v + v xx + 3v x ) 2 φ i = J + I.
Computing I, we obtain

I = ξ i j+1 η i j -v 3 xx + 12vv 2 xx + 64v 3 + 60v 2 v xx φ i -54 ξ i j+1 η i j v 3 x φ i + 27 ξ i j+1 η i j v 2 x v xx φ i -108 ξ i j+1 η i j vv x v xx φ i -216 ξ i j+1 η i j v 2 v x φ i + 216 ξ i j+1 η i j vv 2 x φ i = ξ i j+1 η i j -v 3 xx + 12vv 2 xx + 64v 3 + 60v 2 v xx φ i -54 ξ i j+1 η i j v 3
x φ i + I 1 + I 2 + I 3 + I 4 (4.80)

with

I 1 = 9 ξ i j+1 η i j ∂ x (v 3 x )φ i = -9 ξ i j+1 η i j v 3 x φ ′ i , (4.81)
I 2 = -54 ξ i j+1 η i j v∂ x (v 2 x )φ i = 54 ξ i j+1 η i j ∂ x (vφ i )v 2 x = 54 ξ i j+1 η i j v 3 x φ i + 54 ξ i j+1 η i j vv 2 x φ ′ i , (4.82)
I 3 = -72 ξ i j+1 η i j ∂ x (v 3 )φ i = -72v 3 (ξ i j+1 )φ i (ξ i j+1 ) + 72v 3 (η i j )φ i (η i j ) + 72 ξ i j+1 η i j v 3 φ ′ i , (4.83)
and

I 4 = 108 ξ i j+1 η i j ∂ x (v 2 )v x φ i = -108 ξ i j+1 η i j v 2 ∂ x (v x φ i ) = -108 ξ i j+1 η i j v 2 v xx φ i -108 ξ i j+1 η i j v 2 v x φ ′ i = -108 ξ i j+1 η i j v 2 v xx φ i -36 ξ i j+1 η i j ∂ x (v 3 )φ ′ i = -36v 3 (ξ i j+1 )φ ′ i (ξ i j+1 ) + 36v 3 (η i j )φ ′ i (η i j ) -108 ξ i j+1 η i j v 2 v xx φ i + 36 ξ i j+1 η i j v 3 φ ′′ i . (4.84) I = ξ i j+1 η i j -v 3 xx + 12vv 2 xx + 64v 3 -48v 2 v xx φ i -72v 3 (ξ i j+1 )φ i (ξ i j+1 ) + 72v 3 (η i j )φ i (η i j ) -36v 3 (ξ i j+1 )φ ′ i (ξ i j+1 ) + 36v 3 (η i j )φ ′ i (η i j ) + R, (4.85)
where using that

v C 1 (R) ≤ C ′ S v H 2 (R) (with C ′ S the constant of Sobolev), and v H 2 (R) ∼ u H , the estimate of R leads to |R| ≤ ( φ ′ i L ∞ (R) + φ ′′ i L ∞ (R) )( v L ∞ (R) + v x L ∞ (R) ) ξ i j+1 η i j (v 2 + v 2 x ) ≤ O(L -1/2 ) u H ξ i j+1 η i j (v 2 + v 2 x ).
Similar computations lead to

J = η i j ξ i j -v 3 x + 12vv 2 xx + 64v 3 -48v 2 v xx φ i -72v 3 (ξ i j )φ i (ξ i j ) + 72v 3 (η i j )φ i (η i j ) + 36v 3 (ξ i j )φ ′ i (ξ i j ) -36v 3 (η i j )φ ′ i (η i j ) + O(L -1/2 ) u H η i j ξ i j (v 2 + v 2 x ), (4.86)
ξ i 1 -∞ h i (x)g 2 i (x)φ i (x) = ξ i 1 -∞ -v 3 xx + 12vv 2 xx + 64v 3 -48v 2 v xx φ i -72v 3 (ξ i 1 )φ i (ξ i 1 ) -36v 3 (ξ i 1 )φ ′ i (ξ i 1 ) + O(L -1/2 ) u H ξ i 1 -∞ (v 2 + v 2 x ) (4.87) and +∞ ξ i k i +1 h i (x)g 2 i (x)φ i (x) = +∞ ξ i k i +1 -v 3 xx + 12vv 2 xx + 64v 3 -48v 2 v xx φ i -72v 3 (ξ i ki +1 )φ i (ξ i ki+1 ) + 36v 3 (ξ i ki +1 )φ ′ i (ξ i ki+1 ) + O(L -1/2 ) u H +∞ ξ i k i +1 (v 2 + v 2 x ). (4.88)
Adding (4.85) and (4.86), and summing over j ∈ {1, . . . , k i }, we get (4.2). Let be given N -1 real numbers -∞ = y 1 < y 2 < . . . < y N < y N +1 = +∞ with y i -y i-1 ≥ 2L/3. For i = 1, . . . , N , assume that v = (4 -∂ 2

ξ i k i +1 ξ i 1 h i (x)g 2 i (x)φ i (x)dx = ξ i k i +1 ξ i 1 -v 3 x + 12vv 2 xx + 64v 3 -48v 2 v xx φ i -72 ki j=1 v 3 (ξ i j+1 )φ i (ξ i j+1 ) -72 ki j=1 v 3 (ξ i j )φ i (ξ i j ) + 144 ki j=1 v 3 (η i j )φ i (η i j ) -36 ki j=1 v 3 (ξ i j+1 )φ ′ i (ξ i j+1 ) + 36 ki j=1 v 3 (ξ i j )φ ′ i (ξ i j ) + O(L -1/2 ) u H ξ i k i +1 ξ i 1 (v 2 + v 2 x ). ( 4
F i (•)). Let u ∈ H 1 (R), with y = (1 -∂ 2 x )u ∈ M + (R), that satisfies
x ) -1 u satisfies (4.57)-(4.64), with local extrema on [η i 0 , η i ki+1 ] arranged in decreasing order in the following way:

M i 1 ≥ M i 2 ≥ . . . ≥ M i ki+1 ≥ 0, m i 1 ≥ m i 2 ≥ . . . ≥ m i ki ≥ 0, M i j+1 ≥ m i j , j = 1, . . . , k i . (4.90)
There exist γ 0 > 0 and L 0 > 0 only depending on the speeds (c i ) N i=1 , such that if 0 < γ < γ 0 and L > L 0 > 0, then defining the functional E i (•)'s and F i (•)'s as in (4.52)-(4.55), it holds

F i (u) ≤ 18M i 1 E i (u) -72(M i 1 ) 3 + u 3 H O(L -1/2
g 2 i (x)φ i (x) = E i (u) -12   ki j=0 (M i j+1 ) 2 - ki j=1 (m i j ) 2   + u 2 H O(L -1/2
R h i (x)g 2 i (x)φ i (x)dx = F i (u) -144   ki j=0 (M i j+1 ) 3 - ki j=1 (m i j ) 3   + u 3 H O(L -1/2 ). (4.93)
Now, let us show that h i ≤ 18M i 1 on Ω i . Note that by (4.5) and ( 4.56), one can check that 18M i 1 ≥ c i /4. We rewrite the function h i as

h i (x) =                      -v xx -6v x + 16v, x < η i 0 , -∂ 2 x + 3∂ x + 2 v -3v x + 18v, η i 0 < x < ξ i 1 , -∂ 2 x -3∂ x + 2 v + 3v x + 18v, ξ i j < x < η i j , -∂ 2 x + 3∂ x + 2 v -3v x + 18v, η i j < x < ξ i j+1 , -∂ 2 x -3∂ x + 2 v + 3v x + 18v, ξ i ki+1 < x < η i ki+1 , -v xx + 6v x + 16v, x > η i ki+1 , j = 1, . . . , k i . Then, if x ∈ Ω i \ [η i 0 , η i ki+1 ], using that v xx = 4v -u, ( 3
.39), (4.64) and (4.65), it holds

h i ≤ |v xx | + 6|v x | + 16v ≤ u + 32v ≤ c i 9 .
If η i 0 < x < ξ i 1 , then v x ≥ 0, and using that y = (1 -∂ 2 x )u ≥ 0, it follows from Lemma 2.2 that

h i = -(∂ 2 x + 3∂ x + 2)v -3v x + 18v = -(2 + ∂ x )(4 -∂ 2 x ) -1 (1 + ∂ x )u -3v x + 18v ≤ 18v.
If ξ i j < x < η i j , then v x ≤ 0, and similarly using that y = (1 -∂ 2 x )u ≥ 0, it follows from Lemma 2.2 that

h i = -(∂ 2 x -3∂ x + 2)v + 3v x + 18v = -(2 -∂ x )(4 -∂ 2 x ) -1 (1 -∂ x )u + 3v x + 18v ≤ 18v. Therefore, it holds h i (x) ≤ 18 max x∈Ωi v(x) = 18M i 1 , ∀x ∈ Ω i . (4.94)
Now, taking φ i ≡ 1 on R in (4.66), we have g i L 2 (R) ≤ u H . Also, from the definition of h i , and using (2.3) and Remark 4.1, we have

h i L ∞ (R) ≤ u L ∞ (R) + 32 v L ∞ (R) ≤ O( u H ).
Then, combining (4.92)-(4.94), we obtain

F i (u) -144   ki j=0 (M i j+1 ) 3 - ki j=1 (m i j ) 3   = R h i (x)g 2 i (x)φ i (x)dx + u 3 H O(L -1/2 ) = Ωi h i (x)g 2 i (x)φ i (x)dx + Ω c i h i (x)g 2 i (x)φ i (x)dx + u 3 H O(L -1/2 ) ≤ 18M i 1 Ωi g 2 i (x)φ i (x)dx + h i L ∞ (R) g i 2 L 2 (R) φ i L ∞ (Ω c i ) + u 3 H O(L -1/2 ) ≤ 18M i 1   E i (u) -12   ki j=0 (M i j+1 ) 2 - ki j=1 (m i j ) 2     + u 3 H O(L -1/2 ).
Therefore, using that M i j+1 ≥ m i j and proceeding as in Lemma 3.6 (see (3.42)), we infer that

F i (u) ≤ 18M i 1 E i (u) -72(M i 1 ) 3 + 144 ki j=1 (M i j+1 ) 3 -(m i j ) 3 - 3 2 M i 1 (M i j+1 ) 2 -(m i j ) 2 + u 3 H O(L -1/2 ) ≤ 18M i 1 E i (u) -72(M i 1 ) 3 + u 3 H O(L -1/2 ).
This proves the lemma.

The lemma below is the generalization of Lemma 3.3.

Lemma 4.6 (General Quadratic Identity). Let Z = (z i ) N i=1 ∈ R N with |z i -z i-1 | ≥ L/2, and u ∈ L 2 (R). It holds E(u) - N i=1 E(ϕ ci ) = u -S Z 2 H + 4 N i=1 c i v(z i ) - c i 6 + O(e -L/4 ), (4.95)
where S Z is defined in (4.11) and O(•) only depends on (c i ) N i=1 .

Proof. Let us compute

u -S Z 2 H = R [(1 -∂ 2 x )(u -S Z )][(4 -∂ 2 x ) -1 (u -S Z )] = u 2 H + S Z 2 H -2 R [(1 -∂ 2 x )S Z ][(4 -∂ 2 x ) -1 u] = u 2 H + S Z 2 H -2 N i=1 c i R [(1 -∂ 2 x )ϕ ci (x -z i )]v = u 2 H + S Z 2 H -4 N i=1 c i v(z i ), (4.96)
where we use that (1 -∂ 2 x )ϕ ci (• -z i ) = 2c i δ zi with δ zi the Dirac mass applied at point z i . We also have

S Z 2 H = R [(1 -∂ 2 x )S Z ][(4 -∂ 2 x ) -1 S Z ] = N i=1 R [(1 -∂ 2 x )ϕ ci (x -z i )][(4 -∂ 2 x ) -1 S Z ] = 2 N i=1 c i δ zi , (4 -∂ 2 x ) -1 S Z H -1 ,H 1 = 2 1≤i,j≤N c i δ zi , (4 -∂ 2 x ) -1 ϕ cj (• -z j ) H -1 ,H 1 = 2 N i=1 c i (4 -∂ 2 x ) -1 ϕ ci (0) + 2 1≤i,j≤N i =j c i c j (4 -∂ 2 x ) -1 e -|zi-zj | , (4.97)
where •, • H -1 ,H 1 denote the duality H -1 /H 1 , and we recall that

δ zi ∈ H -1 (R) since δ zi H -1 (R) ≤ C S , with C S the constant appearing in (2.3). Now, using that |z i -z i-1 | ≥ L/2, (4 -∂ 2 x ) -1 e -|zi-zj| = 1 4 R e -2|x ′ -(zi-zj )| e -|x ′ | dx ′ = 1 3 e -|zi-zj | - 1 6 e -2|zi-zj | = O(e -L/4 ), (4.98)
and combining (4.97) and (4.98), for L > L 0 > 0 with L 0 ≫ 1, we get

S Z 2 H = N i=1 E(ϕ ci ) + O(e -L/4
E(u 0 ) - N i=1 E(ϕ ci ) ≤ O(ε 2 ) + O(e -L/4 ), (4.100) |E i (u 0 ) -E(ϕ ci )| ≤ O(ε 2 ) + O(e - √ L ), i = 1, . . . , N, (4.101
)

and |F i (u 0 ) -F (ϕ ci )| ≤ O(ε 2 ) + O(e - √ L ), i = 1, . . . , N, (4.102)
where O(•) only depend on (c i ) N i=1 . Proof. For the first estimate, applying triangular inequality and (1.14), we have

|E(u 0 ) -E(S Z 0 )| = | u 0 H -S Z 0 H | ( u 0 H + S Z 0 H ) ≤ u 0 -S Z 0 H ( u 0 -S Z 0 H + 2 S Z 0 H ) ≤ ε 2 ε 2 + 2 √ 3 N i=1 c i . ( 4
E(u 0 ) - N i=1 E(ϕ ci ) ≤ |E(u 0 ) -E(S Z 0 )| + E(S Z 0 ) - N i=1 E(ϕ ci ) ≤ ε 2 (ε 2 + O(1)) + O(e -L/4 ) ≤ O(ε 2 ) + O(e -L/4 ).
For the second estimate, using the exponential decay of ϕ ci 's and the φ i 's, and the definition of E i (•), we have

|E i (u 0 ) -E(ϕ ci )| ≤ u 0 2 H(Ωi) -ϕ ci 2 H(Ωi) + O(e - √ L ) = u 0 H(Ωi) -ϕ ci H(Ωi) u 0 H(Ωi) + ϕ ci H(Ωi) + O(e - √ L ) ≤     u 0 -S Z 0 H(Ωi) + 1≤j≤N j =i ϕ cj H(Ωi)       u 0 -S Z 0 H + 2 √ 3 N j=1 c j   + O(e - √ L ) ≤ ε 2 + O(e -L/8 ) ε 2 + O(1) + O(e - √ L ) ≤ O(ε 2 ) + O(e - √ L ).
Similarly, for the third estimate, using the exponential decay of ϕ ci 's and the φ i 's, and the definition of F i (•), we have

|F i (u 0 ) -F (ϕ ci )| ≤ Ωi u 3 0 -ϕ 3 ci + O(e - √ L ) ≤ Ωi |u 0 -ϕ ci | u 2 0 + u 0 ϕ ci + ϕ 2 ci + O(e - √ L ) ≤ u 0 -ϕ ci L 2 (Ωi) Ωi u 2 0 + u 0 ϕ ci + ϕ 2 ci 2 1/2 + O(e - √ L ) ≤     u 0 -S Z 0 L 2 (Ωi) + 1≤j≤N j =i ϕ cj L 2 (Ωi)     • O(1) + O(e - √ L ) ≤ ε 2 + O(e -L/8 ) • O(1) + O(e - √ L ) ≤ O(ε 2 ) + O(e - √ L ).
This proves the lemma.

End of the proof of Theorem 1.1

Let u ∈ X ([0, T [), with 0 < T ≤ +∞, be a solution of the DP equation satisfying (1.12)-(1.14) and (4.2) for some t 0 ∈]0, T [. Let M i 1 = v(t 0 , ξ i 1 (t 0 )) = max x∈Ji v(t 0 , x), with J i 's as in (4.8), and δ i = c i /6 -M i 1 . First, from (4.7) and (4.9), we know that for i = 2, . . . , N ,

ξ i 1 (t 0 ) -ξ i-1 1 (t 0 ) ≥ 2L 3 > L 2 .
Applying (4.95) and (4.100) with u(t 0 ), we get

u(t 0 ) - N i=1 ϕ ci (• -ξ i 1 (t 0 ))
δ 2 i M i 1 + c i 3 ≤ 1 4 N i=1 M i 1 ∆ t0 0 E i (u) + O(ε 2 ) + O(L -1/2 ),
and using the Abel transformation with M 0 1 = 0, we obtain

N i=1 δ 2 i M i 1 + c i 3 ≤ 1 4 N i=2 (M i 1 -M i-1 1 )∆ t 0 J i,K + O(ε 2 ) + O(L -1/2 ) , (4.109)
where J i,K (t) is defined in (4.22). From (4.2) we know that u(t 0 ) ∈ U (γ, L/2), on account of Lemma 4.1 there exists X = (x 1 , . . . , xN ) with xi ∈ J i such that E (u(t 0 ) -S X ) ≤ O(γ 2 ), where S X is defined in (4.11). Recalling that v(t 0 , ξ i 1 (t 0 )) = max x∈Ji v(t 0 , x) and using (4.95), we obtain E (u(t 0 ) -S ξ1 ) ≤ O(γ 2 ) + O(e -L/4 ), with ξ 1 = (ξ 1 1 , . . . , ξ N 1 ). From (4.5), we deduce that

v(t 0 ) - N j=1 ρ cj (• -ξ j 1 (t 0 )) L ∞ (R)
≤ O(γ) + O(e -L/8 ).

Thus, we infer that v(x) = N j=1 ρ cj (• -ξ j 1 (t 0 )) + O(γ) + O(e -L/8 ), ∀x ∈ R, and applying this formula with x = ξ i 1 (t 0 ) and using that ξ j 1 (t 0 ) -ξ j-1 1 (t 0 ) > L/2, we get v(ξ i 1 (t 0 )) = We take γ = A( √ ε + L -1/8 ), then M i 1 = c i /6 + O( √ ε) + O(L -1/8 ). Therefore, for 0 < ε < ε 0 and L > L 0 > 0, with ε 0 ≪ 1 and L 0 ≫ 1, it holds 0 < M

δ 2 i M i 1 + c i 3 ≤ O(ε 2 ) + O(L -1/8 ).
Therefore, using that (M i 1 + c i /3) -1 < 3/c i , there exists C > 0 only depending on (c i ) N i=1 such that δ i ≤ C(ε + L -1/4 ), i = 1, . . . , N.

Appendix. Proof of Lemma 4.2

The aim of this subsection is to prove Lemma 4.2. Let us first assume that u is smooth solution. The case u ∈ X ([0, T [) will follow by a density argument.

We compute the time variation of the following energy:

d dt R yvg = R y t vg + R yv t g = I + J.
Applying the operator (1 -∂ 2 x )(•) on both sides of equation (1.7), we get

y t = - 1 2 (1 -∂ 2 x )∂ x u 2 - 3 2 ∂ x u 2
At this stage it is worth noticing that the term R uh x g cancels with the one in J. Finally, we obtain

J = - 2 3 R u 3 g ′ + R uhg ′ + 1 2 R uh x g ′′ . ( 4
d dt R yvg = 2 3 R u 3 g ′ -4 R u 2 vg ′ - 3 2 R u 2 v x g ′′ - 1 2 R u 2 vg ′′′ + R uhg ′ + 1 2 R uh x g ′′ . (4.123)
Now, substituting u by 4v -v xx and using integration by parts, we rewrite the energy as

R yvg = R v (1 -∂ 2 x )(4v -v xx ) g = 4 R v 2 g -5 R vv xx g + R v(∂ 4 x v)g = 4 R v 2 g + K 1 + K 2 .
By computing

K 1 = 5 R ∂ x (vg)v x = 5 R v 2 x g + 5 R vv x g ′ = 5 R v 2 x g + 5 2 R ∂ x (v 2 )g ′ = 5 R v 2 x g - 5 2 R v 2 g ′′ (4.124)
and

K 2 = - R ∂ x (vg)v xxx = - R v x v xxx g - R vv xxx g ′ = K 3 + K 4 (4.125)
with

K 3 = R ∂ x (v x g)v xx = R v 2 xx g + R v x v xx g ′ = R v 2 xx g + 1 2 R ∂ x (v 2 x )g ′ = R v 2 xx g - 1 2 R v 2
x g ′′ (4.126) 40 and 4) .

K 4 = R ∂ x (vg ′ )v xx = R v x v xx g ′ + R vv xx g ′′ = 1 2 R ∂ x (v 2 x )g ′ - R ∂ x (vg ′′ )v x = - 1 2 R v 2 x g ′′ - R v 2 x g ′′ - R vv x g ′′′ = - 3 2 R v 2 x g ′′ - 1 2 R ∂ x (v 2 )g ′′′ = - 3 2 R v 2 x g ′′ + 1 2 R v 2 g (

. 1 )Remark 2 . 1 (

 121 Moreover E(•) and F (•) are conserved by the flow. Control of L ∞ Norm by L 2 Norm). From (2.1), it holdsu(x) = e -x 2 x -∞ e x ′ y(x ′ )dx ′ + e x 2 +∞ x e -x ′ y(x ′ )dx ′ and u x (x) = -e -x 2 x -∞ e x ′ y(x ′ )dx ′ + e x 2 +∞ x e -x ′ y(x ′ )dx ′ ,which lead to |u x (x)| ≤ u(x), ∀x ∈ R.

  ϕ(x) and ρ(x) profiles.

  ϕ ′ (x) and ρ ′ (x) profiles.

Figure 1 :

 1 Figure 1: Variation of peakon and smooth-peakon at initial time with the speed c = 1.

  .13) According to Lemma 3.2, we know that v is close to ρ c (• -ξ) in L ∞ norm with ρ c (0) = c/6. Therefore v must have at least one local maximum on [α, β]. Assume that on [α, β] the function v admits k + 1 points (ξ j ) k+1 j=1 with local maximal values for some integer k ≥ 0, where ξ 1 is the first local maximum point and ξ k+1 the last local maximum point 2 . Then between ξ 1 and ξ k+1 , the function v admits k points (η j ) k j=1

  .40) Now, combining (3.23), (3.30) and (3.40), we get

Remark 4 . 1 (

 41 Distance Between v and the Sum of N Smooth-peakons). From the definition of E(•) and H (see respectively (1.3) and (1.6)), one can clearly see

0

  |ϕ(x -10) -ϕ(x -10 -10 -9 )| 2 1/2 ≈ 10 -9 log x 0 |ϕ(x -10) -ϕ(x -10 -10 -9 )| 2 1/2 ≈ -20.72

Figure 2 :

 2 Figure 2: Distance between two very close peakons (at time t = 10 with respective speeds 1 and 1+10 -10 ).

3 , 4 , 5 ,

 345

Proposition 4 . 1 (

 41 Exponential Decay of the Functional J i,K (t)). Let u ∈ X ([0, T [), with 0 < T ≤ +∞, be a solution of equation (1.1) that satisfies (1.12)-(1.14) and (4.4)-(4.5). There exist γ 0 > 0 and L 0 > 0 only depending on c 1 such that if 0 < γ < γ 0 and L > L 0 > 0, then for any 4 ≤ K √ L,J i,K (t) -J i,K (0) ≤ O(e -L 8K ), ∀t ∈ [0, t 0 ], i = 2, . . . , N. (4.24)The proof of Proposition 4.1 relies on the following Virial type identity. Lemma 4.2 (Virial Type Identity). Let u ∈ X ([0, T [), with 0 < T ≤ +∞, be a solution of equation (1.1) that satisfies (1.12)-(1.14). For any smooth space function g : R → R, it holds

Figure 4 :

 4 Figure 4: Localization-function φ green (x) = ψ(x -15) -ψ(x -65) (at time t = 10) profile. Also, the peakon 4ϕ(x -40) and the smooth-peakon 4ρ(x -40) (at time t = 10 with speed c = 4) profiles. In this example, one can see that φ green is close to 1 on ]25, 55[, and decays exponentially to 0 on R\]10, 70[.

  .89) Finally, adding (4.87)-(4.89), we obtain the lemma.

Lemma 4 . 5 (

 45 Connection Between E i (•) and

c

  i δ i + O(ε 2 ) + O(e -L/4 ). (4.104)In the same way, from (4.91) we get

F 1 ) 3 + 1 )

 131 i (u(t 0 )) ≤ 18M i 1 E i (u(t 0 )) -72(M i 1 ) 3 + O(L -1/2 ), O(L -1/2 ) ,(4.105)by summing over i ∈ {1, . . . , N }. Now, we will use the following notation: for a function f : R + → R, we set∆ t0 0 f = f (t 0 ) -f (0). (4.106)From (4.105) and the fact that E(•) and F (•) are conservation laws for u,3 + 18M i 1 E i (u 0 ) -F i (u 0 ) + O(L -1/2 ). (4.107) Note that, from (4.101) and (4.102), for 0 < ε < ε 0 and L > L 0 > 0 with ε 0 ≪ 1 and L 0 ≫ 1, it holds N i=1 -72(M i 1 ) 3 + 18M i 1 E i (u 0 ) -F i (u 0 ) = -72 107) and (4.108), we get N i=1

ρρc i 6 +

 6 cj (ξ i 1 (t 0 ) -ξ j 1 (t 0 )) + O(γ) + O(e -L/8 ) cj (ξ i 1 (t 0 ) -ξ j 1 (t 0 )) + O(γ) + O(e -L/8 ) = O(γ) + O(e -L/8).

  -1/8 ), and the theorem follows by choosing A = 2C.Remark 4.2 (The Role of the Number of Extrema). In the case where v = (4 -∂ 2 x ) -1 u admits a infinite number of local maximal values on some [α i , β i ] (see (4.57)), with i ∈ {1, . . . , N }, it suffices to change the finite sums over j by infinite sums in Lemmas 3.4-3.5 and Lemmas 4.3-4.4.

  towards a solution of the DP equation emanating from u 0 . (4.44) then follows by the uniqueness result. Combining (4.44) and Remark 4.1, it follows that v n -→

n→+∞ v in C [0, T 0 ]; H 2 (R) ,

(4.45)

W 1,1 (R) is the space of L 1 (R) functions with derivatives in L 1 (R) and BV (R) is the space of function with bounded variation.

In the case of an infinite countable number of local maximal values, the proof is exactly the same.

In the case of an infinite countable number of local maximal values, the proof is exactly the same.

Acknowledgements. The author would like to thank his PhD advisor Luc Molinet for his help and his careful reading of this manuscript.

and substituting y t by this value, I becomes

By computing

and

with

and

Adding (4.113) and (4.114), we get

and adding (4.112) and (4.115), we get

The first two integrals give us

Finally, we obtain

Applying the operator (4 -∂ 2 x ) -1 (•) on both sides of equation (1.7) and using (4.16), we get

Substituting v t by this value, J becomes

By computing

and

and

Using that h xx = -u 2 + h and h xxx = -2uu x + h x , we have