Construction of solutions to the L

Construction of solutions to the L ² -critical KdV equation with a given asymptotic behaviour

Raphaël Côte

Abstract

We consider the critical Korteweg-de Vries equation : ut+ (uxx+u⁵)x= 0, t, x∈R.

LetRj(t, x) =Qc_j(x−xj−cjt)(j= 1, . . . , N) beN soliton solutions to this equation. DenoteU(t)the KdV linear group, and letV ∈H¹be with sufficient decay on the right, that is(1 +x^2+δ₊ ⁰)V ∈L² for someδ0>0.

We construct a solutionu(t)to the critical Korteweg-de Vries equation such that

t→∞lim

‚

‚

‚u(t)−U(t)V −

N

X

j=1

Rj(t)

‚

‚

‚H¹

= 0.

1 Introduction.

1.1 General setting

We consider the critical Korteweg-de Vries equation :

ut+ (uxx+u⁵)x= 0, t, x∈R. (1) It is a special case of the generalized KdV equation :

ut+ (uxx+u^p)x= 0, t, x∈R, (2) wherep≥2. The casep= 2corresponds to the original equation introduced by Korteweg and de Vries [8] in the context of shallow water waves. For bothp= 2 and p= 3, this equation has many applications to Physics : see for example Miura [20], Lamb [10].

There are two formally conserved quantities for solutions to (2) : Z

u²(t) = Z

u²(0) (L² mass), (3)

E(u(t)) = 1 2

Z

u²_x(t)− 1 p+ 1

Z

u^p+1(t) =E(u(0)) (energy). (4) The local Cauchy problem for (2) has been intensively studied by many authors.

Kenig, Ponce and Vega [6] proved the following existence and uniqueness result

2000 Mathematical Subject Classification : Primary 35Q53, Secondary 35B40, 35Q51.

Keywords : Critical and Generalized Korteweg-de Vries equations, wave operator, large data, solitons, linear-non linear interaction.

in H¹(R) : for u₀ ∈ H¹(R), there exist T = T(ku₀k_H1) > 0 and a solution u∈ C([0, T], H¹(R)) to (2) satisfying u(0) = u₀, which is unique in the class YT ⊂C([0, T], H¹(R)). Moreover, ifT1 denotes the maximal time of existence for u, then eitherT1 = +∞ (global solution) orT1 <∞ and ku(t)k_H1 → ∞ as t↑T1 (blow-up solution). For such a solution, one has conservation of mass and energy. In the critical case, problem 1), this result is improved to local well-posedness in L² (see [6] and [7]).

The next problem is to know whether these solutions to (2) are global in time, or blow-up. In the case 2 ≤ p < 5 (sub-critical), all solutions in H¹ are global and uniformly bounded thanks to the conservation laws and the Gagliardo-Nirenberg inequality :

∀v∈H¹(R), Z

|v|^p+1≤κ(p) Z

v²

^p+3₄ Z v_x²

^p−1₄

. (5)

The casep= 5isL²-critical, in the sense that mass remains unaffected by scal- ing. Indeed, u_λ(t, x) =λ^1/6u(λt, λ^1/3x)is also a solution to (1), andku_λk_L2 = kuk_L2. Moreover, existence of finite time blow-up solutions was proved by Merle [19] and Martel and Merle [15]. Thereforep= 5also appears as a critical expo- nent for the long time behaviour of solution to (2).

Forp >5(super-critical case), numerics predict blow-up.

A fundamental property of (2) is the existence of a family of explicit traveling wave solutions. IfQdenotes the only solution (up to translation) of :

Q >0, Q∈H¹(R), Q_xx+Q^p=Q, i.e. Q(x) =

3 cosh²(2x)

1/(p−1)

, then forc >0 the soliton

R_c,x0 =c^1/(p−1)Q(√

c(x−x₀−ct))is a solution to(2).

Solitons are stable inH¹in the sub-critical casep∈[2,5)(see [13], and unstable in thep >5super-critical (see [22]) and p= 5critical case (see [14]).

For p= 2 and p = 3, equation (2) is completely integrable, and thus has very special features. The inverse scattering transform method allows to solve the Cauchy problem in an appropriate space (for example if u₀ ∈ H⁴ and xu₀ ∈ L¹) and the qualitative behaviour of solutions is well understood. For example, given u₀ smooth and with rapid decay, there exist N solitons R_cj,xj such that

u(t)−

N

X

j=1

R_cj,xj(t)

_{L∞(x≥−t1/3)}≤ C

t^1/3 (as t→ ∞).

See for example Schuur [21], Eckhaus and Schuur [4], Miura [20].

However, if p 6= 2 or 3, the inverse scattering transform method does not longer apply, and the description of solutions in the general, non-integrable case is a widely open problem, especially in the critical case. It can be decomposed in two types of problems.

Problem 1 : Asymptotic behavior. In the sub-critical case, given an initial data u0, can we describe the behavior of the out-coming solutionu(t)to (2) ? In the

critical and super-critical cases, doesu(t)blow-up ? Can we determine the blow up rate and profile ?

Problem 2 : Construction of a non-linear wave operator. Given some reasonable behavior at t→ ∞, can we find a solution u(t)to (2) defined for large enough t, with this behavior ? Is there uniqueness foru(t)?

1.2 Recent results on Problems 1. and 2. for the critical KdV equation

From now on, we will focus only on equation (1), that is, theL²-critical case.

Let us now develop some results which will be the base to our result. The first result deals with scattering for small initial data. One wants to prove that given an initial data u₀, small in an adequate functional space, the arising solution to the non-linear equation (1) behaves likes U(t)v₀, the solution to the linear equationu_t+u_xxx= 0, with initial datav₀(U(t)is the linear KdV group). The map u₀7→ v₀ is called the scattering operator. The following result is an easy corollary of Kenig, Ponce and Vega [6].

Scattering operator. Givenu0small enough in L², the out-coming solutionu(t) to (1) is global in time, and there is scattering, in the sense that there exists a function V ∈L² so that

ku(t)−U(t)Vk_L2 →0 as t→ ∞.

This is the description of solutions with initial data around0(inL²), a result which can be understood as stability around 0.

The second point answers the question of behavior of solutions with initial data close to a soliton. As we are in the critical case, one does not have stability : contrary to the sub-critical case (see [14]), one has instability and blow-up. Let us cite a result of Merle [19] and Martel and Merle [16].

Blow-up solutions to (1). There exists α₀ > 0 such that the following is true.

Suppose

E(u)<0 and Z

u(t)²≤ Z

Q²+α0.

Then u(t) blows-up in finite or infinite time T ∈ (0,∞]. Furthermore, there exist λ(t)> C(T−t)^−1/3,ε∈ {−1,1} andx(t)∈Rsuch that

ελ^1/2(t)u(t, λ(t)x+x(t))* Q in H¹-weak as t↑T.

These results are related to Problem 1. Let us now turn to results concerning Problem 2. A surprising result of Martel [11] is the existence and uniqueness of N-solitons in the critical case :

Existence and uniqueness of the N-soliton. Let p ∈ [2,5]. Let N ∈ N, 0 <

c1< . . . < cN, and x1, . . . , xN ∈R. There exist T0 ∈Rand a unique function u∈C([T0,+∞),R), which is aH¹ solution to (2), and such that

u(t)−

N

X

j=1

Q_cj(· −x_j−c_jt) _H1

→0 as t→ ∞.

Furthermore, u∈C^∞([T₀,∞)×R) and convergence takes place inH^s for all s≥0, with an exponential decay :

∀s≥0,∃As /

u(t)−

N

X

j=1

Qc_j(· −xj−cjt) _H1

≤Ase^−γt,

whereγ=σ0

√σ0/32 andσ0= min(c1, c2−c1, . . . , cN−cN−1).

This result appears as a development of monotonicity properties and a dy- namical argument, ideas which where used by Martel and Merle [13] and Martel, Merle and Tsai [17]. It is a surprise that the argument applies also in the critical casep= 5, although it fails in the proof of stability (failure which isn’t due to a lack in the proof, but to true instability : see [14], [16]). The second surprise is the uniqueness of a solution behaving as a sum ofN solitons.

The last result solves the case of a linear behavior, that is the existence of a wave operator (see [3]).

Large data wave operator. Let V ∈ L². There exist T0 ∈ R and a function u∈C([T0,∞), L²)solution to (1) such that

ku(t)−U(t)Vk_L2 →0 as t→ ∞.

Furthermore uis unique in an adapted class.

In the same way that the result of Martel [11] was based on considerations of Martel, Merle and Tsai [17], this result relies on the analysis of Kenig, Ponce and Vega [6].

1.3 Statement of the main result

Our goal in this article is to construct solutions which behave like a sum of a linear term U(t)V, and of N solitons, for the L²-critical Korteweg-de Vries equation (1). Our main result is the following.

Theorem 1 (Nonlinear wave operator for (1)). LetV ∈H¹have sufficient decay on the right, i.e. such that (1 +x₊)^2+δ0V(x)∈L² for some δ₀ >0 (we denote x₊ = max{0, x}).

Let N ∈N,0< c₁< . . . < c_N andx₁, . . . , x_N ∈R. LetR_j(t, x) =Q_cj(x− x_j−c_jt), for j= 1, . . . , N, be N solitons.

Then there exists u^∗ ∈C([T₀,+∞), H¹), for some T₀ ∈R, solution to (1) and such that u^∗(t) is uniformly bounded inH¹ and

u^∗(t)−U(t)V −

N

X

j=1

Rj(t) _H1

→0 as t→ ∞. (6)

Theorem 1 allows to work with large data (V large in L²), which is both surprising and satisfactory. The decay on the right we assume forV is to ensure low interaction with the solitons. This result should be viewed as a step in the solving process of Problem 2.

Remark 1. This result essentially units the linear approach contained in [7] and [3], and the solitons related approach, developed in [18] and [11]. The difficulty is to mix both methods together, so that they do not break down.

An important change in the method of proof when considering [11] is the following. Solitons have an exponential decay, and so integrability (in time) is always automatic. Here the linear termU(t)V will interact with the solitons to produce a polynomial decay in time, which will require to be taken care of.

Remark 2. This result is analogous to that obtained in [2], where a non-linear wave operator is constructed in the sub-critical case p = 4. However, in the sub-critical case, much more decay and smoothness are required on V. This is due to the fact that the linear scattering analysis of [7] is no longer available if p6= 5.

In the sub-critical case,we have to relie on the scattering theory of Hayashi and Naumkin [5]. There it is proved scattering for small datau₀∈H^1,1={u∈ H¹|xu∈H¹} : forp > 3, given such a u₀ the out-coming solution u(t) to (2) is global, satisfies the linear decay rate ku(t)k_L^∞ ≤ Ct^−1/3, and there exists V ∈ L² such that ku(t)−U(t)Vk_L2 → 0 as t → ∞. Their method is a very beautiful clock-work, but breaks down at some point when constructing the non-linear wave operator. To recover from this, the setting must be strengthen, and hence, the conditions onV must be reinforced.

Here, the methods of [11] and [7] can be smoothly adapted to take care of the interaction between non-linear terms (the solitons) and the linear term (U(t)V), to provide an almost sharp result.

Indeed, our smoothness assumptionH¹is a natural setting to work with the solitons, and especially to have bounded energy. On the other side, notice that the decay assumption only concerns the L² level for V, and only decay on the right. The assumption on U(t)V should be understood in this way : to handle the interaction of the solitons, we need one degree of decay on V so that its interference is low enough. To prevent the solitons from interfering too much when handling the linear term U(t)V, we need a second order of decay onV.

An optimal result for our framework would then be(1 +x²₊)V(x)∈L²(dx).

In view of this, our assumption appears to be almost optimal.

Remark 3. The uniqueness of solutions to (1) with a given asymptotic be- haviour of the formU(t)V +R(t)is not clear. Remind that for V = 0, that is, theN-soliton, one has uniqueness in H¹ (see [11]) : it is linked to the fact the constructed solution is smooth and converges exponentially fast in H^s for all s ≥0 (s= 4 would be enough). If V belong to H¹ but not more, this is not possible. However, one might be able to prove uniqueness for smootherV. Remark 4. There are some analogous results for the (critical) non-linear Schrö- dinger equation. See Bourgain and Wang [1], Krieger and Schlag [9], Merle [18].

In [1], a solution to the critical NLS equation with a given blow-up behaviour is constructed : thanks to the conformal transform, this is in fact equivalent to construct a solution to the critical NLS equation which behaves like the sum of a soliton and a linear term. High smoothness and low interaction with the soliton are required on the linear term.

Acknowledgment

I would like to thank my advisor Frank Merle for numerous discussions, Luis Vega for his invitation, and Herbert Koch for bringing my attention to similar questions. This research was supported in part by the European network HYKE as contract HPRN-CT-2002-00282 and by the University of the Basque Country.

2 Strategy of the proof.

Following a usual convention, different positive constants might be denoted by the same letterC.

LetV as in the hypothesis of Theorem 1,0< c1< . . . < cN andx1, . . . , xN ∈ R. Denote the soliton with speedc_j and shiftx_j

Rj(t, x) =Qc_j(x−xj−cjt).

Define alsoR(t) =PN

j=1Rj(t).

LetSn be an increasing sequence of time, so thatSn→ ∞asn→ ∞and

n→∞lim E(U(Sn) +R(Sn)) = lim inf

t→∞ E(U(Sn) +R(Sn)). (7) (such a sequence obviously exists ; the condition on the energy appears when concluding the proof of Theorem 1). Forn >0, we defineun(t), the solution of

u_nt+ (u_nxx+u⁵)x= 0,

un(Sn) =U(Sn) +R(Sn). (8) Equivalently, we introducewn(t)the error term

w_n(t) =u_n(t)−U(t)V −R(t), so thatwn(t)satisfies the equation

( w_nt+w_nxxx+

u⁵−PN

j=1R⁵_j(t)

x

= 0,

w_n(S_n) = 0. (9)

Asu(Sn)∈H¹,un(t)is well defined, at least on a small interval of time around Sn.

The heart of the proof of Theorem 1 is the following result :

Proposition 1 (Uniform estimates). There exist T₀,K₀ and a continuous function η: [1,∞)→R⁺∗, depending on V, with

η(t)↓0 as t→ ∞,

such that the following is true. For all nsuch that Sn ≥T0, the solution un(t) to (8) and the solutionwn(t)to (9) belong toC([T0, Sn], H¹). Furthermore, we have the uniform decay estimate and control (in n) :

∀t∈[T0, Sn], kwn(t)k_L2 ≤η(t), and kwn(t)k_H1 ≤K0. (10) The proof of this proposition requires several steps.

The first remark allows us to further assume smallness onwn(t), in order to get the decay (10).

Proposition 1’ (Reduction of proof ). There exist ε₀ > 0, T₀ ≥ 1 and a decreasing continuous functionη : [1,∞)→R⁺∗, depending onV, with

η(t)↓0 as t→ ∞, such that the following is true. Introduce the norm

kf(t, x)k_N([A,B])=kfk_L5

xL¹⁰_t (t∈[A,B])+ sup

t∈[A,B]

kf(t)k_L2 x.

Let n ∈ N so that Sn ≥ T0. Let In ∈ [T0, Sn] such that kwnk_N([In,S_n]) ≤ ε0. Then in fact,

∀t∈[In, Sn], kwnk_N([t,Sn])≤η(t), and kwn(t)k_H1 ≤K0.

We introduce theL⁵_xL¹⁰_t space as it is necessary in the control of the linear termU(t)V : see [7] for further details.

Proof of Proposition 1 assuming Proposition 1’. This is a continuity argument.

Let

T0= inf{τ:τ ≥1andη(τ)≤ε0}, and define

I_n^∗= infn

τ:τ∈[1, Sn], andkwnk_N([τ,S_n])≤ε0

o .

We now use the continuity the norm L⁵_xL¹⁰_t ∩C⁰H¹ under the flow of (1), (see [7]). As w_n(S_n) = 0, we obtain that the set on which we do the infimum is non-empty, so that I_n^∗< S_n.

Then of course, this allows us to apply Proposition 1’ withI_n =I_n^∗ so that

∀t∈[I_n^∗, Sn], kwnk_N([t,S_n])≤η(t), and kwn(t)kH¹ ≤K0. (11) By minimality ofI_n^∗, ifI_n^∗ >1, we also get that

lim sup

t↓I_n^∗

kw_nk_N([t,Sn])≥ε₀. In particular, this gives

ε0≤lim sup

t↓I_n^∗

kwnk_N([t,Sn])≤lim sup

t↓I^∗_n

η(t)≤η(I_n^∗).

So thatη(I_n^∗)≥ε0.

In any case, we get that I_n^∗ ≤ T0 (as η is decreasing) : (11) allows us to conclude.

Proof of Proposition 1’.

Step 1 : Monotonicity and non-linear tools.We obtainL²estimates on the right.

Let us introduce the cut-off speed

σ0∈(0,1/2 min{c1, c2−c1, . . . , cN −cN−1}), (12) to be determined in the proof of the following Proposition 2 below, and the cut-off function

ψ(x) = 2 πarctan

exp

−

√σ₀ 2 x

, ψ0(t, x) =ψ(x−σ0t−2|x1|). (13) ψ0(t)allows us to separate the solitons interaction from theU(t)V interac- tion.

Proposition 2 (Interaction with the solitons). There existσ₁>0,ε₁,T₁, C₁ and K₀ such that the following is true. If σ₀ ≤σ₁, ε₀ ≤ε₁ and T₀ ≥ T₁, then, for alln∈N and allt∈[In, Sn],

kw(t)k_L2(1−ψ₀(t))≤C₁e^−σ0

√σ0

8 t+C₁kU(t)Vk_L2(1−ψ₀(t))

+C₁(S_n−t+ 1)kU(t)VkL²(1−ψ0(S_n))+C₁ Z Sn

t

kU(t)VkL²(1−ψ0(t))dt, and

kwn(t)kH¹≤K0.

The control of theH¹-norm simply relies on uniform bounds of the energy, and on the smallness assumption onkw(t)kL². The deep result is the first esti- mate.

Essentially we obtain a polynomial decay onkwn(t)kL²(1−ψ0(t)) (instead of an exponential decay in the case of solely soliton). However the good point is that we can choose this polynomial decay to be as fast as we want by lowering the interaction ofU(t)V with the solitons, that is, by requiring sufficient decay on the right forV : see Lemma 2.

Step 2 : Linear theory. Essentially we have to take care of the interaction of U(t)V and wn. For this, we use the linear estimates and the setting of [6] and [7].

Proposition 3 (Ineteraction with the linear term). There existsε2 >0, T2,C2 such that the following is true. Suppose that for someC andδ0>0, we have for all nsuch that Sn≥T2 :

∀t∈[I_n, S_n], kw_n(t) +U(t)Vk_L2(1−ψ₀(t)≤ C t^1+δ0. Then there existsC₂ such that if we denote :

η(t) =C2kU(τ)Vk⁵_L5

xL¹⁰_τ (τ≥t)+C2e^−σ0

√σ0

4 tkU(τ)Vk_L5

xL¹⁰_τ(τ≥t)+C2

t^δ0, we have :

kwn(t)k_N([In,Sn])≤η(t).

Of course,η(t)decreases to 0 as t → ∞, and so satisfies the conditions of Proposition 1’.

Finally, Proposition 2, Lemma 2 and estimates (42) and (43) ensure that the assumptions of Proposition 3 are fulfilled ifV is chosen as in Theorem 1, that is V ∈H¹andx^2+δ₊ ⁰V ∈L². Fixσ₀< σ₁,ε₀= min{ε1, ε₂}andT₀= min{T1, T₂}: this completes the proof of Proposition 1’, and so, of Proposition 1 .

Proof of Theorem 1. From Proposition 1, we are able to prove some compact- ness property in L² on the sequenceun(T0). The limit of a subsequence yields an initial data ϕ0, from which u^∗(t) is the out-coming solution to (1). Then Proposition 1 allows to conclude that

ku^∗(t)−U(t)V −R(t)k_L2 →0.

To obtain the H¹ convergence, we need another argument. We compare E(U(S_n) +R(S_n))andE(u^∗(t)), taking advantage of (7). By developping

E(u^∗(t)) =E(w^∗(t) +U(t)V +R(t)),

and studying carefully all the obtained terms, we finally prove that the error termkw_x^∗(t)kL² →0 ast→ ∞ : this completes the proof of Theorem 1.

The proof of Theorem 1 assuming Proposition 1 is done in Section 3. The rest of the proof completes the proof of Proposition 1’ and thus that of Propo- sition 1. In Section 4., we give some preliminary estimates to be used both in Section 5. and Section 6. Section 5. is devoted the proof of Proposition 2. Finally, Proposition 3 is proved in Section 6.

3 Proof of Theorem 1 assuming Proposition 1

In this section, we assume Proposition 1 holds, and from this we conclude the proof of Theorem 1.

3.1 A compactness result linked with the monotonicity Lemma 5

From Proposition 1, we dispose of a sequence u_n(t) defined on [T₀, S_n] (we dropped the terms withS_n < T₀), solutions to (2), such that

un(Sn) =U(Sn)V +

N

X

j=1

Rj(Sn) =U(Sn) +R(Sn),

and that the following uniform estimates holds (wn(t) =un(t)−U(t)V−R(t)) :

∀n∈N, ∀t∈[T0, Sn], kwn(t)k_L2 ≤η(t) and kw(t)k_H1≤K0. Claim.u_n(T₀)is a compact sequence in the sense that

A→∞lim sup

n∈N

Z

|x|≥A

un2(T0, x)dx= 0.

Proof of the Claim. Indeed, letε >0, and T(ε)such thatη(T(ε))≤√ ε. Then Z

(un(T(ε))−U(T(ε))V −R(T(ε))²≤ε.

LetA(ε)be such thatR

|x|≥A(ε)(U(T(ε))V +R(T(ε)))²(x)dx≤ε; we get Z

|x|≥A(ε)

u_n²(T(ε), x)dx≤2ε.

Let g ∈ C³ a function such that g(x) = 0 if x ≤ 0, g(x) = 1 if x ≥ 2, and furthermore0≤g⁰(x)≤1,0≤g⁰⁰⁰(x)≤1.

Remind that iff ∈C³ does only depend onx, we have d

dt Z

u_n²f =−3 Z

u_n²_xf_x+ Z

u_n²f_xxx+ 2p p+ 1

Z

u_n^p+1f_x.

(See Lemma 5 and its proof). ForC(ε)to be determined later, we then have d

dt Z

un2(t, x)g

x−A(ε) C(ε)

=− 3 C(ε)

Z un2

xg⁰

x−A(ε) C(ε)

+ 1

C(ε)³ Z

u_n²g⁰⁰⁰

x−A(ε) C(ε)

+ 2p

(p+ 1)C(ε) Z

u_n^p+1g⁰

x−A(ε) C(ε)

. As t ≥ T0, un satisfies kun(t)k_H1 ≤ K0+kVk_H1 +PN

j=1kQc_jk_H1 ≤ C⁰. So that :

d dt

Z

un2(t, x)g

x−A(ε) C(ε)

≤ 1 C(ε)

3

Z un2

x(t) + Z

un2(t) + 2p

p+ 1kunk^p−1_L∞

Z un2(t)

≤ 1 C(ε)

3C⁰²+ 2p

p+ 12^(p−1)/2C^0p+1

.

Now choose C(ε) = maxn

1,^T(ε)−T_ε ⁰

3C⁰²+_p+1^2p 2^(p−1)/2C^0p+1o

, from which we derive

d dt

Z

un2(t, x)g

x−A(ε) C(ε)

≤ ε

T(ε)−T₀. And after integration in time betweenT0 andT(ε),

Z

x≥2C(ε)+A(ε)

un2(T0, x)≤ Z

un2(T0, x)g

x−A(ε) C(ε)

≤3ε.

Now considering _dt^d R

un2(t, x)g_−A(ε)−x

C(ε)

, we get in a similar way Z

x≤−2C(ε)−A(ε)

un2(T0, x)≤3ε.

So that if we denoteAε= 2C(ε/6) +A(ε/6), we obtain

∀n∈N, Z

|x|≥Aε

u_n²(T₀, x)≤ε, as claimed.

3.2 Construction of u

and L

convergence to the profile

Now un(T0) is a bounded sequence in H¹(R), and so converges weakly up to a subsequence, to some ϕ0 in H¹(R) (we suppose for convenience that the whole sequence converges weakly). The previous compactness result ensures that the convergence is strong in L²(R). Indeed, let ε > 0, and A such that R

|x|≥Aϕ²₀(x)dx≤εand

∀n∈N, Z

|x|≥A

u_n²(T₀, x)≤ε.

The injectionH¹([−A, A])→L²([−A, A])is compact, so thatR

|x|≤A|u_n(T₀, x)−

ϕ0(x)|²dx→0. We thus derive that lim sup

n→∞

kun(T0)−ϕ0k²_L2(R)≤4ε.

As this is true for all ε >0,u_n(T₀)→ϕ₀ inL²(R).

Denoteu^∗(t)the solution to

u^∗_t + (u^∗_xx+u^∗p)_x= 0, u^∗(T₀) =ϕ₀.

The Cauchy problem being well-posed inL²(R),u^∗ is well defined, at least for t in a neighborhood V of T₀. Now the flow is continuous in L² (in fact it is Lipschitz), so that for allt∈ V, u_n(t)→u^∗(t)inL². As(u_n(t))_n is a bounded sequence inH¹, this proves that the whole sequence converges weakly tou^∗(t) in H¹ :

∀t∈ V, lim

n→∞u_n(t) =u^∗(t) inL²(R)−strong andH¹(R)−weak. (14) Thus, we can take the limit in the estimates (10) (with tfixed), to get

∀t∈ V, ku^∗(t)−U(t)V−R(t)k_L2 ≤η(t), and ku^∗(t)−U(t)V−R(t)k_H1≤K0. This shows that u^∗(t) is H¹ uniformly bounded on V, so that by the Cauchy problem theory and a standard continuity argument,u^∗is defined for allt≥T0. Hence,w^∗(t) is uniformly bounded inH¹, and satisfies the expected L² decay estimate :

kw^∗(t)k_H1≤K0 and kw^∗(t)k_L2≤η(t)→0 as t→ ∞. (15)

3.3 H

convergence of u

(t) to its profile

TheH¹ convergence comes essentially from an analysis of the energyE(u^∗(t)).

From (14),un(t)* u^∗(t)H¹-weak and un(t)→u^∗(t)inL⁶ asn→ ∞, and we deduce that

E(u^∗(T₀))≤lim inf

n→∞ E(u_n(T₀))≤lim inf

n→∞ E(u_n(S_n))

≤lim inf

n→∞ E(U(S_n)V +R(S_n)).

Now, conservation of energy gives forE(u^∗(t)) =E(u^∗(T0)), fort≥T0. By (7), and in view of the previous computation, we have

lim inf

t→∞ E(U(t)V +R(t))−E(u^∗(t))

= lim inf

t→∞ E(U(t)V +R(t))−E(u^∗(T0))

= lim

n→∞E(U(Sn)V +R(Sn))−E(u^∗(T0))

≥0. (16)

Thus, let us estimateE(U(t)V +R(t))−E(u^∗(t)): E(U(t)V +R(t))−E(u^∗(t))

=E(U(t)V +R(t))−E(w^∗(t) +U(t)V +R(t))

=E(U(t)V +R(t))−E(w^∗(t))−E(U(t)V +R(t))− Z

w^∗_x(t)U(t)Vx

− Z

w^∗_x(t)Rx(t)) +1 6

5

X

k=1

C₆^k Z

w^∗(t)^k(U(t)V +R(t))^6−k

=−1 2 Z

|w^∗_x(t)|²− Z

w^∗_x(t)U(t)Vx− Z

w^∗_x(t)Rx(t) +1

6

6

X

k=1

C₆^k Z

w^∗(t)^k(U(t)V +R(t))^6−k. (17) Remind (15) : by interpolationL²-H¹, we get that for allp≥2,kw^∗(t)kL^p→0 as t→ ∞.

Let us first control the second line in (17) : fork= 2, . . . ,6,

Z

w^∗(t)^k(U(t)V +R(t))^6−k

≤ kw^∗(t)k^k_LkkU(t)V +R(t)k^6−k_L∞ =o_t→∞(1).

Fork= 1, we have also

Z

w^∗(t)(U(t)V +R(t))⁵

≤ kw^∗(t)kL²kU(t)V +R(t)kL²kU(t)V +R(t)k⁴_L∞=ot→∞(1).

Now, Z

w^∗_x(t)Rx(t)

= Z

w^∗(t)Rxx(t)

≤ kw^∗(t)kL²kR(t)kH² =ot→∞(1).

The last termR

w^∗_x(t)U(t)V_xrequires a little more attention. Consider the func- tion U(−t)(w^∗(t)) : then kU(−t)(w^∗(t))k_L2 = kw^∗(t)k_L2 → 0 as t → ∞ and kU(−t)(w^∗(t))k_H1 =w^∗(t)k_H1 is uniformly bounded int. Hence, the only pos- sible weak limit ofU(−t)(w^∗(t))in H¹(ast→ ∞) is 0. This proves that :

U(−t)(w^∗(t))→0 inL²−strong as t→ ∞, U(−t)(w_x^∗(t))*0 in L²−weak as t→ ∞.

This proves that Z

w^∗_x(t)U(t)Vx= Z

U(−t)(w^∗_x(t))Vx=ot→∞(1).

We can conclude from (17) that

E(U(t)V +R(t))−E(u^∗(t)) =−1 2 Z

|w^∗_x(t)|²+ot→∞(1), and in view of (16),

0≤lim inf

t→∞ E(U(t)V +R(t))−E(u^∗(t))

≤lim inf

t→∞

−1 2 Z

|w^∗_x(t)|²+o_t→∞(1)

≤ −1 2lim sup

t→∞

Z

|w^∗_x(t)|².

This proves thatkw^∗_x(t)k_L2 →0, and along with (15), we get that ku^∗(t)−U(t)V −R(t)k_H1 →0 as t→ ∞.

This concludes the proof of Theorem 1.

The following is devoted the proof of Proposition 1, or more precisely of Proposition 1’. We will now only work on the interval[I_n, S_n].

4 Preliminaries

4.1 Cut-off functions and localized quantities

We already introduced σ0 ∈ (0,1/2 min{c1, c2−c1, . . . , cN −c_N−1}), and the cut-off function :

ψ(x) = 2

πarctan e⁻

√σ0 2 x

. (13)

We can check thatlim_+∞ψ= 0,lim_−∞ψ= 1, andψis decreasing. Furthermore, by direct computations,

ψ⁰(x) =−

√σ₀ 2πcosh^√_σ

0

2 x, ψ⁰⁰⁰=σ0

4 ψ⁰(x) 1− 2 cosh^√_σ

0

2 x

! , and so,

|ψ⁰⁰⁰(x)| ≤ −σ0

4 ψ⁰(x). (18)

We introduce, forj = 1, . . . , N−1: m_j(t) = c_j+c_j+1

2 t+x_j+x_j+1

2 , m₀(t) =σ₀t−2|x1|, m−1(t) = σ₀

2 t−2|x1|.

So that we can define, forj=−1, . . . , N−1:

ψj(t, x) =ψ(x−mj(t)), ψN(t, x) = 1.

Then we set, for j= 1, . . . , N−1 :

φ0(t) =ψ0(t), φj(t) =ψj(t)−ψ_j−1(t), φN(t) = 1−ψN−1(t).

By construction,Pj

k=1φ_k=ψ_j. Finally, we define some local quantities related to mass and energy :

M_j(t) = Z

u²_t(t)φ_j(t), E_j(t) = Z 1

2u²_x(t)− 1

p+ 1u^p+1(t)

φ_j(t), F_j(t) =E_j(t) + 1

100M_j(t).

Forj≥1, theφ_jseparates the solitonsR_jfrom one another.ψ₀(t)separates the solitons from the linear term U(t)V. The aim of ψ₋₁(t) is different : it provides an interval on which U(t)V is small inH¹ an so inL^∞ (see Lemma 2 hereafter). This will be crucially used in the almost monoticity Lemma 5 (it is in fact the only place whereψ₋₁(t)plays a role).

Observe that kU(t)VkL^∞ ≤ CkVk_L1t^−1/3, so that pointwise smallness on U(t)V is automatic ifV ∈L¹. However, this hypothesis is not part of Theorem 1.

4.2 Preliminary bounds on w

(t)

Notice that from the uniform bound on the energy, we get a uniform control on wn(t) for t ∈ [In, Sn]. This is the purpose of the following lemma. This preli- miniary result will be very important in the proof of the almost monotonicity Lemma 5.

Lemma 1 (Bound on theH¹ norm of wn(t)). There existsK0independent of ε0∈]0, κ(6)^−1/4](remind (5)), such that

∀n∈N, ∀t∈[In, Sn], kwn(t)k_H1 ≤K0. (19) In particular, thekw(t)kL^∞ can be made arbitrarily small :

∀n∈N,∀t∈[In, Sn], kw(t)kL^∞ ≤p

ε0K0. (20) Remark that this lemma gives the second estimate of Proposition 1’.

Proof. We combine smallness ofwn(t)inL² along with uniform bounds (inn) onE(un). The energy is preserved so thatE(un(Sn)) =E(un(t)). Then we have

E(u_n(S_n)) =E(U(S_n)V +R(S_n))

≤C Z

|U(Sn)Vx|²+C

N

X

j=1

Z Qc_j2

x+C Z

|U(Sn)V|⁶+C

N

X

j=1

Z

Q⁶_cj ≤C.

So that the energy E(un(t)) is uniformly bounded (in n). Now we have the following.

Claim. Let f, ε ∈ H¹, with kεkL² ≤ κ(6)^−1/4. Then there is a function G:R⁺→R⁺ such that

kεk_H1 ≤3E(f+ε) +G(kfk_H1).

To conclude, it suffice to apply the claim forε=wn(t)andf =U(t)V+R(t) (whoseH¹-norm is uniformly bounded int).

Let us prove the claim. Indeed, we compute : E(f+ε) = 1

2 Z

(f +ε)²_x−1 6

Z

(f +ε)⁶

= 1 2

Z f_x²+

Z

fxεx+1 2

Z

ε²_x−1 6

5

X

k=0

C₆^k Z

ε^kf^6−k−1 6

Z ε⁶. Now, we have R

f_x² ≤ kfk²_H1, |R

fxεx| ≤ kfk_H1kεxk_L2, R

f⁶ ≤ kfk⁶_H1, and

|R

εf⁵| ≤ kfk⁵_H1kεk_L2. For k = 2, . . . ,5, we have the Gagliardo-Nirenberg in- equality (whose sharp constant isκ(k)) :

Z

ε^kf^6−k ≤κ(k)kfk^6−k_L∞kεk^k/2+1_L2 kεxk^k/2−1_L2 .

Fork = 6, the Gagliardo-Nirenberg inequality also applies, but gives an expo- nent 2 forkεk_L2 :

1 6

Z

ε⁶≤ κ(k)

6 kεk⁴_L2kεxk²_L2 ≤ 1 6 Z

ε²_x.

So that we get from the energy equality : 1

2 Z

ε²_x≤E(f+ε) +kfk²_H1+kfk_H1kεxk_L2+kfk⁶_H1+kfk⁵_H1kεk_L2

+

5

X

k=2

kfk^6−k_L∞kεk^k/2+1_L2 kεxk^k/2−1_L2 +1 6

Z ε²_x. This can be rewritten as (kεk_L2 ≤1)

1

3kεxk²_L2 ≤E(f+ε) + 2⁵(kfkH¹+kfk⁶_L∞)(1 +kεxk^3/2_L2 ).

Ifa²≤K₁+K₂a^3/2, then obviously a²≤K₁+K₂⁴, so that we get kεxk²_L2 ≤3E(f+ε) + 3·2⁶(kfkH¹+kfk²⁴_L∞).

4.3 Estimates of U (t)V on the right

We now obtain bounds for U(t)V on the right, which is will be crucial for the monotonicity Lemma 5, and also in Section 5 (analysis of the interaction of the linear termU(t)V).

Lemma 2 (U(t)V estimates on the right). Let f ∈L², then

kU(t)fkL²(1−ψ−1(t))≤ kfkL²(1−ψ−1(t/2))→0 as t→ ∞. (21) In particular, iff ∈H¹, then

sup

x≥m−1(t)

|U(t)f(x)|²≤4kfk_L2kf_xk_L2(1−ψ−1(t/2))→0 as t→ ∞. (22) Suppose that(1+x^q₊)f(x)∈L²(dx), for someq >0. Then there exists a constant C=C(σ0, x1)independent off such that

∀t≥1, kU(t)fk_L2(1−ψ0(t))≤ C

t^qk(1 +x^q₊)f(x)k_L2(dx). (23) If (1 +x^1/2₊ )f(x)∈L²(dx), we have furthermore that

Z

t≥0

kU(t)fk²_L2(1−ψ₀(t))dt≤Ck(1 +x^1/2₊ )f(x)k²_L2(dx)<∞. (24) We will apply this result toV andVx.

Proof. The key remark is thatU(t)“pushes” theL²-mass on the left. Let ϕ= ψ₋₁ orϕ=ψ₀. We compute :

d dτ

Z

|U(2τ−t)f|²ϕ(τ)

= 2 Z

(U(2τ−t)f)_τU(2τ−t)f ϕ(τ) + Z

|U(2τ−t)f|²ϕ_τ(τ)

=−4 Z

U(2τ−t)fxxxU(2τ−t)f ϕ(τ) + Z

|U(2τ−t)f|²ϕτ(τ)

= 4 Z

U(2τ−t)fxxU(2τ−t)fxϕ(τ) + 4 Z

U(2τ−t)fxxU(2τ−t)f ϕx(τ) +

Z

|U(2τ−t)f|²ϕτ(τ)

=−6 Z

|U(2τ−t)fx|²ϕx(τ)−4 Z

U(2τ−t)fxU(2τ−t)f ϕxx(τ) +

Z

|U(2τ−t)f|²ϕ_τ(τ)

=−6 Z

|U(2τ−t)f_x|²ϕ_x(τ) + Z

|U(2τ−t)f|²(2ϕ_xxx(τ) +ϕ_τ(τ)).

Asψxxx≤ ^σ₄⁰|ψx|, andψx<0, we have that, forϕ=ψ₋₁ orψ0, ϕx(τ)<0 and 2ϕxxx(τ) +ϕτ(τ)≥0.

So that τ 7→R

U(2τ−t)f(x)²ϕ0(τ, x)dx is an increasing function of τ. In par- ticular, when comparing for τ=tandτ =t/2 (t≥0), we have :

∀t≥0, Z

|U(t)f|²ϕ(t)≥ Z

f²ϕ0(t/2).

As the flowU(t)preserves theL²-mass, we get in each caseϕ=ψ−1 orψ0:

∀t≥0, Z

|U(t)f|²(x)(1−ψ₋₁(t, x))dx≤ Z

f²(x)(1−ψ₋₁(t/2, x))dx, (25) Z

|U(t)f|²(x)(1−ψ₀(t, x))dx≤ Z

f²(x)(1−ψ₀(t/2, x))dx. (26) (25) immediatly gives (21). Let x≥m₋₁(t). Then for y ≥ x, 1−ψ₋₁(t, y)≥ 1−²_πarctan(1) = ¹₂. Thus,

|U(t)f(x)|²=−2 Z ∞

y

U(t)f(y)U(t)f_x(y)dy

≤2 Z ∞

y

|U(t)f(y)|²dy

^1/2Z ∞ y

|U(t)fx(y)|²dy ^1/2

≤8 Z ∞

y

|U(t)f(y)|²(1−ψ₋₁(t, y))dy 1/2

× Z ∞

y

|U(t)fx(y)|²(1−ψ₋₁(t, y))dy ^1/2

≤8kU(t)fk_L2(1−ψ−1(t))kU(t)fxk_L2(1−ψ−1(t))

≤8kfk_L2(1−ψ−1(t/2))kfxk_L2(1−ψ−1(t/2)). This is (22).

We will now use (26). Suppose that for someq >0,(1 +x^q₊)f(x)∈L²(dx).

Then fort≥1, Z

f²(1−ψ₀(t/2)) = Z

x≤σ0t/4

f²(1−ψ₀(t/2)) + Z

x≥σ0t/4

f²(1−ψ₀(t/2))

≤ sup

x≤σ0t/4

(1−ψ0(t/2, x)) Z

f²+ σ0t

4

−2qZ

x≥σ0t/4

x^2qf²

≤C(x0)e^−σ0

√σ0

4 tkfk²_L2+C(σ0)t^−2qkx^q₊fk²_L2. And we get

∀t≥1, kU(t)fkL²(1−ψ0(t))≤ C

t^qk(1 +x₊)^qfkL², which is (23).

Suppose now that(1 +x^1/2₊ )f(x)∈L²(dx). Then Z ∞

t=0

Z

x

|U(t)f(x)|²(1−ψ0(t, x))dxdt≤ Z ∞

t=0

Z

x

f²(x)(1−ψ0(t/2, x))dxdt

≤ Z

x

f²(x) Z ∞

t=0

(1−ψ₀(t/2, x))dtdx

≤C Z

f²(x)1 +x₊ σ0

dx

≤Ck(1 +x^1/2₊ )f(x)k²_L2(dx), and this proves (24).

5 Control of the interaction of w

(t) with the soli- tons

This section is devoted to the proof of Proposition 2. We develop arguments very similar to those of [17] and [16].

5.1 Modulation close to the asymptotic profile

Lemma 3. There exist T1 large enough and ε1>0 small enough such that if T1≥T1 andε0≤ε1, the following is true.

There exist2N C¹ functionsyj, γj: [In, Sn]→Rsuch that if we denote : R˜j(t, x) =Q_γj(t)(x−yj(t)), R(t, x) =˜

N

X

j=1

R˜j(t, x),

˜

wn(t) =un(t, x)−U(t)V −R(t, x),˜ we have for all j= 1, . . . , N :

Z

˜

w_n(t, x) ˜R_{j x}(t, x)dx= 0 and Z

˜

w_n(t, x) ˜R_j³(t, x)dx= 0.

Moreover, there exists C11 such that : kw˜n(t)kL²+

N

X

j=1

|γj(t)−cj|+

N

X

j=1

|yj(t)−xj−cjt| ≤C11ε0, (27)

|y_j⁰(t)−cj|+|γ_j⁰(t)| ≤C11e^−σ0

√σ0

2 t+C11kU(t)Vk_L2(1−ψ0(t))

+C₁₁ Z

˜

w_n²(t)e^{−√σ0|x−cjt|}

^1/2 . (28) Proof. The existence of the modulation is essentially an application of the im- plicit function theorem. Consider theC^∞ functional

F : [In, Sn]×L²×Rⁿ×Rⁿ→Rⁿ×Rⁿ,

(t, u,(y_j)_j,(γ_j)_j)7→(F_1j(t, u,(y_j)_j,(γ_j)_j), F_2j(t, u,(y_j)_j,(γ_j)_j)), with

F1j(t, u,(yj)j,(γj)j) = Z

(u−U(t)V −R(t)) ˜˜ R_{j x}(t)dx, F2j(t, u,(yj)j,(γj)j) =

Z

(u−U(t)V −R(t)) ˜˜ Rj 3(t)dx,

locally on a neighborhood of the curve y_j(t) = x_j +c_jt, γ_j(t) = c_j, u = U(t)V+R(t). To expressyj, γjin function ofu, t, we apply the implicit function theorem stated in the Appendix : let us prove that∂y_j,γ_jFis invertible at points (t, U(t)V +R(t), xj +cjt)j,(cj)j), compute ∂uF, and do some uniform (in t) estimates.

For allt,αbeing yj orγk we compute

∂αF1j(t) =− Z

(∂αR˜j)(t) ˜R_{j x}(t) + Z

(u−U(t)V −R˜j(t))(∂αR˜_{j x})(t),

∂αF2j(t) = Z

(∂αR˜j)(t) ˜R_j³(t) + 3 Z

(u−U(t)V −R˜j(t))(∂αR˜j)(t) ˜R²_j(t), and

(∂_yjR˜_j)(t, x) =−R˜_{j x}(t, x), (∂_γjR˜_j)(t, x) = 1

4cj

R˜_j(t, x) + 1 2cj

(x−x_j−c_jt) ˜R_{j x}(t, x).

Letu, y_j andγ_j be such that

ku−U(t)V −R(t)k_L2+

N

X

j=1

|y_j|+|γ_j| ≤ε₀. We get that

∂_yjF_1j− Z

Q_cj²_x

≤Cε₀,

(recallkQc_jk_L2=kQk_L2) and fork6=j, using the exponential decay :

|∂y_kF1j(t)| ≤Ce^−σ

√σ0

4 t+Cε0,

|∂ykF_2j(t)|+|∂γkF_1j(t)|+|∂γkF_2j(t)| ≤Ce^−σ

√σ0

4 t+Cε₀. NowQis an even function, so that

|∂γ_jF1j(t)| ≤Cδ, |∂y_jF2j(t)| ≤Cε0.