Large data wave operator for the generalized

Korteweg-de Vries equations

Raphael Cote

Abstract

We consider the generalized Korteweg-de Vries equations :

ut + (uxx + up)x = 0, t, x ∈ R,

for p ∈ (3,∞). Let U(t) be the associated linear group. Given V in theweighted Sobolev space H2,2 = {f ∈ L2| ‖(1+ |x|)2(1−∂2

x)f‖L2 <∞},possibly large, we construct a solution u(t) of the generalized Korteweg-de Vries equation such that :

limt→∞

‖u(t)− U(t)V ‖H1 = 0.

We also prove uniqueness of such a solution in an adequate space.In the L2-critical case (p = 5), this result can be improved to any

possibly large function V in L2 (with convergence in L2).

1 Introduction.

1.1 Recall of known results.

We consider the generalized Korteweg-de Vries equation :

ut + (uxx + up)x = 0, t, x,∈ R, (1)

where p ≥ 2. The case p = 2 corresponds to the original equation introducedby Korteweg and de Vries [6] in the context of shallow water waves. For bothp = 2 and p = 3, this equation has many applications to Physics : see forexample Miura [14], Lamb [7].

There are two formally conserved quantities for solutions to (1) :∫u2(t) =

∫u2(0) (L2 mass), (2)

E(u(t)) =12

∫u2

x(t)− 1p+ 1

∫up+1(t) = E(u(0)) (energy). (3)

Mathematical Subject Classification : Primary 35Q53, Secondary 35B40.Keywords : Generalized Korteweg-de Vries equations, wave operator, large data.

1

The local Cauchy problem for (1) has been intensively studied by many au-thors. Kenig, Ponce and Vega [5] proved the following existence and unique-ness result in H1(R) : for u0 ∈ H1(R), there exist T = T (‖u0‖H1) > 0 and asolution u ∈ C([0, T ],H1(R)) to (1) satisfying u(0) = u0, which is unique insome class YT ⊂ C([0, T ],H1(R)). For such solution, one has conservationof mass and energy. Moreover, if T1 denotes the maximal time of existencefor u, then either T1 = +∞ (global solution) or T1 <∞ and ‖u(t)‖H1 →∞as t ↑ T1 (blow-up solution).

For p = 2 and p = 3, equation (1) is completely integrable, and thushas very special features. The inverse scattering transform method allows tosolve the Cauchy problem in an appropriate space (for example if ‖(1 +|x|2)5u0‖C4 < ∞) and to find the asymptotic behavior of solutions ast → ±∞ : see for example Schuur [16], Eckhaus and Schuur [2], Miura[14]. However, if p 6= 2 or 3, the inverse scattering transform method doesnot longer apply, and the description of solutions as t → +∞ in the non-integrable case is an open problem. Let us recall some results which are notbased on the inverse scattering transform method.

In the case 2 ≤ p < 5, all solutions in H1 are global and uniformlybounded due to the conservations laws and the Gagliardo-Nirenberg inequal-ity :

∀v ∈ H1(R),∫|v|p+1 ≤ C(p)

(∫v2

) p+34(∫

v2x

) p−14

.

The case p = 5 is L2-critical, in the sense that the mass remains unaffectedby scaling. If :

ut + (uxx + u5)x = 0, t, x,∈ R. (4)

Then uλ(t, x) = λ1/6u(λt, λ1/3x) is also a solution to (4), and ‖uλ‖L2 =‖u‖L2 . In this case, the local existence result of [5] is improved to initial datain L2 (instead of H1). However, existence of finite time blow-up solutionswas proved by Merle [13] and Martel and Merle [10]. Therefore p = 5 alsoappears as a critical exponent for the long time behavior of solutions to (1).

Another problem which was studied by many authors is scattering forsmall initial data in an appropriate functional space, see for example [17],[15], [1], [4]. Let us recall the result of Hayashi and Naumkin [4]. Introducethe following weighted Sobolev spaces :

Hs,m = {φ ∈ S ′| ‖φ‖Hs,m = ‖(1 + |x|2)m/2(1− ∂2x)s/2φ‖L2 <∞}. (5)

Let p > 3. Given u0 small enough in H1,1, the out-coming solution u(t)is global in time, and there is scattering, in the sense that there exists afunction V ∈ L2 so that :

‖u(t)− U(t)V ‖L2 → 0.

2

where U(t) denotes the linear operator for the KdV equation, i.e. v(t) =U(t)V satisfies vt + vxxx = 0, v(0) = V . This is the description of solutionsaround with initial data around 0 (in H1,1).

Now, there exist also special explicit traveling wave solutions called soli-tons. If Q denotes the unique solution (up to translation) of :

Q > 0, Q ∈ H1(R), Qxx +Qp = Q, i.e. Q(x) =

(p+ 1

2 cosh2(p−12 x)

) 1p−1

,

then for c > 0, the soliton

Rc,x0 = c1

p−1Q(√c(x− x0 − ct)) is a solution to (1).

We should notice that for p > 3, solitons do not appear in the small dataanalysis of [4], as :

‖Rc,x0‖H1,1 ≥ c0, for some uniform constant c0 > 0.

Indeed :∫(Rc,x0)

2x = c

p+32(p−1)

∫Q2

x, and∫

(xRc,x0)2 = c

7−3p2(p−1)

∫(xQ)2.

For p > 3, p+32(p−1) > 0 but 7−3p

2(p−1) < 0, so that if ‖Rc,x0‖H1 → 0, then c → 0and thus ‖Rc,x0‖H0,1 → ∞. (Notice that H1,1 is not sharp from this pointof view).

Description of solutions around a sum of decoupled solitons is available :Martel and Merle [9], Martel, Merle and Tsai [11] proved stability in H1

and asymptotic stability (in L2(x ≥ ct) for c > 0) of a sum of decoupledsolitons, in the sub-critical case 2 ≤ p < 5 (in the critical case p = 5, onehas blow-up around a soliton [10]).

Our goal is to construct solutions to (1) with a given linear asymptoticbehavior : that is the construction of a wave operator with respect to thefree KdV operator U(t). This problem is reciprocal to (linear) scattering forsmall initial data.

Let p > 3, and V ∈ H2,2 (without smallness assumption). We constructa solution u(t) to (1), defined for large enough times, and such that u(t)−U(t)V → 0 in H1 as t → ∞. Furthermore, u(t) is unique in an adequatespace.

In the L2-critical case (p = 5), we obtain an optimal result, in the sensethat for V ∈ L2, we construct u(t) solution to (4) such that u(t)−U(t)V → 0in L2. Again V need not be small in L2.

The philosophy underlying these results is the following : the tools neededto prove global existence for small data can be applied successfully to con-struct solutions with a given linear profile, small or large.

3

In the case of non-linear Schrodinger equations, there are many resultsconcerning the construction of wave operators. For a review, see e.g. Ginibreand Velo [3].

1.2 Statement of the results.

There are two main results : one in the general case p > 3 which uses theframework of Hayashi and Naumkin [4], and one in the critical case p = 5,using the framework of Kenig, Ponce and Vega [5].

Let p > 3. Fix once for all the three constants :

γ ∈ (0,min{1/2, (p−3)/3}), α =12−γ ∈ (0, 1/2) and δ =

p− 3− 2γ3

> 0.

(6)Following the framework of Hayashi and Naumkin [4], we will use the nota-tion D = ∂x = ∂

∂x for the partial differentiation with respect to the spacevariable x, and :

Dαφ = F−1ξαe−(iπ/2)(1+α)φ.

As in [4], we will use the following two operators :

J tφ = U(t)xU(−t)φ = (x− 3t∂2x)φ, and Itφ = xφ+ 3t

∫ x

−∞∂tφ(t, y)dy.

We write J t and It to emphasize that we will always consider norms at afixed time t although J t and It are space-time operators.

Our working spaces will be defined through the time dependent M t0

norm :

Ht = {φ ∈ L2(R)|M t0(φ) = ‖φ‖H1 + ‖DJ tφ‖L2 + ‖DαJ tφ‖L2 <∞}.

J t only appears in the norm, since it is convenient to do linear estimates (see[4], Lemma 2.3). But we introduced It because it is easier to handle whendoing energy methods estimates. Notice that M0

0 is very similar to ‖ · ‖H1,1 .Different positive constants might be denoted by the same letter C.We now state our results.

Theorem 1 (Large data wave operator). Let p > 3, and V ∈ H2,2.Then :

1. There exist T0 = T0(‖V ‖H2,2) ≥ 1 and a unique u ∈ C([T0,∞),Ht)solution to (1) so that :

M t0(u(t)− U(t)V ) → 0 as t→∞.

4

2. Moreover, there exists a constant C independent of time and V , sothat :

∀t ≥ T0, M t0(u(t)− U(t)V ) ≤ C(1 + ‖V ‖p

H2,2)t−δ.

In particular, ‖u(t)− U(t)V ‖H1 ≤ Ct−δ.

Remark. The scattering result of Hayashi and Naumkin [4] is the following :for small initial data in H1,1, the associated solution u(t) is global in time,satisfies ‖u(t)‖L∞ ≤ Ct−1/3 (linear decay rate), and there exists a scatteringfunction V such that u(t)− U(t)V → 0 in L2 as t→∞.

Our point here is the reciprocal problem : we construct a solution u witha given scattering state U(t)V . We do not need any smallness assumption ;however some integration by parts do not work as well as in [4], so we needto assume V ∈ H2,2, with basically a convergence result in H1,1.

This result can be extended to a more general non-linearity. For example :

Corollary 1. Let us consider the equation :

ut + (uxx + f(u))x = 0. (7)

where f : R→ R is C1, with f(0) = 0 and f ′(x) = O(xp−1) as x→ 0 (andp > 3). Given V ∈ H2,2, there exist T0 = T0(‖V ‖H2,2) ≥ 1 and a uniqueu ∈ C([T0,∞),Ht) solution to (7) so that :

M t0(u(t)− U(t)V ) → 0 as t→∞.

Moreover, there exists a constant C independent of time and V , so that :

∀t ≥ T0, M t0(u(t)− U(t)V ) ≤ C(1 + ‖V ‖p

H2,2)t−δ.

The proof of the corollary follows from that of Theorem 1. In particular,although our proof is done for the focusing power case f(x) = xp, it is alsotrue for the defocusing case f(x) = −|x|p−1x.

In the critical case p = 5, we prove an analogous result for V ∈ L2 :

Theorem 2 (Wave operator in the critical case). Let p = 5. For anyV ∈ L2, there exist T0 = T0(V ) ∈ R and u ∈ C0([T0,∞), L2) solution to thecritical KdV equation (4), so that :

‖u(t)− U(t)V ‖L2 → 0.

u is unique in the class {u|u ∈ L∞t L2x ∩ L5

xL10t and ∂xu ∈ L∞x L2

t }.

Remark. The theorem is true if one weakens the hypothesis to V be suchthat ‖U(t)V ‖L5

xL10t

+ ‖∂xU(t)V ‖L∞x L2t<∞.

As previously, our proof extends to the defocusing case ut+(uxx−u5)x =0.

5

The proofs of the two results strongly rely on the scattering analysis of[4] and [5]. However, the arguments developed in each case are of completelydifferent nature, this is why the proofs will be presented in a separate way :Section 2 gives the strategy of the proofs, Section 3 is devoted to the proofof Theorem 1, and Section 4, to that of Theorem 2.

Acknowledgment

I would like to express my gratitude to my advisor Frank Merle for hisavailability and his constant encouragements, and to Luis Vega and YvanMartel for enlightening discussions. I would also thank the referee for hiscareful reading of the manuscript.

2 Strategy of the proofs.

2.1 The general case p > 3.

We shall always consider w(t) = u(t) − U(t)V . Thus we are looking for wsatisfying the equivalent conditions :{

wt + wxxx + (w + U(t)V )px = 0,

M t0(w(t)) → 0 as t→∞.

(8)

To construct w, we will use the following approximation scheme. Let(Sn)n∈N be an increasing sequence in R such that Sn → ∞ as n → ∞.Define wn(t) as a solution to :{

wnt + wnxxx + (wn + U(t)V )px = 0,

wn(Sn) = 0,(9)

defined on a maximal interval of the form (In, Sn] : it corresponds to un

solution to (1) with initial condition un(Sn) = U(Sn)V . Since V ∈ H2,2,U(Sn)V ∈ H2, so un exists and is the unique H2 solution : the same is truefor wn.

Our method is then to prove that in fact :

1. In ≤ T0 independent of n : we can define wn on an interval [T0, Sn]whose lower bound T0 is fixed.

2. We have uniform (in n) decay estimates for wn on the interval [T0, Sn].

(Of course, if 3 < p < 5, there is global existence in H1, and thusIn = −∞ is automatic). To prove this, we will make an intensive useof the tools developed by Hayashi and Naumkin : this is the heart ofthe proof, and it is done in Proposition 1.

6

3. We then prove that the sequence wn(t) converges to a certain w(t) (inC0

t L2x).

4. By weak limit, we improve the regularity of w(t), to conclude that wis a strong solution to (8).

This does the existence part of the theorem. For the uniqueness part, westudy again the L2 difference of a solution with the one we constructed,and show, with a Gronwall type argument, that these two solutions coincidewhere they are defined.

This scheme of proof is very similar to that of [8] and [12]. In [8], Martelalso study the problem of constructing solutions with a given asymptoticbehavior : there it is proved that given a sum of solitons R(t), there exist aunique u(t) solution to (1) with 2 ≤ p ≤ 5 such that ‖u(t) − R(t)‖H1 → 0as t → ∞, and furthermore, the convergence takes place in all the Sobolevspaces Hs with an exponential decay. Remark that Martel deeply used ideasof the stability for a sum of decoupled solitons [11].

2.2 The L2-critical case (p = 5).

In the critical case, we do not need H1,1 regularity. Since Kenig, Ponce andVega [5] obtained global existence for small data in L2, we use their setting.

The proof goes in this way : denoting as usual w(t) = u(t)− U(t)V , wewrite formally that an eventual w should solve a fixed point problem withthe condition w(t) → 0 as t→∞.

The fixed point problem is in fact very similar to that of the Cauchyproblem, so that we can reuse the linear estimates proved by Kenig, Ponceand Vega [5] for their global existence theorem for small data.

The fact that w(t) → 0 is our smallness condition, which allows us tohave a contracting map, and thus a fixed point.

3 Proof of Theorem 1 : p > 3.

We start by reminding the linear estimates of [4] which will be used through-out the proof.

3.1 Linear estimates.

Let p > 3. Remind our notations : three fixed constants γ ∈]0,min{(p −3)/3, 1/2}[, α = 1/2 − γ, δ = (p − 3 − 2γ)/3 > 0, the operator J tφ =xφ− 3t∂2

xφ = U(t)xU(−t)φ, and our working norm:

M t0(φ) = ‖φ‖H1 + ‖DαJ tφ‖L2 + ‖DJ tφ‖L2 .

7

First a few remarks on M t0. Of course M0

0 (φ) ≤ C‖φ‖H1,1 . Second, note thatJ tU(t)V = U(t)xV (and U(t) is a Hs isometry), so that if V ∈ H1,1, wehave the uniform control in t :

M t0(U(t)V ) ≤ C‖V ‖H1,1 . (10)

We now remind the linear results obtained in [4] (Lemma 2.2), in a slightlyimproved form.

Lemma 1. Let t > 0 and φ be a function so that M t0(φ) is bounded. Then

for r > 4 :

‖φ‖Lr ≤ C

(1 + t)1/3−1/(3r)M t

0(φ).

And one also has the pointwise inequalities :

|φ(x)| ≤ CM t0(φ)

(1 + t)1/3

(1 +

∣∣∣ xt1/3

∣∣∣)− 14, |φx(x)| ≤ CM t

0(φ)t2/3

(1 +

∣∣∣ xt1/3

∣∣∣) 14.

As a simple consequence, for V ∈ H1,1, we have similar decay estimateson U(t)V .

Proof. See [4], Lemma 2.2 and its proof (especially inequalities 2.16, 2.17.and 2.18). For completeness, the proof is given in Appendix A.

We will also need the polarized version of Lemma 2.3 of [4] :

Lemma 2. Let h, k : R → R. Then the following inequalities are valid iftheir right-hand side is bounded :

‖Dαkp‖L2 ≤ C‖kp−1‖L2(‖kkx‖1/2L∞ + ‖k‖3γ

L∞‖kkx‖(1−3γ)/2L∞ ),

‖Dα|k|p−1hx‖L2 ≤ C(‖Dαh‖L2 + ‖hx‖L2)(‖k‖p−3L∞ ‖kkx‖L∞

+‖k‖p−3−2γL∞ ‖k‖2γ

L2‖kkx‖L∞ + ‖k‖p−3+2γL∞ ‖kkx‖1−γ

L∞ ).

Proof. See [4], Lemma 2.3 and its proof (case σ = 0).

3.2 Uniform estimates on wn.

Recall that wn is the solution to the problem :{wnt + wnxxx + (wn + U(t)V )p

x = 0,wn(Sn) = 0,

(9)

where Sn →∞ is an increasing sequence of times. Using estimates developedin [4], we have the following proposition for wn :

8

Proposition 1 (Uniform estimates on wn). Let p > 3. There existsT0 ≥ 1 independent of n, so that for all n ∈ N such that Sn ≥ T0, wn ∈C([T0, Sn],H1(R)), and :

∀t ∈ [T0, Sn], M t0(wn(t)) ≤ C(1 + ‖V ‖p

H2,2)t−δ (11)

where δ = (p− 3 + 2γ)/3 > 0 and C are independent of n and V .

Of course Proposition 1 is the heart of the proof of Theorem 1.

Proof. Define I∗n ≥ 1 minimal so that :

∀t ∈ [I∗n, Sn], M t0(wn(t)) ≤ 1. (12)

Lemma 3. Suppose we can prove :

∀t ∈ [I∗n, Sn], M t0(wn(t)) ≤ C(1 + ‖V ‖p

H2,2)t−δ.

Then I∗n ≤ T0 independent of n and Proposition 1 holds true :

∀t ∈ [T0, Sn], M t0(wn(t)) ≤ C(1 + ‖V ‖p

H2,2)t−δ.

Proof of Lemma 3. This follows from a continuity argument. First, due toTheorem 3, proved in Appendix B, t 7→M t

0(un(t)) is upper semi-continuous :since M t

0(wn(Sn)) = 0, I∗n < Sn.Let T0 ≥ 1 be such that C(1 + ‖V ‖p

H2,2)T−δ0 ≤ 1. If I∗n > T0, C(1 +

‖V ‖pH2,2)I∗n

−δ < 1, our hypothesis gives that :

∀t ∈ [I∗n, Sn], M t0(wn(t)) ≤ C(1 + ‖V ‖p

H2,2)t−δ ≤ C(1 + ‖V ‖pH2,2)I∗n

−δ < 1.

By our upper semi-continuity argument (again Theorem 3), there wouldbe a t′ < I∗n so that for t ∈ [t′, Sn], M t

0(wn(t)) ≤ 1. This contradicts theminimality of I∗n, and so I∗n ≤ T0.

The decay estimate follows immediately.

So it is enough to prove :

∀t ∈ [I∗n, Sn], M t0(wn(t)) ≤ C(1 + ‖V ‖p

H2,2)t−δ,

assuming (12). Recall :

M t0(φ) = ‖φ‖L2 + ‖Dφ‖L2 + ‖DαJ tφ‖L2 + ‖DJ tφ‖L2 .

We will now estimate each one of the 4 norms involved. Let us denote K =1 + ‖V ‖H2,2 . We will successively prove that for t ∈ [I∗n, Sn], we have :

(i) ‖wn‖L2 ≤ CKpt−(p−3)/3.

(ii) ‖wn‖H1 ≤ CKpt−(p−3)/3.

9

(iii) ‖DαItwn‖L2 ≤ CEpt−δ.

(iv) ‖Itwnx‖L2 ≤ CKpt−(p−3)/3.

(v) M t0(w(t)) ≤ CKpt−δ.

Remark that we first do estimates on It (step (iii) and (iv)), since it easierto handle when doing energy methods estimates. In step (v) we shall go backto estimates on J t.

Before we do this, we have to notice that Lemma 1 applies for t ∈ [I∗n, Sn]to give :

|wn(t, x)| ≤ Ct−1/3

(1 +

|x|t1/3

)− 14

, |wnx(t, x)| ≤ Ct−2/3

(1 +

|x|t1/3

) 14

.

Then we can add up U(t)V in these estimates to get :

|(U(t)V + wn(t))(x)| ≤ CKt−1/3

(1 +

|x|t1/3

)− 14

, (13)

|(U(t)V + wn(t))x(x)| ≤ CKt−2/3

(1 +

|x|t1/3

) 14

, (14)

‖U(t)V + wn(t)‖Lr ≤ CKt−1/3−1/(3r), r > 4. (15)

Proof of (i).Let us multiply (9) by wn and integrate in x :

12d

dt

∫w2

n = −∫

(U(t)V +wn)pxwn = −p

∫(U(t)V +wn)x(U(t)V +wn)p−1w.

So (after simplification by ‖wn‖L2) :

d

dt‖wn‖L2 ≤ ‖U(t)V + wn‖L2‖(U(t)V + wn)(U(t)V + wn)x‖L∞

‖U(t)V + wn‖p−3L∞ .

For t ∈ [I∗n, Sn], we get :

d

dt‖wn‖L2 ≤ C(K + ‖wn(t)‖L2) ·

K2

t· Kp−3

t(p−3)/3≤ CKpt−

p3 .

Thus, by integration in time on [t, Sn], using wn(Sn) = 0, this gives :

‖wn(t)‖L2 ≤ C(p)Kp(t−(p−3)/3 − S−(p−3)/3

n

)≤ CKpt−(p−3)/3. (16)

Proof of (ii).Let us differentiate (9) with respect to x :

wnxt + wnxxxx + (U(t)V + wn)pxx = 0.

10

Now, multiply by wnx and integrate in x. After an integration by part, weget :

12d

dt

∫wn

2x =

∫(U(t)V + wn)p

xwnxx

= p

∫wnxx(U(t)V + wn)x(U(t)V + wn)p−1.

The point is to rule out wnxx, term that we cannot control : for this we willsplit the second term and do integrations by parts. First :

2∫wnxxwnx(U(t)V + wn)p−1 = −p

∫wn

2x(U(t)V + wn)x(U(t)V + wn)p−2,

which we can easily control. Indeed :

p

∣∣∣∣∫ wnxxwnx(U(t)V + wn)p−1

∣∣∣∣≤ p‖wnx‖2

L2‖(U(t)V + wn)x(U(t)V + wn)‖L∞‖U(t)V + wn‖p−3L∞

≤ CKp−1‖wnx(t)‖2L2t

p3 ≤ CKp‖wnx‖L2t

p3 .

(as ‖wn‖H1 ≤M t0(w(t)) ≤ 1). For the second term :∫

wnxx(U(t)V )x(U(t)V + wn)p−1 = −∫wnx(U(t)V )xx(U(t)V + wn)p−1

−∫wnx(U(t)V )x(U(t)V + wn)p−2(U(t)V + wn)x.

The first integral would be troublesome (with the double derivative onU(t)V ), this is why we made the hypothesis V ∈ H2,2 (in [4], this phe-nomenon is avoided because the integration by parts works better). Thesecond integral is fine. So we can do the estimate (note that we use theL∞ decay estimate (16) on (U(t)V )xx = (U(t)Vx)x, as Vx ∈ H1,1), for allt ∈ [I∗n, Sn] :

p

∣∣∣∣∫ wnxx(U(t)V )x(U(t)V + wn)p−1

∣∣∣∣≤ C‖wnx‖L2‖(U(t)V +wn)(U(t)Vx)x‖L∞‖U(t)V +wn‖L2‖U(t)V +wn‖p−3

L∞

+C‖wnx‖L2‖(U(t)V )x‖L2‖(U(t)V +wn)(U(t)V +wn)x‖L∞‖U(t)V +wn‖p−3L∞

≤ CKp‖wnx‖L2t−p/3.

(we used estimate (16), and ‖wn‖H1 ≤ 1). So finally, after simplifying by‖wnx‖L2 :

d

dt‖wnx(t)‖L2 ≤ CKpt−p/3.

11

After integration between t and Sn (wn(Sn) = 0) :

‖wnx(t)‖L2 ≤ CKp(t−(p−3)/3 − S−(p−3)/3

n

).

And so, we neglect the Sn term, and when adding up with (16), we obtain :

‖wn(t)‖H1 ≤ C(p)Kpt−(p−3)/3. (17)

Proof of (iii).We now turn to the estimates on Itwn. Let us denote L = ∂t + ∂xxx

the linear KdV operator, and remind some commutation relations of thedifferent operators involved . Remind the definition of the dilation operatorItφ = xφ+ 3t

∫ x−∞ φtdx and of J tφ = xφ− 3t∂xxφ. Then :

Itφ− J tφ = 3t∫ x

−∞Lφdx.

We have the following commutation relations :

[L, J t] = 0, [L, It]φ = 3∫ x

−∞Lφdx, [J t, ∂x] = [It, ∂x] = −Id.

Again notice that ItU(t)V − J tU(t)V = 3t∫ x−∞ LU(t)V dx = 0 so that

‖DαItU(t)V ‖L2 + ‖DItU(t)V ‖L2 ≤ C‖V ‖H1,1 .Let f : R → R be a C1 function and φ be such that f ′(φ)Itφx has a

sense. Then :

It(f(φ)x) = xf(φ)x + 3tf(φ)t = xf ′(φ)φx + 3tf ′(φ)φt = f ′(φ)Itφx.

We will use this formula for f(x) = xp and φ = U(t)V + wn(t).We compute :

LItwn = ItLwn + 3∫ x

−∞Lwn = −It(U(t)V + wn)p

x − 3(U(t)V + wn)p

= −p(U(t)V + wn)p−1It(U(t)V + wn)x − 3(U(t)V + wn)p

= −p(U(t)V + wn)p−1(It(U(t)V + wn))x − (3− p)(U(t)V + wn)p.

Apply the operator Dα, multiply by DαItwn and integrate in space (remindthat [Dα, L] = 0 and (Lφ, φ) = 1

2ddt

∫φ2) :

d

dt‖DαItwn‖2

L2 ≤ ‖DαItwn‖L2‖Dα(U(t)V +wn)p−1(It(U(t)V +wn))x‖L2

+ C‖DαItwn‖L2‖Dα(U(t)V + wn)p‖L2 . (18)

12

We now apply Lemma 2 with k = U(t)V +wn and h = It(U(t)V +wn). Sothat (along with the linear estimates (13), (14), (15)) :

‖kkx‖L∞ ≤ CK2t−1, ‖k‖L2 ≤ CK, ‖k‖L∞ ≤ CKt−13 ,

‖kp−1‖L2 = ‖k‖p−1

L2(p−1) ≤ CKp−1t(p−1)(− 1

3+ 1

6(p−1)) ≤ CKp−1t−(2p−3)/6,

and :‖Dαh‖L2 ≤ CK, ‖hx‖L2 ≤ CK.

We can compute :

‖Dα(U(t)V + wn)p−1(It(U(t)V + wn))x‖L2

≤ CKp(t−(p−3)/3+1 + t−(p−3−2γ)/3+1 + t−(p−3+2γ)/3−1+γ)≤ CKpt−(p−2γ)/3.

(as t ≥ I∗n ≥ 1; the point being (p− 2γ)/3 > 1). And for the second term :

‖Dα(U(t)V + wn)p‖L2 ≤ CKp(t−(2p−3)/6−1/2 + t−(2p−3)/6−γ−(1−3γ)/2)≤ CKpt−p/3+γ/2.

Finally, after simplification by ‖DαItwn‖L2 , (18) gives :

d

dt‖DαItwn‖L2 ≤ CKpt−(p−2γ)/3.

And as before, we integrate on [t, Sn] :

‖DαItwn(t)‖L2 ≤ CKpt−(p−3−2γ)/3. (19)

Proof of (iv).Again, we compute :

LIt(wnx) = ItLwnx + 3Lwn = −It(U(t)V + wn)pxx − 3(U(t)V + wn)p

x

= −(It(U(t)V + wn)px)x − 2(U(t)V + wn)p

x

= −p(U(t)V + wn)p−1(It(U(t)V + wn)x)x

−p(p− 1)(U(t)V + wn)p−2(U(t)V + wn)xIt(U(t)V + wn)x

−2p(U(t)V + wn)p−1(U(t)V + wn)x. (20)

We want to multiply by Itwnx and integrate in space. There is essentiallyone troublesome term, the double derivative one. We split it :

(U(t)V + wn)p−1(It(U(t)V + wn)x)x

= (U(t)V + wn)p−1(ItU(t)Vx)x + (Itwnx)x). (21)

13

The second term will have only first order terms after integration by parts,which is fine :∣∣∣∣∫ (U(t)V + wn)p−1(Itwnx)xI

twnx

∣∣∣∣=

(p− 1)2

∣∣∣∣∫ (U(t)V + wn)p−2(U(t)V + wn)x(Itwnx)2∣∣∣∣

≤ C‖(U(t)V + wn)(U(t)V + wn)x‖L∞‖U(t)V + wn‖p−3L∞ ‖I

twnx‖2L2

≤ CKp−1t−p/3‖Itwnx‖2L2 .

However the first term on the right hand side of (21) requires extra regularityon V . Indeed

(ItU(t)Vx)x = (J tU(t)Vx)x = (U(t)xVx)x,

and (xVx ∈ H1,1), M t0(U(t)(xVx)) ≤ ‖V ‖H2,2 so that by Lemma 1 :

|(ItU(t)Vx)x(x)| ≤ C‖V ‖H2,2

t2/3

(1 +

∣∣∣ xt1/3

∣∣∣) 14.

We can now estimate (essentially in the same way that for the H1 estimate) :∣∣∣∣∫ (U(t)V + wn)p−1(ItU(t)Vx)xItwnx

∣∣∣∣≤ ‖U(t)V + wn‖p−3

L∞ ‖U(t)V + wn‖L2

‖(U(t)V + wn)(ItU(t)Vx)x‖L∞‖Itwnx‖L2

≤ CKp−1tp/3‖Itwnx‖L2 .

The remaining two terms in (20) are simpler and can be treated directly(after multiplication by Itwnx and integration in x) :∣∣∣∣∫ (U(t)V + wn)p−2(U(t)V + w)xI

t(U(t)V + wn)xItwnx

∣∣∣∣≤ ‖U(t)V + wn‖p−3

L∞ ‖(U(t)V + wn)(U(t)V + wn)x‖L∞

‖It(U(t)V + wn)x‖L2‖Iwnx‖L2

≤ CKptp/3‖Itwnx‖L2 ,∣∣∣∣∫ (U(t)V + wn)p−2(U(t)V + wn)xItwnx

∣∣∣∣≤ ‖U(t)V + wn‖p−3

L∞ ‖(U(t)V + wn)(U(t)V + wn)x‖L∞

‖U(t)V + wn‖L2‖Itwnx‖L2

≤ CKptp/3‖Itwnx‖L2 .

Hence, (after simplifying by ‖Itwnx‖L2) we get :

d

dt‖Itwnx‖L2 ≤ CKp−1(K + ‖Itwnx‖L2)tp/3 ≤ CKpt−p/3.

14

And Gronwall’s Lemma (between t and Sn) gives :

‖Itwnx(t)‖L2 ≤ CKpt−(p−3)/3. (22)

Proof of (v).We now have to go back to

J twn(t) = Itwn(t)− 3t∫ x

−∞Lwn(t)dx′ = Itwn(t) + 3t(U(t)V + wn(t))p.

Using the commutation relations, we get :

‖DαJ twn‖L2 + ‖DJ twn‖L2 ≤ ‖DαItwn‖L2 + 3t‖Dα(U(t)V + wn(t))p‖L2

+ ‖Itwnx‖L2 + ‖wn‖L2 + 3pt‖(U(t)V + wn(t))p−1(U(t)V + wn(t))x‖L2 .

Now, using again Lemma 2 with k = U(t)V + wn(t) :

‖Dα(U(t)V + wn(t))p‖ ≤ C‖kp−1‖L2(‖kkx‖1/2L∞ + ‖k‖3γ

L∞‖kkx‖(1−3γ)/2L∞ )

≤ CKpt−p/3+γ/2.

And as usual :

‖(U(t)V + wn)p−1(U(t)V + wn)x‖L2

≤ ‖U(t)V + wn‖p−3L∞ ‖(U(t)V + wn)(U(t)V + wn)x‖L∞‖U(t)V + wn‖L2

≤ CKptp/3.

So using these last two inequalities, along with (16), (19) and (22), we get :

‖DαJ twn‖L2 + ‖DJ twn‖L2 ≤ CKpt(p−3+2γ)/3. (23)

Recall δ = (p− 3− 2γ)/3 > 0. So adding up (17) and (23) gives :

∀t ∈ [I∗n, Sn], M t0(wn(t)) ≤ CKpt−δ.

3.3 Construction and uniqueness of u.

Proof of Theorem 1. Existence of u.Proposition 1 provides us with a sequence wn(t) solution to (9) satisfying

uniform estimates in n :

∀t ∈ [T0, Sn], M t0(wn(t)) ≤ CKpt−δ.

In particular, for t ∈ [T0, Sn], estimates (13) and (14) are valid.Let us prove that for all k ∈ N, (wn)n≥k is a convergent sequence in

C0([T0, Sk], L2(R)). For this we show that (wn)n≥k is a Cauchy sequencein C0([T0, Sk], L2(R)) : let us consider vn,m = wn − wm in L2. Without

15

loss of generality, we can suppose that m > n ≥ k. First : ‖vn,m(Sn)‖L2 ≤CKpS

−(p−3)/3n (see (16)).

vn,m satisfies (we denote v = vn,m for simplicity in the computations) :

vt + vxxx + (U(t)V + wn)px − (U(t)V + wm)p

x = 0. (24)

We multiply by v and integrate in x :

12d

dt

∫v2 =

∫v((U(t)V + wn)p

x − (U(t)V + wm)px).

Now, for any functions φ and ψ :

φpx − ψp

x = pφp−1φx − pψp−1ψx = pφp−1(φ− ψ)x + p(φp−1 − ψp−1)ψx,

and :|φp−1 − ψp−1| ≤ C|ψ − φ|(|φ|p−2 + |ψ|p−2).

So that with φ = U(t)V + wn, ψ = U(t)V + wm (after integration by partson the first term) :

12d

dt

∫v2 = −p− 1

2

∫v2(U(t)V + wn)p−2(U(t)V + wn)x

+∫v(U(t)V + wn)p−1 − (U(t)V + wm)p−1)(U(t)V + wm)x.

Treating each term separately (for t ∈ [T1, Sn]) :∣∣∣∣∫ v2(U(t)V + wn)p−2(U(t)V + wn)x

∣∣∣∣ ≤ CKp−1t−p/3‖v‖2L2 ,∣∣∣∣∫ v(U(t)V + wn)p−1 − (U(t)V + wm)p−1)(U(t)V + wm)x

∣∣∣∣≤ C

∫v2(|U(t)V + wn|p−2 + |U(t)V + wm|p−2|(U(t)V + wm)x|

≤ CKp−1t−p/3‖v‖2L2 .

So that :d

dt‖v‖2

L2 ≤ CKp−1t−p/3‖v‖2L2 .

Now, using Gronwall’s Lemma on [t, Sn] :

‖vn,m(t)‖L2 ≤ C‖vn,m(Sn)‖L2 ≤ CS−(p−3)/3n .

This proves that (wn)n≥k is a Cauchy sequence in the space C([T0, Sk], L2)and so converges to a certain w(t, x). Since this can be done for arbitrarilylarge n (and Sn → ∞), w ∈ C([T0,∞), L2) is the only possible weak limitof (wn)n∈N.

16

Given a fixed t ≥ T0, M t0(wn(t)) ≤ CKpt−δ, so by weak limit :

M t0(w(t)) ≤ CKpt−δ <∞.

Now wn satisfies (9), hence :

wn(t) = wn(T0) +(∫ t

T0

U(t− s)(wn(s))pds

)x

.

wn(T0) → w(T0) in L2 and wn(t) → w(t) in L2. Furthermore, wn → win Cb([T0, t], L2) and wn is a bounded sequence in L∞([T0, t],H1) (for allt ≥ T0), so that by interpolation wp

n → wpn in C([T0, t], L2). Thus :(∫ t

T0

U(t− s)(wn(s))pds

)x

→(∫ t

T0

U(t− s)(w(s))pds

)x

in H−1.

Hence, w satisfies the integral formulation of (8). As a consequence, whendefining u(t) = w(t)+U(t)V , u ∈ Cb([T0,∞),H1) satisfies the conditions ofTheorem 1.

Uniqueness of u.Let u be another solution. We switch to w(t, x) = u(t, x)− (U(t)V )(x).

Then of course : {wt + wxxx + (U(t)V + w)p

x = 0,M t

0(w(t)) → 0 as t→∞.

Introduce T1 such that for t ≥ T1, M t0(w(t)) ≤ 1. Let us introduce again

v(t, x) = w(t, x)− w(t, x). Then :

vt + vxxx + (U(t)V + w)px − (U(t)V + w)p

x = 0,

which is basically identical to (24). So multiplying by v and integrating inx, one can do the same computations as previously, so that for t ≥ T1 :∣∣∣∣ ddt‖v(t)‖2

L2

∣∣∣∣ ≤ C(V )t−p/3‖v‖2L2 .

By Gronwall’s Lemma between t and τ :

‖v(t)‖L2 ≤ C(V )‖v(τ)‖L2 .

By hypothesis, M τ0 (v(τ)) → 0 so that ‖v(τ)‖L2 → 0, and letting τ → ∞

gives :∀t ≥ T1, v(t, x) = 0.

So w(t, x) = w(t, x), u(t, x) = u(t, x) for t ≥ T1 and by H1 uniqueness, forall t such that u and u are defined, u(t, x) = u(t, x).

17

4 Proof of Theorem 2 : the critical case p = 5.

4.1 Linear estimates.

Let us now set p = 5, and remind the linear estimates in [5].

Lemma 4. Let φ be a function of space and ψ be a function of time andspace. Then the following inequalities hold, assuming their right-hand sideterm is bounded :

1. ‖∂xU(t)φ‖L∞x L2t ([T,∞)) ≤ ‖φ‖L2

x.

2. supt≥T

∥∥∥∥∂x

∫ ∞

tU(t− s)ψ(s)ds

∥∥∥∥L2

x

≤ ‖ψ‖L1xL2

t ([T,∞)).

3.∥∥∥∥∂2

xx

∫ ∞

tU(t− s)ψ(s)ds

∥∥∥∥L∞x L2

t ([T,∞))

≤ ‖ψ‖L1xL2

t ([T,∞)).

4. ‖U(t)φ‖L5xL10

t ([T,∞)) ≤ C‖φ‖L2x.

5.∥∥∥∥∫ ∞

tU(t− s)ψ(s)ds

∥∥∥∥L5

xL10t ([T,∞))

≤ ‖ψ‖L

5/4x L

10/9t ([T,∞))

.

Proof. See [5]. In the proof, T = −∞ ; for general T , the estimates 1. and4. are clear, and for estimates 2., 3. and 5., replace ψ by ψ1t≥T .

4.2 Construction of u.

Proof of Theorem 2. Let us do formal computations first : suppose u is thedesired solution, and write :

u(t) = U(t)V + w(t), i.e. w(t) = u(t)− U(t)V, limt→∞

‖w‖L2 = 0.

u satisfies :

u(τ) = U(τ)u(0) + ∂x

∫ τ

0U(τ − s)u5(s)ds

= U(τ − t)u(t) + ∂x

∫ τ

tU(τ − s)u5(s)ds.

Thus, w satisfies :

U(τ)V+w(τ) = U(τ−t)(U(t)V+w(t))+∂x

∫ τ

tU(τ−s)(U(s)V+w(s))5(s)ds.

U(τ)V cancels and we obtain after composing by U(t− τ) :

U(t− τ)w(τ) = w(t) + ∂x

∫ τ

tU(t− s)(U(s)V + w(s))5(s)ds.

18

Now, let τ → ∞, ‖U(t − τ)w(τ)‖L2 = ‖w(τ)‖L2 → 0, so we obtain w as afixed point :

w(t) = −∂x

∫ ∞

tU(t− s)(U(s)V + w(s))5(s)ds.

This explains the following scheme : we will show that such fixed pointexists by a contracting map argument. Let us introduce the function Φ.

Φ : w 7→ −∂x

∫ ∞

tU(t− s)(U(s)V + w(s))5ds. (25)

Our goal is to find a fixed point for Φ.Let us introduce the following norm :

‖φ‖XT= ‖φ‖L5

xL10t ([T,∞)) + ‖∂xφ(t)‖L∞x L2

t ([T,∞)).

From estimate 5. of Lemma 4, we get :

‖Φ(w)(t)‖L5xL10

t ([t,∞)) ≤ ‖(U(t)V + w(t))4(U(t)V + w(t))x‖L5/4x L

10/9t ([t,∞))

≤ ‖(U(t)V + w(t))4‖L

5/4x L

10/4t ([t,∞))

‖(U(t)V + w(t))x‖L∞x L2t ([t,∞))

≤ C‖U(t)V + w(t)‖4L5

xL10t ([t,∞))‖∂x(U(t)V + w(t))‖L∞x L2

t ([t,∞)).

By estimate 3. :

‖∂xΦ(w)(t)‖L∞x L2t ([t,∞)) ≤ ‖(U(t)V + w(t))5‖L1

xL2t ([t,∞))

≤ ‖U(t)V + w(t)‖5L5

xL10t ([t,∞))

Adding up the last two estimates (and using |a + b|k ≤ 2k−1(|a|k + |b|k),|a4b| ≤ |a|5 + |b|5):

‖Φ(w)‖XT≤ C‖U(t)V + w(t)‖4

L5xL10

t ([T,∞))

×(‖U(t)V + w(t)‖L5

xL10t ([T,∞)) + ‖∂x(U(t)V + w(t))‖L∞x L2

t ([T,∞))

)≤ 23(‖U(t)V ‖4

L5xL10

t ([T,∞)) + ‖w‖4XT

)(‖U(t)V ‖XT+ ‖w‖XT

)

≤ C(‖U(t)V ‖4L5

xL10t ([T,∞))‖U(t)V ‖XT

+ ‖U(t)V ‖XT‖w‖4

XT+ ‖w‖5

XT).

Let us prove the following simple lemma :

Lemma 5. Let V be such that U(t)V ∈ X0 (in particular, if V ∈ L2). Then‖U(t)V ‖L5

xL10t ([T,∞)) → 0 as T →∞.

Proof. Consider

f(τ, x) = ‖U(t)V ‖5L10

t [τ,∞), and h(x) = ‖U(t)V ‖5L10

t [0,∞) = f(0, x).

19

f and h are finite x-a.e. as∫x f <∞ and

∫x h <∞. Then h ∈ L1

x and for allτ ≥ 0, 0 ≤ f(τ, x) ≤ h(x). Now, since f is an exhausting integral, we have :

x− a.e., limτ→∞

f(τ, x) = 0.

Lebesgue’s dominated convergence theorem applies : limτ→∞ ‖f(τ, x)‖L1x

=0, which is exactly limτ→∞ ‖U(t)V ‖L5

xL10t [τ,∞) = 0.

As V ∈ L2, by the previous lemma, ‖U(t)V ‖L5xL10

t ([T,∞)) → 0 as T →∞.Moreover, ‖U(t)V ‖XT

≤ C‖V ‖L2 . So our estimate writes :

‖Φ(w)‖XT≤ C(‖U(t)V ‖4

L5xL10

t ([T,∞))‖V ‖L2 + ‖V ‖L2‖w‖4XT

+ ‖w‖5XT

). (26)

This shows that for T0 large enough, there exists δ > 0 so that Φ mapsBXT0

(0, δ) to itself :

Φ : BXT0(0, δ) → BXT0

(0, δ).

The same computations show that for T0 large enough, δ > 0 small enough,Φ : BXT0

(0, δ) → BXT0(0, δ) is a contraction. Thus, Φ has a unique fixed

point, which we denote v.v = Φ(v) writes :

v(t) = −∂x

∫ ∞

tU(t− s)(U(s)V + v(s))5ds,

and by construction u(t) = U(t)V + v(t) is a solution to (4). Now by (26),if δ has been chosen small enough, we have :

‖v‖XT≤ C‖U(t)V ‖4

L5xL10

t ([T,∞))‖V ‖L2 → 0 as T →∞.

And by estimate 2. of Lemma 4 :

‖v‖L∞t ([T,∞),L2x) ≤ ‖U(t)V + v‖5

L5xL10

t ([T,∞))

≤ C(‖U(t)V ‖5L5

xL10t ([T,∞)) + ‖v‖5

XT) → 0 as T →∞.

To conclude, v satisfies the decay estimate ‖v(t)‖L2 → 0 and moreover :

‖v‖L∞t ([T,∞))L2x

+ ‖v‖L5xL10

t ([T,∞)) + ‖∂xv‖L∞x L2t ([T,∞)) → 0 as T →∞.

Appendix A.

For the sake of completeness, we present the proof of lemma 1 of [4].

20

Proof of Lemma 1. Let us note v(t, x) = U(−t)φ. We have the identity :

φ(x) = U(t)v =1π<∫ ∞

0eipx+ip3t/3v(t, p)dp

=1

π 3√t<∫ ∞

0eiqη+iq3/3

(v(t, χ) +

(v

(t,q3√t

)− v(t, χ)

))dq

=13√t<Ai

(x3√t

)v(t, χ) +R(t, x),

where we made the change of variable q = p 3√t and η = x/ 3

√t, and we

introduce χ =√−x/t if x ≤ 0 and χ = 0 if x ≥ 0, and :

R(t, x) =1

π 3√t<∫ ∞

0eiqη+iq3/3

(v

(t,q3√t

)− v(t, χ)

)dq.

We now estimate R(t, x), and consider two cases : x ≥ 0 and x ≤ 0. Forthis, we will do appropriate integration by parts to use the decay related tothe oscillatory integral.

Let us introduce a final notation µ =√|η|. We have the identity :

eiqη+iq3/3 =1

1 + iq(q2 + µ2)∂

∂q(qeiqη+iq3/3). (27)

Consider the case x ≥ 0, that is χ = 0. We can do an integration by partsusing (27) in the remainder term R :

R(t, x) =1

π 3√t<∫ ∞

0

(iq(q2 + µ2)(v(t, q/ 3

√t)− v(t, 0))

1 + iq(q2 + µ2)

− q3√tvp

(t,q3√t

))eiqη+iq3/3dq

1 + iq(q2 + µ2).

Using the identity

‖DαJ tφ‖L2 = ‖U(t)Dα(xU(−t)φ)‖L2 = ‖Dα(xv)‖L2 = ‖|p|αvp‖L2 ,

and the Cauchy-Schwarz inequality, we get :

|v(t, p)− v(t, 0)| ≤∫ p

0|vp(t, ρ)|dρ ≤

(∫ p

0|ρ|−2αdρ

∫ p

0|v2

p(t, ρ)ρ2α|dρ

) 12

≤ C|p|γ‖|p|αvp(t, p)‖L2 ≤ C|p|γ‖DαJ tφ‖L2 . (28)

On the other hand, using the change of variable z = q(1+µ2), a computationshows that, for any a, b > 0 such that 3b− a > 1, we have :

∞∫0

qadq

(1 + q(q2 + µ2))b≤

µ+1∫0

2qadq

(1 + q(1 + µ2))b+

∞∫µ+1

qadq

1 + q3b≤ C

(1 + µ)3b−a−1.

(29)

21

(we used the inequality 1 + q(q2 + µ2) ≥ 12(1 + q(1 + µ2)) for the first term

and q2 ≤ q2 + µ2 for the second). Thus we can estimate :

|R(t, x)| ≤ C3√t

∫ ∞

0

(|v(t, q/ 3√t)− v(t, 0)|+ q/ 3

√t|vp(t, q/ 3

√t)|)dq

|1 + iq(q2 + µ2)|

≤ C

t(1+γ)/3‖DαJ tφ‖L2

∫ ∞

0

qγdq

1 + q(q2 + µ2)

+C3√t2

(∫ ∞

0q2αv2

p

(t,q3√t

)dq

∫ ∞

0

q1+2γdq

(1 + q(q2 + µ2))2

)1/2

≤ C‖DαJ tφ‖L2

t(1+γ)/3(1 + µ)2−γ. (30)

Let us now consider the case x ≤ 0, that is, η = −µ2 ≤ 0 (recall µ =√|x|/ 6

√t, χ = µ/ 3

√t). We will now use the identity :

eiqη+iq3/3 =1

1 + i(q2 − µ2)(q + µ)∂

∂q((q − µ)eiqη+iq3/3). (31)

We integrate by parts the remainder term R :

R(t, x) =1

π 3√t<∫ ∞

0

(i(q − µ)2(3q + µ)

1 + i(q2 − µ2)(q + µ)

(v

(t,q3√t

)− v(t, χ)

)−q − µ

3√tvp

(t,q3√t

))eiqη+iq3/3dq

1 + i(q + µ)(q − µ)2.

A computation similar to (29) gives (provided that 3c− a− b− 1 > 0) :∫ ∞

0

|q − µ|aqbdq

(1 + (q − µ)2(q + µ))c≤

∫ 2(1+µ)

0+∫ ∞

2(1+µ)

≤ C

∫ 1+µ

−1−µ

|z|a(1 + µ)bdz

1 + (z2(1 + µ))c+∫ ∞

1+µ

|z|a+bdz

1 + z3c

≤ C(1 + µ)a+b+1−3c.

(with the change of variable z = q − µ and z′ = z√

1 + µ). Thus, by theCauchy-Schwarz inequality, we obtain :

|R(t, x)| ≤ C3√t

∫ ∞

0

(∣∣∣∣v(t, q3√t)− v(t, χ)

∣∣∣∣+ |q − µ|3√t

∣∣∣∣vp

(t,q3√t

)∣∣∣∣)× dq

|1 + i(q − µ)2(q + µ)|

≤ C

t(1+γ)/3‖DαJ tφ‖L2

(∫ ∞

0

|q − µ|γdq1 + (q − µ)2(q + µ)

+(∫ ∞

0

(q − µ)2dq(1 + (q − µ)2(q + µ))2q2α

)1/2)

≤ C‖DαJ tφ‖L2

t(1+γ)/3(1 + µ)2−γ. (32)

22

(recall α = 1/2− γ)). It remains to bound ‖v(t, ·)‖L∞ :

|v(t, p)| ≤∫R

|vp(t, ρ)|dρ ≤∫|ρ|≥1

|vp(t, ρ)|dρ+∫|ρ|≤1

|vp(t, ρ)|dρ

≤

(∫|ρ|≥1

|v2pρ

2|dρ∫|ρ|≥1

dρ

ρ2

) 12

+

(∫|ρ|≤1

|v2pρ

2α|dρ∫|ρ|≤1

dρ

ρ2α

) 12

≤ C(‖DJ tφ‖L2 + ‖DαJ tφ‖L2). (33)

Thus, adding up (30), (32), along with the estimate |Ai(η)| ≤ C(1+ |η|)−1/4

and (33), we obtain (recall η = x/ 3√t) :

|φ(x)| ≤ C

t1/3(1 + |η|)−1/4M t

0(φ), (34)

which is the first pointwise estimate. The Lr-estimate follows :

‖φ‖Lr ≤ C

t1/3M t

0(φ)

(∫ (1 +

|x|3√t

)r/4

dx

)1/r

≤ C

t1/3−1/(3r)M t

0(φ)(∫

(1 + |η|)r/4 dη

)1/r

≤ C

t1/3−1/(3r)M t

0(φ).

Now we switch to the derivative φx :

φx(x) =i

π<∫ ∞

0eiqη+iq3t/3v

(t,q3√t

)qdq.

Using identity (27), we obtain analogously to (30), in the domain x ≥ 0(η = µ2 ≥ 0) :

|φx(x)| ≤ C

t2/3

∫ ∞

0

|v(t, q/ 3√t)|qdq

|1 + iq(q2 + µ2)|+C

t

∫ ∞

0

|vp(t, q/ 3√t)|q2dq

|1 + iq(q2 + µ2)|

≤ C

t2/3(‖v(t)‖L∞ + ‖|p|αvp(t, p)‖L2)

≤ C

t2/3(‖DJ tφ‖L2 + ‖DαJ tφ‖L2). (35)

(we used the Cauchy-Schwarz inequality as in (30), and in the last inequality,we used again (33)). In the domain x ≤ 0 (η = −µ2 ≤ 0), we use identity

23

(31) to get analogously to (32) :

|φx(x)| ≤ C3√t2

∫ ∞

0

(|v(t,q3√t

)+|q − µ|

3√t

∣∣∣∣v(t, q3√t)∣∣∣∣)

× qdq

|1 + i(q − µ)2(q + µ)|

≤ C3√t2 ‖v‖L∞

∫ ∞

0

qdq

|1 + i(q − µ)2(q + µ)|

+C3√t2 ‖|p|

αvp(t, p)‖L2

(∫ ∞

0

(q − µ)2q2αdq

(1 + (q − µ)2(q + µ))2

) 12

.

The integral of the second term can be estimated by our regular computa-tions, but we have to be more careful with the first term :

∞∫0

qdq

1 + (q − µ)2(q + µ)≤ C

2(µ+1)∫0

(1 + µ)dq1 + (q − µ)2µ

+ C

∞∫2(µ+1)

(q − µ)dq1 + (q − µ)3

≤ C√

1 + µ

∫dz

1 + z2+

C

1 + µ≤ C

√1 + µ.

So that we obtain, for x ≤ 0, the estimate :

|φx(x)| ≤ C3√t2

(‖DJ tφ‖L2 + ‖DαJ tφ‖L2)√

1 + µ. (36)

Finally, combining (35) and (36) gives the second pointwise estimate (recallη = x/ 3

√t) :

|φx(x)| ≤ CM t0(φ)

t2/3(1 + η)1/4. (37)

Appendix B.

To conclude, we present a proof of a local existence theorem in M t0, which is

needed in the discussion of Theorem 1. The proof is done for forward times,but of course, it is also true backwards.

Theorem 3. Let t0 ∈ R and u0 ∈ H1(R) such that M t00 (u0) < ∞. Then

there exist T = T (M t00 (u0)) and a unique solution u : [t0, t0 + T )×R→ R

to : {ut + uxxx + (up)x = 0,u|t=t0 = u0.

Furthermore :

∀t ∈ [t0, T + t0[, M t0(u(t)) <∞, and lim sup

t↓t0M t

0(u(t)) ≤M t00 (u0).

24

Proof. Uniqueness is straightforward as there is already uniqueness in H1.For the existence part, we proceed by regularization. Let un

0 ∈ H3 withcompact support such that :

M t00 (un

0 − u0) → 0.

(un0 exists by standard density arguments). In particular, we can suppose

that :∀n, M t0

0 (un0 ) ≤ 2M t0

0 (u0) = K.

For every un0 , the local existence theorem inH3 ensures the existence of un(t)

on an interval [t0, t0 + Tn]. Remind that existence in H3 has the same timespan as in H1. Since the initial data sequence (un

0 )n is uniformly boundedin H1, Tn = Tn(‖un

0‖H1) ≥ T ∗ > 0.The point in working in H3 is that It(un(t)) = J t(un(t))− t(un)p is well

defined in H1. Indeed, one computes :∣∣∣∣ ddt∫x(un)2

∣∣∣∣ = ∣∣∣∣−3∫

(unx)2 − 2p

p+ 1

∫(un)p+1

∣∣∣∣ ≤ C‖un(t)‖H1 ,

so that ∀t ∈ [t0, t0 + T ∗), ‖√xun(t)‖L2 < ∞ (recall u0(t0) has compact

support, so that quantity is initially well defined at time t0). In the sameway, the derivatives in time of ‖xun‖L2 + ‖

√xun

x(t)‖L2 are controlled by‖un(t)‖H2 , and that of ‖xun

x‖L2 by ‖un(t)‖H3 . So finally, the H3 bound onun(t) ensures that for t ∈ [t0, t0 + T ∗] :

‖J tun(t)‖H1 ≤ ‖xun(t)‖H1 + t‖unxx(t)‖H1 <∞.

And for Itun(t) :

‖It(un(t))‖H1 ≤ ‖J t(un(t))‖H1 + t‖(un(t))p‖H1 <∞.

So now it is possible to do the a priori computations of Naumkin and Hayashi[4] for un(t). Let I = [t0, t0+T ∗n) such that for all t ∈ I,M t

0(un(t)) ≤ 2K (and

I maximal for this property) : by H3 continuity, T ∗n > 0. Their computationsgive (see equations 3.3, 3.8 and 3.9 of [4]).

d

dt‖un‖2

H1 ≤ CM t

0(un(t))p−2

t2/3(1 + t)(p−2)/3‖un‖2

H1 ≤CKp

t2/3(1 + t)(p−2)/3, (38)

and similar estimates for ‖DαItun‖L2 and ‖DItun‖L2 .Let T be such that :∫ t0+T

t0

CKp

t2/3(1 + t)(p−2)/3dt ≤ K

3.

(There are 3 estimates). By a standard continuity argument, T ∗ ≥ T ∗n ≥ T(independent of n), and :

∀n, ∀t ∈ [t0, t0 + T ], M t0(un(t)) ≤ 2K.

25

In the same way, (38) gives that t 7→M t0(un(t)) is equicontinuous (in n).

Now, standard computations show that un(t) is a Cauchy sequence inC([t0, t0 + T ], L2), so it converges to u(t). Indeed, with v = un − um :

vt + vxxx + (un)px − (um)p

x = 0.

Multiply by v, and integrate in space :

d

dt‖v‖2

L2(t) ≤(2K)p

t2/3(1 + t)(p−2)/3‖v‖2

L2(t).

And by Gronwall’s Lemma (taking the supremum) :

‖v‖C0([t0,t0+T ],L2) ≤ CKp‖v(0)‖L2 .

But v(0) → 0 as m,n→∞ (in H1, so in L2).u(t) can also be seen as weak limit in H1, and in M t

0, of un(t) (for tfixed) : therefore we get the bounds in H1 and M t

0 for u(t). Finally, we takethe H−1-limit in the integral formulation :

un(t) = S(t− t0)un0 +

∫ t

t0

S(t− τ)(un)px(τ)dτ.

There is no problem for S(t− t0)un0 , and for the second term :∫ t

t0

S(t−τ) ((un)px(τ)− (um)p

x) dτ =(∫ t

t0

S(t− τ) ((un)p(τ)− (um)p) dτ)

x

.

(un)p(τ)− (um)p → 0 in C0([t0, t0 + T ], L2) : so we can take the limit in L2

in the integral, and then in the H−1 sense for the whole term.u(t) is a solution to (1) in the integral sense, u(t0) = u0, and u ∈

L∞([t0, t0 + T ],H1) : so u is the unique C0([t0, t0 + T ],H1) solution. Byweak limit :

∀t ∈ [t0, t0 + T [, M t0(u(t)) ≤ 2K, and lim sup

t↓t0M t

0(u(t)) ≤M t00 (u0).

References

[1] Michael Christ and Michael I. Weinstein, Dispersion of small amplitudes solutionsof the generalized Korteweg-de Vries equation, Journal of Functional Analysis 100(1991), 87–109.

[2] Wiktor Eckhaus and Peter Schuur, The emergence of solutions of the Korteweg-de Vries equation form arbitrary initial conditions, Mathematical Methods in theApplied Sciences 5 (1983), 97–116.

[3] Jean Ginibre and Giorgio Velo, Long range scattering for some Schrodinger relatednon-linear systems, Preprint (2004).

26

[4] Nakao Hayashi and Naumkin Pavel I., Large time asymptotic of solutions to thegeneralized Korteweg-De Vries equation, Journal of Functional Analysis 159 (1998),110–136.

[5] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering resultfor the generalized Korteweg-De Vries equation via contraction principle, Communi-cation in Pure and Applied Mathematics 46 (1993), 527–620.

[6] D. J. Korteweg and G. de Vries, On the change of form of long waves advancingin a rectangular canal, and on a new type of long stationnary waves, PhilosopicalMagazine 539 (1895), 422–443.

[7] George L. Jr. Lamb, Element of soliton theory, John Wiley & Sons, New York, 1980.

[8] Yvan Martel, Asymptotic n-soliton-like solutions of the subcritical and critical gen-eralized Korteweg-de Vries equations, American Journal of Mathematics (2005).

[9] Yvan Martel and Frank Merle, Asympotic stability of solitons for subcritical gener-alized KdV equations, Archives for Rational Mechanical and Analysis 157 (2000),219–254.

[10] , Blow up in finite time and dynamics of blow up solutions for the L2-criticalgeneralized KdV equations, Journal of the American Mathematical Society 15 (2002),617–664.

[11] Yvan Martel, Frank Merle, and Tai-Peng Tsai, Stability and asymptotic stability in theenergy space of the sum of n solitons for subcritical gKdV equations, Communicationsin Mathematical Physics 231 (2002), 347–373.

[12] Frank Merle, Construction of solutions with exactly k blow-up points for theSchrodinger equation with critical nonlinearity, Communications in MathematicalPhysics 129 (1990), no. 2, 223–240.

[13] , Existence of blow-up solutions in the energy space for the critical general-ized Korteweg-de Vries equation, Journal of the American Mathematical Society 14(2001), 555–578.

[14] Robert M. Miura, The Korteweg-de Vries equation : a survey of results, SIAM Review18 (1976), 412–459.

[15] Gustavo Ponce and Luis Vega, Nonlinear small data scattering for the generalizedKorteweg-de Vries equation, Journal of Functional Analysis 90 (1990), 445–457.

[16] Peter Schuur, Asymptotic analysis of solitons problems, Lectures Notes in Mathemat-ics 1232 (1986).

[17] Walter A. Strauss, Dispersion of low-energy waves for two conservative equations,Archives for Rational Mechanical and Analysis 55 (1974), 86–92.

Raphael CoteDepartement de Mathematiques et ApplicationsEcole normale superieure45, rue d’Ulm, 75230 Paris Cedex 05, Franceraphael.cote@ens.fr

27