Well-posedness for the generalized Benjamin-Ono equations with arbitrary large initial data in the critical space

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Well-posedness for the generalized Benjamin-Ono equations with arbitrary large initial data in the critical

space

Stéphane Vento

To cite this version:

Stéphane Vento. Well-posedness for the generalized Benjamin-Ono equations with arbitrary large initial data in the critical space. 2008. �hal-00296637�

Well-posedness for the generalized Benjamin-Ono equations with arbitrary

large initial data in the critical space

St´ephane Vento, Universit´e Paris-Est,

Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, 5 bd. Descartes, Cit´e Descartes, Champs-Sur-Marne,

77454 Marne-La-Vall´ee Cedex 2, France E-mail: stephane.vento@univ-paris-est.fr

Abstract. We prove that the generalized Benjamin-Ono equations ∂_tu+ H∂_x²u±u^k∂_xu = 0, k ≥ 4 are locally well-posed in the scaling invariant spaces ˙H^sk(R) where s_k = 1/2−1/k. Our results also hold in the non- homogeneous spaces H^sk(R). In the case k = 3, local well-posedness is obtained inH^s(R),s >1/3.

Keywords: NLS-like equations, Cauchy problem AMS Classification: 35Q55, 35B30, 76B03, 76B55

1 Introduction

In this paper we pursue our study of the Cauchy problem for the generalized Benjamin-Ono equations

∂_tu+H∂_x²u±u^k∂_xu= 0, x, t∈R,

u(x, t = 0) =u₀(x), x∈R, (gBO) with k an integer ≥ 3 and with H the Hilbert transform defined via the Fourier transform by

Hf =F⁻¹(−isgn(ξ) ˆf(ξ)), f ∈ S^′(R). (1.1) The Hilbert transform is a real operator, and consequently we look for real-valued solutions. In view of (1.1), we see that H is nothing but −i

on positive frequencies and +ion negative ones. A very close equation to (gBO) is then the derivative nonlinear Schr¨odinger equation

∂_tu−i∂_x²u±u^k∂_xu= 0. (1.2) for which all our results remain true. Furthermore, (gBO) and (1.2) enjoy the same linear estimates, see Section 3.

A remarkable feature of (gBO) is the following scaling invariance: if u(t, x) is a solution of the equation on [−T,+T], then for any λ > 0, u_λ(t, x) = λ^1/ku(λ²t, λx) also solves (gBO) on [−λ⁻²T,+λ⁻²T] with ini- tial datau_λ(0, x) and moreover

kuλ(·,0)kH˙^s =λ^s+k1−12ku(·,0)kH˙^s.

Hence the ˙H^s(R) norm is invariant if and only ifs=s_k= 1/2−1/k and we may expect well-posedness in ˙H^sk(R).

When k= 1, (gBO) is the ordinary Benjamin-Ono equation derived by Benjamin [1] and later by Ono [15] as a model for one-dimensional waves in deep water. The Cauchy problem for the Benjamin-Ono equation has been extensively studied these last years, see [17, 16, 6]. In [18], Tao introduced a gauge transformation (a kind of Cole-Hopf transformation) which amelio- rate the derivative nonlinearity, and get the well-posedness of this equation inH^s(R),s≥1. Recently, combining a gauge transformation together with a Bourgain’s method, Ionescu and Kenig [5] shown that one could go down to L²(R), which seems to be the critical space for the Benjamin-Ono equa- tion. Note also that Burq and Planchon [4] have obtained well-posedness in H^s(R), s >1/4 by similar methods. It is worth noticing that all these results have been obtained by compactness methods. On the other hand, Molinet, Saut and Tzvetkov [14] proved that, for all s ∈ R, the flow map u₀ 7→uis not of classC²from H^s(R) toH^s(R). Furthermore, building suit- able families of approximate solutions, Koch and Tzvetkov proved in [11]

that the flow map is actually not even uniformly continuous on bounded sets ofH^s(R), s >0. This explains why a Picard iteration scheme fails to solve the Benjamin-Ono equation in Sobolev spaces.

In the case of the modified Benjamin-Ono equation (k= 2), Kenig and Takaoka [10] have recently obtained the global well-posedness in the energy space H^1/2(R). This have been proved thanks to a localized gauge trans- formation combined with a space-time L² estimate of the solution. It is

important to note that this result is far from that given by the scaling index s₂ = 0. However, it is known to be sharp since the solution map u₀ 7→ u is notC³ inH^s(R) as soon as s <1/2 (see [13]).

In the case k= 3, the local well-posedness is known in H^s(R), s >1/3 for small initial data [13] but only in H^s(R), s >3/4 for large initial data.

In [19], we showed that (gBO) isC⁴-ill-posed inH^s(R),s <1/3, in the sense that the flow-mapu₀ 7→ufails to beC⁴. We prove here that well-posedness occurs inH^s(R),s >1/3, and without smallness assumption on the initial data.

Concerning the case k ≥4, global well-posedness inH^s(R), s > s_k was derived for small initial data by Molinet and Ribaud in [13]. Later, by means of a gauge transformation, the same authors [12] removed the size restriction on the data and showed well-posedness in H^1/2(R), whatever the value of k. By a refinement of their method, we reached in [19] the well-posedness inH^s(R), s > s_k, but for high nonlinearities only (k≥12 in fact). On the other hand, in the particular casek= 4, Burq and Planchon [3] proved the local well-posedness in the critical space ˙H^1/4(R). Inspired by their works, we extend in this paper the well-posedness to ˙H^sk(R) for anyk≥4, and our method is flexible enough to get the result in the non-homogeneous space H^sk(R). A standard fixed point argument allows us to construct a unique solution in a subspace of ˙H^sk(R) with a continuous flow-mapu₀ 7→u. Recall that Biagioni and Linares [2] proved using solitary waves, that this map cannot be uniformly continuous in ˙H^sk(R). In the surcritical case s < s_k, we also know that the solution-map (if it exists) fails to beC^k+1 inH^s(R), see [13].

2 Notations and main results

2.1 Notations

For A and B two positive numbers, we write A .B if it exists c >0 such that A ≤cB. Similarly define A& B, A ∼B ifA ≥ cB and A .B .A respectively. When the constantcis large enough, we writeA≪B. For any f ∈ S^′(R), we useFf or ˆf to denote its Fourier transform. For 1≤p≤ ∞, L^p is the standard Lebesgue space and its space-time versions L^p_xL^q_T and L^q_TL^px (T >0) are endowed with the norms

kfk_L^p_xL^q

T =

kfk_L^q_t([−T;T])

L^px(R) and kfk_L^q

TL^px =

kfk_L^p_x(R)

L^q_t([−T;T]).

The pseudo-differential operator D^α_x is defined by its Fourier symbol |ξ|^α. We will denote by P₊ and P₋ the projection on respectively the positive and the negative spatial Fourier modes. Thus one has

iH=P₊−P₋.

Let η ∈ C₀^∞(R), η ≥ 0, supp η ⊂ {1/2 ≤ |ξ| ≤ 2} with P_∞

−∞η(2^−jξ) = 1 for ξ 6= 0. We set p(ξ) = P

j≤−3η(2^−jξ) and consider, for all j ∈ Z, the operator Q_j defined by

Q_j(f) =F⁻¹(η(2^−jξ) ˆf(ξ)).

We adopt the following summation convention. Any summation of the form r . j, r ≫ j,... is a sum over the r ∈ Z such that 2^r . 2^j..., thus for instance P

r.j =P

r:2^r.2^j. We define then the operators Q_.j =P

r.jQr, Q_≪j =P

r≪jQ_r, etc. For 1 ≤p, q, r ≤ ∞ and s ∈R, let ˙Bp^s,r(L^q_T) be the homogeneous Besov space equipped with the norm

kfkB˙_p^s,r(L^q_T)= X

j∈Z

[2^jskQjfk_L^p_xL^q

T]^r1/r

.

Finally for s ∈ R and θ ∈ [0,1], we define the solution space ˙S^s,θ (where lives our solutionu) and the nonlinear space ˙N^s,θ (where lives the nonlinear termu^k∂_xu) by

S˙^s,θ = ˙B^s+4^{3θ−14 ,2} 1−θ

(L_T^2θ), N˙^s,θ = ˙B^s+4^{1−3θ4 ,2} 3+θ

(L

2 2−θ

T ).

2.2 Main results

We first state our well-posedness results in the casek≥4.

Theorem 2.1. Let k ≥4 and u₀ ∈ H˙^sk(R). There exists T = T(u₀) > 0 and a unique solutionu of (gBO) such thatu∈Z˙_T with

Z˙_T =C([−T,+T],H˙^sk(R))∩X˙^sk∩L^k_xL^∞_T .

Moreover, the flow map u₀7→ u is locally Lipschitz from H˙^sk(R) toZ˙_T. In the non-homogeneous case, one has the following result.

Theorem 2.2. Let k ≥ 4 and u₀ ∈ H^s(R), s ≥ s_k. There exists T = T(u₀)>0 and a unique solution u of (gBO) such that u∈Z_T with

Z_T =C([−T,+T], H^s(R))∩X^s∩L^k_xL^∞_T .

Moreover, the flow map u₀7→ u is locally Lipschitz from H^s(R) toZ_T. Remark 2.1. We only obtain the Lipschitz continuity of the map u0 7→u in Theorems 2.1 and 2.2 inH˙^sk(R) (resp. H^s(R)). As noticed in the intro- duction, the solution map given by Theorem 2.1 is not uniformly continuous from H˙^sk(R) to C([−T, T],H˙^sk(R)). Moreover, when s < s_k, the flow map in Theorem 2.2 is no longer of class C^k+1 in H^s(R). It is not clear wether the map given by Theorems 2.1 and 2.2 is C^k+1 or not.

Remark 2.2. The spaces X˙^sk and X^s will be defined in Section 3 and are directly related with the linear estimates for the linear Benjamin-Ono equation.

The main tools to prove Theorems 2.1 and 2.2 are the sharp Kato smoothing effect and the maximal in time inequality for the free solution V(t)u₀ where V(t) = e^itH∂x2. Recall that for regular solutions, (gBO) is equivalent to its integral formulation

u(t) =V(t)u₀∓ Z _t

0

V(t−t^′)(u^k(t^′)∂_xu(t^′))dt^′. (2.1) It is worth noticing that (gBO) provides a perfect balance between the derivative nonlinear term on one hand, and the available linear estimates on the other hand. Heuristically, one may use (2.1) to write

kD_x^sk+1/2uk_L^∞_{x L}²

T +kuk_Lk

xL^∞_T .ku0kH˙^sk +kD_x^sk−1/2∂x(u^k+1)k_L1 xL²_T

.ku₀kH˙^sk +kD_x^sk+1/2uk_L^∞

x L²_Tkuk^k_Lk xL^∞_T

and perform a fixed point procedure. Unfortunately, such an argument fails for several reasons:

• First, it is not clear wether the second inequality holds true or not. In- deed, we used the fractional Leibniz rule (see the Appendix in [9], [12]) at the end pointsL^p,p= 1,∞. However, this inequality becomes true if one works in the associated Besov spaces ˙B∞^sk+1/2,2(L²_T)∩B˙^0,2_k (L^∞_T ) and provides sharp well-posedness for small initial data, see [13].

• The termkV(t)u₀k_L^k_xL^∞

T will be small only ifku0kH˙^sk is small as well, even for smallT. Nevertheless, as noticed in [3], if we consider instead the difference V(t)u₀−u₀, then its L^k_xL^∞_T -norm is small provided we restrict ourselves to a small interval [−T, T] (see Lemma 3.5).

• We also need to get a better share of the derivative in the nonlinear term. By a standard paraproduct decomposition, we see that the worst contribution in∂_xu^k+1 is given byπ(u, u) where

π(f, g) =X

j

∂xQj((Q≪jf)^kQ∼jg).

The main idea is then to inject this term (or more preciselyπ(V(t)u₀, u)) in the linear part of the equation to get the variable-coefficient Schr¨odinger equation

∂_tu+H∂_x²u+π(V(t)u₀, u) =f (2.2) where f will be a well-behaved term. Linear estimates for equation (2.2) are obtained by the localized gauge transform

w_j =e^{i2R−∞x (Q≪ju0)k}P₊Q_ju, j∈Z.

Now we turn to the casek= 3. By similar considerations, we obtain the following result.

Theorem 2.3. Let k = 3 and u₀ ∈ H^s(R), s > 1/3. There exists T = T(u₀)>0 and a unique solutionu of (gBO) such that u∈Z_T with

Z_T =C([−T,+T], H^s(R))∩X^s∩L³_xL^∞_T .

Moreover, the flow map u₀7→ u is locally Lipschitz from H^s(R) toZ_T. This paper is organized as follows. In Section 3, we recall some sharp estimates related with the linear operator V(t), and we derive linear esti- mates for equation (2.2). Section 4 is devoted to the casek≥4. Finally, we prove Theorem 2.3 in Section 5.

3 Linear estimates

3.1 Estimates for the linear BO equation

This section deals with the well-known linear estimates for the Benjamin- Ono equation. Note that all results stated here hold as well for the Schr¨odinger operator S(t) =e^it∂x2.

The following lemma summarizes the main estimates related to the group V(t). See for instance [7, 8] for the proof.

Lemma 3.1. Let ϕ∈ S(R), then kV(t)ϕk_L^∞

TL²_x . kϕk_L², (3.1) kD^1/2_x V(t)ϕk_L^∞

x L²_T . kϕk_L2, (3.2) kD_x^−1/4V(t)ϕk_L⁴_xL^∞_T . kϕk_L². (3.3) Moreover, if T ≤1 and j≥0,

kQ≤0V(t)ϕk_L²_xL^∞

T .kQ≤0ϕk_L² (3.4)

2^−j/2kQjV(t)ϕk_L²_xL^∞_T .kQjϕk_L² (3.5) Definition 3.1. A triplet (α, p, q) ∈R×[2,∞]² is said to be 1-admissible if (α, p, q) = (1/2,∞,2) or

4≤p <∞, 2< q≤ ∞, 2 p +1

q ≤ 1

2, α= 1 p+ 2

q −1

2. (3.6) By Sobolev embedding and interpolation between estimates (3.2) and (3.3) we obtain the following result.

Proposition 3.1 ([12]). If(α, p, q)is 1-admissible, then for allϕin S(R), kD_x^αV(t)ϕk_L^p_xL^q

T .kϕk_L². (3.7)

Now we define our resolution spaces.

Definition 3.2. Let k ≥4 and s ∈ R be fixed. For 0 < ε ≪ 1, we define the spaces X˙^s= ˙S^s,ε∩S˙^s,1 endowed with the norm

kukX˙^s =kukS˙^s,ε+kukS˙^s,1.

At this stage it is important to remark that ˙X^s does not contain any L^∞_T component. As a consequence, for each u∈X˙^s and η >0 fixed, we can choose T =T(u) such thatkukX˙^s < η.

In the case k = 3, we shall require the following result which is not covered by Proposition 3.1.

Lemma 3.2 ([12]). Let 0< T ≤1 and s >1/3. Then it holds that

kV(t)ϕk_L³_xL^∞_T .kϕkH^s, ∀ϕ∈ S(R). (3.8)

We next state the L^pxL^q_T and L^q_TL^px estimates for the linear operator f 7→Rt

0V(t−t^′)f(t^′)dt^′.

Lemma 3.3 ([12]). Letα∈R, and2< p, q ≤ ∞such that for allϕ∈ S(R), kD_x^αV(t)ϕk_L^p_xL^q

T .kϕk_L². Then for all f ∈ S(R²),

D^1/2_x

Z t 0

V(t−t^′)f(t^′)dt^′

L^∞_TL²_x .kfk_L1

xL²_T, (3.9)

D_x^α+1/2 Z t

0

V(t−t^′)f(t^′)dt^′

L^pxL^q_T .kfk_L1

xL²_T. (3.10) Similarly, if

kD_x^αV(t)ϕk_L^p_xL^q

T .kϕk_H^s for anyϕ∈ S(R), then

D^α+1/2_x Z _t

0

V(t−t^′)f(t^′)dt^′

L^pxL^q_T .khD_xi^sfk_L1

xL²_T. (3.11) We shall need the following Besov version of Lemma 3.3.

Lemma 3.4. Let k≥4. For all f ∈ S(R)²,

Z t 0

V(t−t^′)f(t^′)dt^′

L^k_xL^∞_T .kfkN˙^sk,1.

Proof. Note that the triplets (1/2,∞,2) and (−sk, k,∞) are both 1-admissible.

In particular we deduce

Z _T

−T

D_x^1/2V(−t^′)h(t^′)dt^′

L² .khk_L1

xL²_T, ∀h∈ S(R²),

which is the dual estimate of (3.7) for (α, p, q) = (1/2,∞,2). Since L² = B˙₂^0,2, we infer

Z T

−T

D_x^1/2V(−t^′)h(t^′)dt^′

L² .khk_B˙^0,2

1 (L²_T), ∀h∈ S(R²).

The usualT T^∗ argument provides

Z _T

−T

V(t−t^′)f(t^′)dt^′

L^k_xL^∞_T .kfk_B˙−1/k,2 1 (L²_T).

We can conclude with the Christ-Kiselev lemma for reversed norms (Theo- rem B in [3]).

3.2 Linear estimates for equation (2.2)

Here and hereafter we take k ≥4, the special case k= 3 will be discussed in Section 5.

Next lemma will be crucial in the proof of our main results.

Lemma 3.5. Let k ≥ 4 and u₀ ∈ H˙^sk. For any η > 0, there exists T = T(u0) such that

kV(t)u₀−u₀k_L^k_xL^∞

T < η.

Proof. LetN >0 to be chosen later. One has

kV(t)u₀−u₀k_Lk

xL^∞_T . X

|j|<N

kQ_j(V(t)u₀−u₀)k_Lk xL^∞_T +



 X

|j|>N

kQ_ju₀k²_H˙sk





1/2

.

Note thatv=V(t)u₀−u₀ solves the equation

∂_tv+H∂_x²v=−H∂_x²u₀ with zero initial data. ThusV(t)u₀−u₀=R_t

0V(t−t^′)H∂_x²u₀dt^′ and X

|j|<N

kQj(V(t)u₀−u₀)k_L^k_xL^∞

T . X

|j|<N

2^2j

Z _t

0

V(t^′)Q_ju₀dt^′ L^k_xL^∞_T

.T X

|j|<N

2^2jkV(t)Q_ju₀k_Lk xL^∞_T

.T2^2Nku0kH˙^sk.

It suffices now to choose sufficiently largeN and then T small enough.

Let us turn back to the nonlinear (gBO) equation. The sign of the nonlinearity is irrelevant in the study of the local problem, and we choose for convenience the plus sign.

Using standard paraproduct rearrangements, we can rewrite the nonlin-

ear term in (gBO) as follows:

∂_xQ_j(u^k+1) =∂_xQ_j( lim

r→∞(Q_<ru)^k+1)

=∂_xQ_jX^∞

−∞

(Q_<r+1u)^k+1−(Q_<ru)^k+1

=∂xQj

X^∞

−∞

Qru(Q_.ru)^k

=∂_xQ_j X

r∼j

Q_ru(Q_≪ru)^k

+∂_xQ_j X

r&j

(Q_∼ru)²(Q_.ru)^k−1

=∂_xQ_j((Q≪ju)^kQ∼ju)−g_j. We set

π(f, g) =X

j

∂_xQ_j((Q_≪jf)^kQ_∼jg) so that (gBO) reads

∂_tu+H∂_x²u+π(u, u) =g(t, x) with

g=X

j

gj.

Setting

f =π(u_L, u)−π(u, u) +g

where u_L = V(t)u₀ is the solution of the free BO equation, we see that (gBO) is equivalent to

∂_tu+H∂_x²u+π(u_L, u) =f(t, x). (3.12) We intend to solve (gBO) by a fixed point procedure on the Duhamel for- mulation of (3.12):

u(t) =U(t)u₀− Z _t

0

U(t−t^′)f(t^′)dt^′, whereU(t)ϕ is solution to

∂_tu+H∂_x²u+π(V(t)u₀, u) = 0, u(0) =ϕ.

It is worth noticing thatU(t) depends on the datau0.

Setting u_j =Q_ju andf_j =Q_jf, we get from (3.12) that

∂_tu_j+H∂_x²u_j+∂_x((u0,≪j)^ku_j) =∂_x[((u0,≪j)^k−(uL,≪j)^k)u_j]−∂_x[Q_j,(uL,≪j)^k]u∼j+f_j and we will denote by R_j the right-hand side. Now take the positive fre-

quencies and setv_j =P₊u_j:

i∂tvj +∂_x²vj +i∂x((u0,≪j)^kvj) =iP+Rj. Withb_≪j = ¹₂(u_0,≪j)^k, we obtain

i∂_tv_j+ (∂_x+ib_≪j)²v_j =g_j (3.13) with

g_j =−i∂xb_≪j.v_j−b²_≪jv_j+iP₊R_j. (3.14) Lemma 3.6. Letvj be a solution to (3.13) with initial datav0,j∈H˙^sk∩H˙^s. Then there exists C=C(u₀) such that

kvjkX˙^s ≤Ckv0,jkH˙^s +CkgjkN˙^s,1. Proof. We definewj by

w_j =e^iRxb≪jv_j. Then we easily check thatw_j solves

i∂_tw_j+∂_x²w_j =e^iRxb≪jg_j.

From the well-known linear estimates on the Schr¨odinger equation (Lemmas 3.1-3.3) we infer

k∂xw_jk_L^∞_xL²

T .ke^iRxb≪jv_0,jkH˙^1/2 +kgjk_L1 xL²_T. Since∂_xw_j =e^iRxb≪j(∂_xv_j+b_≪jv_j), we have

k∂xv_jk_L^∞_xL²

T .k∂xw_jk_L^∞_{x L}²

T +kb≪jv_jk_L^∞_{x L}²

T

.k∂xw_jk_L^∞_{x L}²

T + 2^−jkb≪jkL^∞k∂xv_jk_L^∞_{x L}²

T.

On the other hand, we can make 2^−jk(u0,≪j)^kk_L^∞ as small as desired by choosing the implicit constantJ =J(u₀) in u_0,≪j large enough:

2^−jk(u0,<j−J)^kk_L^∞ .2^−j2^j−Jku₀k^k_Lk .c(u₀)2^−J ≪1.