Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity

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Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity

Stéphane Vento

To cite this version:

Stéphane Vento. Sharp well-posedness results for the generalized Benjamin-Ono equation with high

nonlinearity. 2007. �hal-00137752�

hal-00137752, version 1 - 21 Mar 2007

BENJAMIN-ONO EQUATION WITH HIGH NONLINEARITY

ST´EPHANE VENTO

Abstract. We establish the local well-posedness of the generalized Benjamin- Ono equation∂tu+H∂_x²u±u^k∂xu= 0 inH^s(R),s >1/2−1/kfork≥12 and without smallness assumption on the initial data. The conditions >1/2−1/k is known to be sharp since the solution mapu0 7→uis not of classC^k+1 on H^s(R) fors <1/2−1/k. On the other hand, in the particular case of the cubic Benjamin-Ono equation, we prove the ill-posedness inH^s(R),s <1/3.

1. Introduction and statement of the results

1.1. Introduction. Our purpose in this paper is to study the initial value problem for the generalized Benjamin-Ono equation

(gBO)

∂

t

u + H∂

_x²

u ± u

^k

∂

x

u = 0, x, t ∈ R , u(x, t = 0) = u

0

(x), x ∈ R ,

where k ∈ N \ {0}, H is the Hilbert transform defined by Hf (x) = 1

π pv 1 x ∗ u

(x) = F

⁻¹

− i sgn(ξ) ˆ f (ξ) (x)

and with initial data u

0

belonging to the Sobolev space H

^s

( R ) = (1−∂

_x²

)

^−s/2

L

²

( R ).

The case k = 1 was deduced by T.B. Benjamin [1] and later by H. Ono [14] as a model in internal wave theory. The Cauchy problem for the Benjamin-Ono equation has been extensively studied. It has been proved in [16] that (BO) is globally well- posed (i.e. global existence, uniqueness and persistence of regularity of the solution) in H

^s

( R ) for s ≥ 3, and then for s ≥ 3/2 in [15] and [5]. Recently, T. Tao [17] proved the well-posedness of this equation for s ≥ 1 by using a gauge transformation. More recently, combining a gauge transformation with a Bourgain’s method, A.D. Ionescu and C.E. Kenig [4] shown that one could go down to L

²

( R ), and this seems to be, in some sense, optimal. It is worth noticing that all these results have been obtained by compactness methods. On the other hand, L. Molinet, J.-C. Saut and N. Tzvetkov [10] proved that, for all s ∈ R , the flow map u

0

7→ u is not of class C

²

from H

^s

( R ) to H

^s

( R ). Furthermore, building suitable families of approximate solutions, H. Koch and N. Tzvetkov proved in [9] that the flow map is not even uniformly continuous on bounded sets of H

^s

( R ), s > 0. As an important consequence of this, since a Picard iteration scheme would imply smooth dependance upon the initial data, one see that such a scheme cannot be used to get solutions in any space continuously embedded in C([0, T ], H

^s

( R )).

Key words and phrases. NLS-like equations, Cauchy problem.

1

For higher nonlinearities, that is for k ≥ 2, the picture is a little bit different. It turns out that one can get local well-posedness results throught a Picard iteration scheme but for small initial data only. This seems mainly due to the fact that the smoothing properties of the linear group V (·) associated to the linear (BO) equation is just sufficient to recover the lost derivative in the nonlinear term, but does not allow to get the required contraction factors. On the other hand, for large initial data, one can prove local well-posedness by compactness methods together with a gauge transformation. Unfortunately, this usually requires more smoothness on the initial data. We summurize now the known results about the Cauchy problem for (gBO) equations when k ≥ 2.

In the case of the modified Benjamin-Ono equation (k = 2), C.E. Kenig and H.

Takaoka [8] have recently obtained the global well-posedness in the energy space H

^1/2

( R ). This have been proved thanks to a localized gauge transformation com- bined with a L

²_xT

estimate of the solution. This result is known to be sharp since the solution map u

0

7→ u is not C

³

in H

^s

( R ), s < 1/2 (see [12]).

For (gBO) with cubic nonlinearity (k = 3), the local well-posedness is known in H

^s

( R ), s > 1/3 for small initial data [12] but only in H

^s

( R ), s > 3/4, for large initial data. Moreover, the ill-posedness has been proved in H

^s

( R ), s < 1/6 [12].

In this paper, we show the ill-posedness of the cubic Benjamin-Ono equation in H

^s

( R ), s < 1/3, which turns out to be optimal according to the above results.

When k ≥ 4, by a scaling argument, one can guess the best Sobolev space in which the Cauchy problem is locally well-posed, that is, the critical indice s

c

such that (gBO) is well-posed in H

^s

( R ) for s > s

c

and ill-posed for s < s

c

. Recall that if u(x, t) is a solution of the equation then u

λ

(x, t) = λ

^1/k

u(λx, λ

²

t) (λ > 0) solves (gBO) with initial data u

λ

(x, 0) and moreover

ku

λ

(·, 0)k

H˙^s

= λ

^s+k1−12

ku(·, 0)k

H˙^s

.

Hence the ˙ H

^s

( R ) norm is invariant if and only if s = s

k

= 1/2 − 1/k and one can conjecture that s

c

= s

k

.

In the case of small initial data, this limit have been reached by L. Molinet and F. Ribaud [12]. This result is almost sharp in the sense that the flow map u

0

7→ u is not of class C

^k+1

from H

^s

( R ) to C([0, T ], H

^s

( R )) at the origin when s < s

k

, [11]. This lack of regularity is also described by H.A. Biagioni and F. Linares in [2] where they established, using solitary waves, that the flow map is not uniformly continuous in ˙ H

^sk

( R ), k ≥ 2.

For large initial data, the local well-posedness of (gBO) is only known in H

^s

( R ), s ≥ 1/2, whatever the value of k. This have been proved in [11] by using the gauge transformation

(1.1) u 7−→

^G

P

+

(e

^{−iR−∞x uk}

u),

together with compactness methods. Note also that very recently, in the particular case k = 4, N. Burq and F. Planchon [3] derived the local well-posedness of (gBO) in the homogeneous space ˙ H

^1/4

( R ).

In this paper, our aim is to improve the results obtained in [11] for large initial

data. We show that for all k ≥ 12, (gBO) is locally well-posed in H

^s

( R ), s > s

k

.

Our proofs follow those of [11] : we perform the gauge transformation w = G(u)

of a smooth solution u of (gBO) and derive suitable estimates for w. The main interest of this transformation is to obtain an equation satisfied by w where the nonlinearity u

^k

u

x

is replaced by terms of the form P

+

(u

^k

P

−

u

x

) in which one can share derivatives on u with derivatives on u

^k

. Working in the surcritical case, this allows to get a contraction factor T

^ν

in our estimates. It is worth noticing that ν = ν(s) verifies lim

s→sk

ν(s) = 0, and this explains why our method fails in the critical case s = s

k

. On the other hand, the restriction k ≥ 12 appears when we estimate the integral term

P

+

e

^{−iR−∞x uk}

u Z

x

−∞

u

^k−2

Hu

xx

(see section 3.2). This term doesn’t seem to have a ”good structure” since the bad interaction

Q

j

u Z

x

−∞

(P

j

u)

^k−2

HP

j

u

xx

forbids the share of the antiderivative R

x

−∞

with other derivatives.

1.2. Main results. Our main results read as follows.

Theorem 1. Let k ≥ 12 and u

0

∈ H

^s

( R ) with s > 1/2 − 1/k. Then there exist T = T (s, k, ku

0

k

H^s

) > 0 and a unique solution u ∈ C([0, T ]; H

^s

( R )) of (gBO) such that

kD

^s+1/2_x

uk

_L^∞_xL²

T

< ∞, (1.2)

kD

^s−1/4_x

uk

L⁴_xL^∞_T

< ∞, (1.3)

kP

0

uk

L²_xL^∞_T

< ∞.

(1.4)

Moreover, the flow map u

0

7→ u is Lipschitz on every bounded set of H

^s

( R ).

As mentioned previously, these results are in some sense almost sharp. However, the critical case s = s

k

remains open. We will only consider the most difficult case, that is the lowest values for s. More precisely we will prove Theorem 1 for s

k

< s < 1/2.

In the case k = 3, we have the following ill-posedness result.

Theorem 2. Let k = 3 and s < 1/3. There does not exist T > 0 such that the Cauchy problem (gBO) admits an unique local solution defined on the interval [0, T ] and such that the flow map u

0

7→ u is of class C

⁴

in a neighborhood of the origin from H

^s

( R ) to H

^s

( R ).

This result implies that we cannot solve (gBO) with k = 3 in H

^s

( R ), s < 1/3 by a contraction method on the Duhamel formulation. Recall that for small initial data [12], we have local well-posedness in H

^s

( R ) for s > 1/3. In view of this, we can conjecture that (gBO) is locally well-posed in H

^s

( R ), s > 1/3.

The remainder of this paper is organized as follows. In section 2, we first derive

some linear estimates on the free evolution operator associated to (gBO) and we

define our resolution space. Then we give some technical lemmas which will be

used for nonlinear estimates. In section 3 we introduce the gauge transformation

and derive the needed nonlinear estimates. The section 4 is devoted to the proof

of Theorem 1. Finally we prove our ill-posedness result in the Appendix.

The author is grateful to Francis Ribaud for several useful comments on the subject.

1.3. Notations. For two positive numbers x, y, we write x . y to mean that there exists a C > 0 which does not depend on x and y, and such that x ≤ Cy. In the sequel, this constant may depend on s and k. We also use ν = ν(s, k) to denote a positive power of T which may differ at each occurrence.

Our resolution space is constructed thanks to the space-time Lebesgue spaces L

^p_x

L

^q_T

and L

^q_T

L

^p_x

endowed for T > 0 and 1 ≤ p, q ≤ ∞ with the norm

kf k

_L^p_xL^q

T

= kf k

_L^q

T([0;T])

L^px(R)

and kf k

_L^q

TL^px

= kf k

_L^p_x(R)

L^q_T([0;T])

. When p = q we simplify the notation by writing L

^p_xT

.

The well-known operators F (or ˆ ·) and F

⁻¹

(or ˇ ·) are the Fourier operators defined by ˆ f (ξ) = R

R

e

^−ixξ

f (x)dx. The pseudo-differential operator D

^α_x

is defined by its Fourier symbol |ξ|

^α

. Let P

+

and P

−

be the Fourier projections to [0, +∞[

and ] − ∞, 0]. Thus one has

iH = P

+

− P

−

.

Let η ∈ C

₀^∞

( R ), η ≥ 0, supp η ⊂ {1/2 ≤ |ξ| ≤ 2} with P

∞

−∞

η(2

^−k

ξ) = 1 for ξ 6= 0.

We set p(ξ) = P

j≤−3

η(2

^−j

ξ) and consider, for all k ∈ Z , the operators Q

k

and P

k

respectively defined by

Q

k

(f ) = F

⁻¹

(η(2

^−k

ξ) ˆ f (ξ)) and P

k

(f ) = F

⁻¹

(p(2

^−k

ξ) ˆ f (ξ)).

Therefore we have the standard Littlewood-Paley decomposition

(1.5) f = X

j∈Z

Q

j

(f ) = P

0

(f ) + X

j≥−2

Q

j

(f ) = P

0

(f ) + ˜ P (f ).

We also need the operators P

≤k

f = X

j≤k

Q

j

f, P

≥k

f = X

j≥k

Q

j

f.

We finally introduce the operators ˜ P

+

= P

+

P ˜ and ˜ P

−

= P

−

P ˜ in order to obtain the smooth decomposition

(1.6) f = ˜ P

−

(f ) + P

0

(f ) + ˜ P

+

(f ).

2. Linear estimates and technical lemmas

2.1. Linear estimates and resolution space. Recall that (gBO) is equivalent to its integral formulation

(2.1) u(t) = V (t)u

0

∓ 1 k + 1

Z

t 0

V (t − τ)∂

x

(u

^k+1

)(τ)dτ,

where V (t) = F

⁻¹

e

^itξ|ξ|

F is the generator of the free evolution. Let us now gather the well-known estimates on the group V (·) in the following lemma.

Lemma 1. Let ϕ ∈ S( R ), then

kV (t)ϕk

L^∞_TL²_x

. kϕk

L²

, (2.2)

kD

_x^1/2

V (t)ϕk

L^∞_xL²_T

. kϕk

L²

, (2.3)

kD

_x^−1/4

V (t)ϕk

_L⁴_xL^∞_T

. kϕk

_L²

.

(2.4)

Moreover, for 0 < T < 1, we have

kP

0

V (t)ϕk

L²_xL^∞_T

. kP

0

ϕk

L²

. (2.5)

The estimate (2.2) is straightforward whereas the proof of the Kato smoothing effect (2.3) and the maximal in time inequality (2.4) can be found in [6]. Estimate (2.5) has been proved in [7].

These estimates motivate the definition of our resolution space.

Definition 1. For s

k

< s < 1/2, we define the space X

_T^s

= {u ∈ S

^′

( R

²

), kuk

X_T^s

<

∞} where 0 < T < 1 and

(2.6) kuk

X_T^s

= kuk

L^∞_TH^s_x

+ kD

_x^s+1/2

uk

_L^∞_xL²_T

+ kD

^s−1/4_x

uk

L⁴_xL^∞_T

+ kP

0

uk

L²_xL^∞_T

. Thus lemma 1 implies immediately that for all ϕ ∈ S( R ) and 0 < T < 1,

(2.7) kV (t)ϕk

X_T^s

. kϕk

H^s

.

We now give some families of norms which are controlled by the X

_T^s

norm. This will be usefull to derive some nonlinear estimates in the sequel.

Definition 2. A triplet (α, p, q) ∈ R ×[2, ∞]

²

is said to be 1-admissible if (α, p, q) = (1/2, ∞, 2) or

(2.8) 4 ≤ p < ∞, 2 < q ≤ ∞, 2 p + 1

q ≤ 1

2 , α = 1 p + 2

q − 1 2 . Proposition 1. If (α − s, p, q) is 1-admissible, then for all u in X

_T^s

, (2.9) kD

_x^α

uk

L^pxL^q_T

. kuk

X_T^s

.

Proof : The inequality

(2.10) kD

^s+1/2_x

uk

L^∞_xL²_T

. kuk

X_T^s

yields the result when (α, p, q) = (1/2, ∞, 2). Assume now (α, p, q) 6= (1/2, ∞, 2).

Let r ∈ [4; p]. Then according to Sobolev embedding theorem, kD

^s+1/r−1/2_x

uk

L^r_xL^∞_T

. kD

_x^s−1/4

uk

L⁴_xL^∞_T

. kuk

X^s_T

. By interpolation with (2.10) we get for all 0 ≤ θ ≤ 1

kD

x^{s+12−(1−1r)θ}

uk

_L^r/θ

x L^2/(1−θ)_T

. kuk

X_T^s

.

We deduce (2.9) by taking θ = r/p since the assumption r ≥ 4 is equivalent to

2

p

+

¹_q

≤

¹₂

.

We list now all the norms needed for the nonlinear estimates.

Corollary 1. For u ∈ X

_T^s

, the following quantities are bounded by kuk

X_T^s

. N

1

= kuk

_L^p_xL^∞_T

, 4 ≤ p ≤ (

¹₂

− s)

⁻¹

, N

2

= T

^−ν

kuk

_L^3k

xT

, N

3

= T

^−ν

kuk

_L^k/(1−s)

x L^2k/s_T

, N

4

= T

^−ν

kuk

L^{k( 13+s)}

−1 x L^{k( 13−}

s 2)−1 T

, N

5

= T

^−ν

kuk

L^3k/4sx L^{k( 12−}

2s 3)−1 T

, N

6

= T

^−ν

kuk

L^k(1−

s 3)−1 x L^6k/s_T

, N

7

= T

^−ν

kuk

_Lk+1

x L^2k(k+1)_T

, N

8

= T

^−ν

kuk

L^{(k−1)( 56−}

s 3)−1

x L^{(k−1)( 2}

s 3−1

6)−1 T

, N

9

= kD

^1−2s+6ε_x

uk

L^{( 32−3s)}

−1

x L^1/3ε_T

, N

10

= kD

^s_x

uk

L⁶_xT

, N

11

= kD

^s+1/2−3εx

uk

L^1/εx L^{( 12−2ε)}

−1 T

, N

12

= kD

^1/2x

uk

L^3/sx L^{( 12−}

2s 3)−1 T

,

where ε, ν > 0 are small enough.

Proof :

(i) Let 4 ≤ p ≤ (

¹₂

− s)

⁻¹

. By separating low and high frequencies, kuk

L^pxL^∞_T

. kP

0

uk

L²_xL^∞_T

+ k P D ˜

^s+1/p−1/2_x

uk

L^pxL^∞_T

. kuk

X_T^s

.

Here we used that ˜ P is continuous on L

^p_x

L

^q_T

, 1 ≤ p, q ≤ ∞, and the 1- admissibility of (1/p − 1/2, p, ∞).

(ii)-(vii) We evaluate the norm of the form N = kuk

_L^p_xL^q_T

with p > 2 and q < ∞. Fix δ > 0 small enough so that α = s − s

k

− 2δ > 0 and

¹_q

− δ > 0. Then using the previous decomposition, Bernstein and H¨ older inequalities, we get

N . T

^ν

kP

0

uk

L²_xL^∞_T

+ T

^ν

k P D ˜

_x^α

uk

L^pxL^{( 1q−δ)}

−1 T

.

One complete the proof by noticing that the triplet (α − s, p, (

¹_q

− δ)

⁻¹

) is 1-admissible.

(viii) Following the same idea, we write N

8

. T

^ν

kP

0

uk

_L²_xL^∞_T

+ T

^ν

k P D ˜

k

k−1(s−sk−2^δ_k)

x

uk

L^{(k−1)( 56−}

s 3)−1

x L^{(k−1)( 2}

s 3−1

6−δ)−1 T

for an appropriate δ > 0. Once again, (

_k−1^k

(s − s

k

− 2

_k^δ

) − s, (k − 1)(

⁵₆

−

s

3

)

⁻¹

, (k − 1)(

^2s₃

−

¹₆

− δ)

⁻¹

) is 1-admissible.

(ix)-(xii) Note finally that the triplets (1 − 3s + 6ε, (

₂³

− 3s)

⁻¹

, 1/3ε), (0, 6, 6), (1/2 − 3ε, 1/ε, (

₂¹

− 2ε)

⁻¹

) and (1/2 − s, 3/s, (

¹₂

−

^2s₃

)

⁻¹

) are 1-admissible.

We now turn to the non-homogenous estimates. Let us first recall the following result found in [11].

Lemma 2. Let (α

1

, α

2

) ∈ R

²

, (ν

1

, ν

2

) ∈ R

²₊

, and 1 ≤ p

1

, q

1

, p

2

, q

2

≤ ∞ such that for all ϕ ∈ S( R ),

kD

^α_x¹

V (t)ϕk

_L^p_x¹_L^q1

T

. T

^ν1

kϕk

L²

, kD

^α_x²

V (t)ϕk

_L^p_x²_L^q2

T

. T

^ν2

kϕk

L²

. Then for all f ∈ S( R

²

),

(2.11) D

^α_x²

Z

t

0

V (t − τ)f (τ)dτ

_L∞

TL²_x

. T

^ν2

kf k

_L^p¯2 xL^q_T^¯2

,

(2.12) D

^αx^1+α2

Z

t 0

V (t − τ)f (τ)dτ

_Lp1

x L^q_T¹

. T

^ν1+ν2

kf k

_L^p_x^¯2_L^q¯2

T

provided min(p

1

, q

1

) > max(¯ p

2

, q ¯

2

) or (q

1

= ∞ and p ¯

2

, q ¯

2

< ∞), where p ¯

2

and q ¯

2

are defined by 1/ p ¯

2

= 1 − 1/p

2

and 1/ q ¯

2

= 1 − 1/q

2

. Using lemma 2 we infer the following result.

Lemma 3. For all f ∈ S( R

²

), the quantity Z

t

0

V (t − τ )f (τ)dτ

X_T^s

can be esti- mated by

(2.13) kf k

L^{( 56+}

s3)−1 x L^{( 56−}

2s3)−1 T

, kD

^s_x

f k

_L^6/5

xT

, kD

_x^s−1/2

f k

_L¹_xL²

T

, kD

^s+1/4_x

f k

_L^4/3

x L¹_T

. Moreover,

(2.14) D

_x^s+1/2

Z

t

0

V (t − τ)f (τ)dτ

L^∞_xL²_T

. kD

_x^s

f k

L¹_TL²_x

.

Proof : (2.13) follows from (2.11)-(2.12) since the triplets (s, (

¹₆

−

₃^s

)

⁻¹

, (

¹₆

+

2s

3

)

⁻¹

), (0, 6, 6), (1/2, ∞, 2) and (−1/4, 4, ∞) are 1-admissible. Inequality (2.14) is proved in [11], proposition 2.8.

2.2. Technical lemmas. In this subsection, we recall some useful lemmas which allow to share derivatives of various expressions in L

^p_x

L

^q_T

norms. One can find proofs of lemmas 4-8 in [11, 7].

Here f and g denote two elements of S( R ).

Lemma 4. If α > 0 and 1 < p, q < ∞, then kD

_x^α

(f g)k

L^pxL^q_T

. kf k

_L^p_x¹_L^q1

T

kD

_x^α

gk

_L^p_x²_L^q2

T

+ kgk

_L^p˜1

x L^q_T^˜1

kD

^α_x

f k

_L^p˜2 x L^q_T^˜2

where 1 < p

1

, p

2

, q

2

, p ˜

1

, p ˜

2

, q ˜

2

< ∞, 1 < q

1

, q ˜

1

≤ ∞, 1/p

1

+1/p

2

= 1/ p ˜

1

+1/ p ˜

2

= 1/p and 1/q

1

+ 1/q

2

= 1/ q ˜

1

+ 1/ q ˜

2

= 1/q.

Moreover the cases (p

1

, q

1

) = (∞, ∞) and (˜ p

1

, q ˜

1

) = (∞, ∞) are allowed.

Lemma 5. If 0 < α < 1 and 1 < p, q < ∞ then kD

^α_x

F(f )k

_L^p_xL^q

T

. kF

^′

(f )k

_L^p_x¹_L^q1

T

kD

_x^α

f k

_L^p_x²_L^q2

T

where 1 < p

1

, p

2

, q

2

< ∞, 1 < q

1

≤ ∞, 1/p

1

+ 1/p

2

= 1/p and 1/q

1

+ 1/q

2

= 1/q.

Lemma 6. If 0 < α < 1, 0 ≤ β < 1 − α and 1 < p, q < ∞, then kD

^β_x

([D

^α_x

, f]g)k

_L^p_xL^q_T

. kgk

_L^p_x¹_L^q1

T

kD

_x^α+β

f k

_L^p_x²_L^q2

T

where 1 < p

1

, q

1

, p

2

, q

2

< ∞, 1/p

1

+ 1/p

2

= 1/p and 1/q

1

+ 1/q

2

= 1/q.

Moreover, if β > 0 then q

1

= ∞ is allowed.

Lemma 7. If α > 0, β ≥ 0 and 1 < p, q < ∞ then kD

^α_x

P

+

(f P

−

D

_x^β

g)k

L^pxL^q_T

. kD

^γ_x¹

f k

_L^p_x¹_L^q1

T

kD

^γ_x²

gk

_L^p_x²_L^q2

T

where 1 < p

1

, q

1

, p

2

, q

2

< ∞, 1/p

1

+ 1/p

2

= 1/p, 1/q

1

+ 1/q

2

= 1/q and γ

1

≥ α, γ

1

+ γ

2

= α + β.

As in [11], we introduce the bilinear operator G defined by G(f, g) = F

⁻¹

1

2 Z

R

ξ

1

(ξ − ξ

1

)

iξ [sgn(ξ

1

) + sgn(ξ − ξ

1

)] ˆ f (ξ

1

)ˆ g(ξ − ξ

1

)dξ

1

.

We easily verify that

(2.15) G(f, f) = ∂

_x⁻¹

(f

x

Hf

x

) = ∂

_x⁻¹

(−i(P

+

f

x

)

²

+ i(P

−

f

x

)

²

) and

(2.16) G(f, g) = ∂

_x⁻¹

(−iP

+

f

x

P

+

g

x

+ iP

−

f

x

P

−

g

x

).

Lemma 8. If 0 ≤ α ≤ 1 and 1 < p, q < ∞ then kD

_x^α

G(f, g)k

L^pxL^q_T

. kD

_x^γ1

f k

_L^p_x¹_L^q1

T

kD

^γ_x²

gk

_L^p_x²_L^q2

T

where 0 ≤ γ

1

, γ

2

≤ 1, γ

1

+ γ

2

= α + 1, 1 < p

1

, q

1

, p

2

, q

2

< ∞, 1/p

1

+ 1/p

2

= 1/p and 1/q

1

+ 1/q

2

= 1/q.

We will also need the following lemma in order to treat low frequencies in the

integral term.

Lemma 9. If α ≥ 0 and 1 ≤ p, q ≤ ∞ then kP

0

(f D

^α_x

g)k

L^pxL^q_T

. kD

^γ_x¹

f k

_L^p_x¹_L^q1

T

kD

^γ_x²

gk

_L^p_x²_L^q2

T

+ kP

0

f k

_L^p˜1

x L^q_T^˜1

kD

^α_x

P

0

gk

_L^p˜2 x L^q_T^˜2

where γ

1

, γ

2

≥ 0, α = γ

1

+γ

2

, 1 < p

i

, q

i

, p ˜

i

, q ˜

i

< ∞, 1/p

1

+1/p

2

= 1/ p ˜

1

+1/ p ˜

2

= 1/p and 1/q

1

+ 1/q

2

= 1/ q ˜

1

+ 1/ q ˜

2

= 1/q.

Proof : We split the product f D

^α_x

g as follows :

(2.17) f D

^α_x

g = P

+

f P

+

D

^α_x

g + P

+

f P

−

D

_x^α

g + P

−

f P

+

D

_x^α

g + P

−

f P

−

D

^α_x

g.

It is sufficient to consider the contribution of the first two terms. For the first one, we remark that

P

0

[P

+

f P

+

(D

_x^α

g)] = P

0

[P

0

(P

+

f )P

0

(P

+

D

^α_x

g)]

and thus using the continuity of P

0

on L

^p_x

L

^q_T

,

kP

0

[P

+

f P

+

(D

_x^α

g)]k

L^pxL^q_T

. kP

0

(P

+

f )P

0

(P

+

D

^α_x

g)k

L^pxL^q_T

. kP

0

f k

_L^p˜1

x L^q_T^˜1

kD

^α_x

P

0

gk

_L^p˜2 x L^q_T^˜2

.

For the second term in (2.17) we have typically contributions of the form P

0

[P

0

(P

+

f )P

0

(P

−

D

_x^α

g)]

which are treated as above, and P

0

[ ˜ P

+

f P ˜

−

D

^α_x

g]. Using decomposition (1.5), one can write

P

0

( ˜ P

+

f P ˜

−

D

^α_x

g) = P

0

X

j∈Z

Q

j

( ˜ P

+

f )P

j

( ˜ P

−

D

^α_x

g) + X

j∈Z

P

j

( ˜ P

+

f )Q

j

( ˜ P

−

D

^α_x

g) +P

0

X

|p|≤2

X

j∈Z

Q

j

( ˜ P

+

f )Q

k−j

( ˜ P

−

D

^α_x

g) .

By a careful analysis of the various localisations, we get P

0

( ˜ P

+

f P ˜

−

D

^α_x

g) = P

0

h X

|p|.1

X

j∈Z

Q

j

( ˜ P

+

f )Q

j+p

( ˜ P

−

D

^α_x

g) i .

Here we define the operators Q

^λ_j

= 2

^−λj

D

^λ_x

Q

j

. It follows that P

0

( ˜ P

+

f P ˜

−

D

_x^α

g) = P

0

h X

|p|.1

X

j∈Z

Q

^−γ_j ¹

( ˜ P

+

D

_x^γ1

f )Q

^γ_j+p¹

( ˜ P

−

D

^γ_x²

g) i .

Thus using Cauchy-Schwarz and H¨ older inequalities, and Littlewood-Paley theo- rem,

kP

0

( ˜ P

+

f P ˜

−

D

^α_x

g)k

_L^p_xL^q_T

. X

|p|.1

h X

j∈Z

|Q

^−γ_j ¹

P ˜

+

D

^γ_x¹

f |

²

1/2

X

j∈Z

|Q

^γ_j¹

P ˜

−

D

_x^γ2

g|

²

1/2

i

L^pxL^q_T

. X

j∈Z

|Q

^−γ_j ¹

P ˜

+

D

^γ_x¹

f |

²

¹2

L^px¹L^q_T¹

X

j∈Z

|Q

^γ_j¹

P ˜

−

D

^γ_x²

g|

²

¹2

L^px²L^q_T²

. kD

^γ_x¹

f k

_L^p_x¹_L^q1

T

kD

^γ_x²

gk

_L^p_x²_L^q2

T

.

3. Nonlinear estimates

3.1. Gauge transformation. By a rescaling argument, it is sufficient to solve

(3.1) u

t

+ Hu

xx

= 2u

^k

u

x

(equation with minus sign in front of the nonlinearity could be treated in the same way). If u ∈ C([0, T ]; H

^∞

( R )) is a smooth solution, we define the gauge transfor- mation

¹

(3.2) w = P

+

(e

^−iF

u), F = F (u) = Z

x

−∞

u

^k

(y, t)dy.

The rest of this subsection is devoted to the proof of the following estimate.

Proposition 2. Let be k ≥ 12 and s

k

< s < 1/2. Let u ∈ C([0, T ]; H

^∞

( R )) be a solution of the Cauchy problem associated to (3.1) with initial data u

0

∈ H

^∞

( R ).

Then there exist ν = ν(s, k) > 0 and a positive nondecreasing polynomial function p

k

such that

kuk

X_T^s

. ku

0

k

H^s

+ T

^ν

p

k

(kuk

X_T^s

)kuk

X_T^s

+(ku

0

k

^k_H^s

+ T

^ν

p

k

(kuk

X_T^s

)kuk

X_T^s

)kD

_x^s+1/2

wk

_L^∞_xL²_T

. (3.3)

Proof : We start by splitting u according to (1.6). Then, using that |P

+

u| =

|P

−

u| (since u is real), we deduce

(3.4) kuk

X_T^s

. kP

0

uk

X_T^s

+ k P ˜

+

uk

X_T^s

.

For the low frequencies, we use the Duhamel formulation of (gBO), lemma 3 and (2.7) to get

kP

0

uk

X_T^s

. kP

0

u

0

k

H^s

+ kP

0

D

_x^1/2

u

^k+1

k

_L¹_xL²

T

. ku

0

k

H^s

+ ku

^k+1

k

_L¹_xL²

T

. ku

0

k

H^s

+ T

^ν

kuk

^k+1

L^k+1_x L^2k(k+1)_T

. ku

0

k

H^s

+ T

^ν

kuk

X^s_T

.

Now we consider the second term in the right-hand side of (3.4). As mentioned in [11], ˜ P

+

u satisfies the dispersive equation

∂

t

( ˜ P

+

u) + H∂

_x²

( ˜ P

+

u) = ˜ P

+

(e

^iF

u

^k

w

x

) − P ˜

+

(e

^iF

u

^k

∂

x

P

−

(e

^−iF

u)) + i P ˜

+

(u

^2k+1

).

Thus, according to lemma 3 k P uk ˜

X_T^s

. kV (t)u

0

k

X_T^s

+

Z

t 0

V (t − τ) ˜ P

+

(e

^iF

u

^k

w

x

)(τ)dτ

_Xs T

+kD

^s_x

u

^2k+1

k

_L^6/5

xT

+ k P ˜

+

(e

^iF

u

^k

∂

x

P

−

(e

^−iF

u))k

L^{( 56+}

s3)−1 x L^{( 56−}

2s3)−1 T

. ku

0

k

H^s

+ A + B + C.

Obviously,

B . ku

^2k

k

_L3/2 xT

kD

^s_x

uk

_L⁶

xT

. kuk

^2k_L3k

xT

kD

_x^s

uk

_L⁶

xT

. T

^ν

kuk

^2k+1_Xs T

.

1we can also setF= ¹₂Rx

−∞u^kin the non-rescaled caseut+Huxx=u^kux.

Term C has a structure P

+

(f P

−

g

x

) thus by lemma 7 C . kD

_x^1/2

(e

^iF

u

^k

)k

_L^6/5

x L³_T

kD

_x^1/2

(e

^−iF

u)k

L^3/sx L^{( 12−}

2s 3)−1 T

. C

1

C

2

. Using lemmas 4-5, we infer C

1

. kD

^1/2_x

u

^k

k

_L^6/5

x L³_T

+ kD

_x^1/2

e

^iF

k

L^{( 12−s)}

−1

x L^2/s_T

ku

^k

k

L^{( 13+s)}

−1 x L^{( 13−}

s 2)−1 T

. kuk

^k−1

L^{(k−1)( 56−}

s 3)−1

x L^{(k−1)( 2}

s 3−1

6)−1 T

kD

^1/2_x

uk

L^3/sx L^{( 12−}

2s 3)−1 T

+kD

^−1/2_x

(u

^k

e

^iF

)k

L^{( 12−s)}

−1

x L^2/s_T

kuk

^k

L^{k( 13+s)}

−1 x L^{k( 13−}

s 2)−1 T

. T

^ν

kuk

^k_XT^s

+ T

^ν

ku

^k

k

_L^1/(1−s)

x L^2/s_T

kuk

^k_XT^s

. T

^ν

kuk

^k_X^s

T

+ T

^ν

kuk

^2k_X^s

(3.5)

T

and in the same way C

2

. kD

^1/2x

uk

L^3/sx L^{( 12}

−2s 3)−1 T

+ kD

^1/2x

e

^−iF

k

L^{( 4}

s 3−1

2)−1 x L^{( 12−}

2s 3)−1 T

kuk

L^{( 12−s)}

−1

x L^∞_T

. kD

^1/2_x

uk

L^3/sx L^{( 12−}

2s 3)−1 T

+ ku

^k

k

L^3/4sx L^{( 12−}

2s 3)−1 T

kuk

L^{( 12−s)}

−1

x L^∞_T

. kuk

X^s_T

+ T

^ν

kuk

^k+1_Xs T

. (3.6)

Combining (3.5) and (3.6), C is bounded by C . T

^ν

(kuk

^k+1_Xs

T

+ kuk

^2k+1_Xs

T

+ kuk

^3k+1_Xs

T

) . T

^ν

p

k

(kuk

X^s_T

)kuk

X_T^s

. In order to study the contribution of A, we decompose e

^iF

u

^k

w

x

as

e

^iF

u

^k

w

x

= D

^1/2x

(e

^iF

u

^k

HD

^1/2x

w) − [D

^1/2x

, e

^iF

u

^k

]HD

^1/2x

w.

Therefore, according to lemma 3, and using the fact that ˜ P

+

is continuous on L

¹_x

L

²_T

, A . kD

^s_x

(e

^iF

u

^k

HD

^1/2_x

w)k

L¹_xL²_T

+ k[D

^1/2_x

, e

^iF

u

^k

]HD

_x^1/2

wk

L^{( 56+}

s 3)−1 x L^{( 56−}

2s 3)−1 T

. A

1

+ A

2

.

Note that A

1

cannot be treated by lemma 4, so we use lemma A.13 in [7]. This leads to

A

1

. kD

_x^s

(e

^iF

u

^k

)k

L⁽¹⁻

s 3)−1

x L^3/2s_T

kD

_x^1/2

(e

^−iF

u)k

L^3/sx L^{( 12−}

2s 3)−1 T

+ ku

^k

k

L¹_xL^∞_T

kD

^s+1/2_x

wk

_L^∞_xL²

T

. A

11

C

2

+ A

^k₁₂

kD

^s+1/2_x

wk

_L^∞_xL²

T

.

By lemma 4 we bound the contribution of A

11

by A

11

. kD

_x^s

u

^k

k

L⁽¹⁻

s 3)−1

x L^3/2s_T

+ kD

^s_x

e

^iF

k

_L^3/2s

x L^6/s_T

ku

^k

k

_L^1/(1−s)

x L^2/s_T

. kuk

^k−1

L^{(k−1)( 56−}

s 3)−1

x L^{(k−1)( 2}

s 3−1

6)−1 T

kD

^s_x

uk

_L⁶_xT

+ kuk

^k

L^k(1−

s 3)−1

x L^6k/s_T

kuk

^k

L^k/(1x ^−s)L^2k/s_T

. T

^ν

kuk

^k_XT^s

+ T

^ν

kuk

^2k_XT^s

.

To treat A

12

= kuk

L^k_xL^∞_T

we use the Duhamel formulation of (gBO) and lemma 2, A

12

. kV (t)u

0

k

L^k_xL^∞_T

+

Z

t 0

V (t − τ)∂

x

u

^k+1

(τ)dτ

_Lk

xL^∞_T

. ku

0

k

H^s

+ kD

^s_x^k+1/2+3ε

u

^k+1

k

L⁽¹x^−ε)−1L^{( 12+2ε)}

−1 T

and setting ε

^′

=

¹₃

(s − s

k

) − ε > 0 it follows that kD

^s_x^k+1/2+3ε

u

^k+1

k

L^(1−ε)x ⁻¹L^{( 12+2ε)}

−1 T

. kD

^s+1/2−3εx ^′

uk

L^1/εx ^′L^{( 12−2ε}

′)−1 T

ku

^k

k

L⁽¹⁻

1

3(s−sk))−1

x L

3 2(s−sk)−1 T

. T

^ν

kuk

X_T^s

kuk

^k

L^k(1−

1

3(s−sk))−1

x L^∞_T

. T

^ν

kuk

^k+1_Xs T

.

Finally, according to lemma 6 we write A

2

. kD

_x^1/2

(e

^iF

u

^k

)k

_L^6/5

x L³_T

kD

_x^1/2

(e

^−iF

u)k

L^3/sx L^{( 12−}

2s 3)−1 T

. C

1

C

2

. T

^ν

p

k

(kuk

X_T^s

)kuk

X_T^s

, witch complete the proof of (3.3).

3.2. Estimate of kD

^s+1/2x

wk

L^∞_xL²_T

. Now our aim is to estimate the term kD

^s+1/2x

wk

L^∞_xL²_T

which appears in (3.3). More precisely we will prove the following proposition.

Proposition 3. Let k ≥ 12 and s

k

< s < 1/2. For all solution u ∈ C ([0, T ]; H

^∞

( R )) of (3.1) with initial data u

0

∈ H

^∞

( R ), we have the following bound,

(3.7) kD

_x^s+1/2

wk

_L^∞_xL²_T

. p

k

(ku

0

k

H^s

)ku

0

k

H^s

+ T

^ν

p

k

(kuk

X_T^s

)kuk

X_T^s

where p

k

is a positive nondecreasing polynomial function.

Proof : Following [11], we see that w satisfies the equation w

t

+ Hw

xx

= P

+

[2e

^−iF

(−ku

^k

P

−

u

x

− iP

−

u

xx

)]

−ik(k − 1)P

+

e

^−iF

u Z

x

−∞

u

^k−2

u

x

Hu

x

. (3.8)

Thus using the Duhamel formulation of (3.8) and lemma 3 we infer kD

^s+1/2_x

wk

L^∞_xL²_T

. kD

^s+1/2_x

V (t)w(0)k

L^∞_xL²_T

+kP

+

[2e

^−iF

(−ku

^k

P

−

u

x

− iP

−

u

xx

)]k

L^{( 56+}

s 3)−1 x L^{( 56−}

2s 3)−1 T

+ D

^s+1/2_x

Z

t

0

V (t − τ)P

+

e

^−iF

u Z

x

−∞

u

^k−2

u

x

Hu

x

dτ

L^∞_xL²_T

.

(3.9)