Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation

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Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation

Luc Molinet, Didier Pilod

To cite this version:

Luc Molinet, Didier Pilod. Global well-posedness and limit behavior for a higher-order Benjamin-

Ono equation. Communications in Partial Differential Equations, Taylor & Francis, 2012, 37 (11),

pp.2050-2080. �hal-00637981�

HIGHER-ORDER BENJAMIN-ONO EQUATION

LUC MOLINET^†AND DIDIER PILOD^‡

†LMPT, Universit´e Fran¸cois Rabelais Tours, F´ed´eration Denis Poisson-CNRS, Parc Grandmont, 37200 Tours, France.

email: [email protected]

‡UFRJ, Instituto de Matem´atica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP: 21945-970, Rio de Janeiro, RJ, Brazil.

email: [email protected]

Abstract. In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation

(0.1) ∂tv−bH∂x²v−aǫ∂x³v=cv∂xv−dǫ∂x(vH∂xv+H(v∂^xv)),

is globally well-posed in the energy spaceH¹(R). Moreover, we study the limit behavior when the small positive parameterǫ tends to zero and show that, under a condition on the coefficientsa,b,candd, the solutionvǫto (0.1) converges to the corresponding solution of the Benjamin-Ono equation.

1. Introduction

Considered here is the following higher-order Benjamin-Ono equation (1.1) ∂

t

v − b H ∂

_x²

v − aǫ∂

_x³

v = cv∂

x

v − dǫ∂

x

(v H ∂

x

v + H (v∂

x

v)),

where x, t ∈ R , v is a real-valued function, a, b, c and d are positive constants, ǫ > 0 is a small positive parameter and H is the Hilbert transform, defined on the line by

(1.2) H f (x) = p.v. 1

π Z

R

f (y) x − y dy.

The equation above corresponds to a second order approximation of the unidirec- tional evolution of weakly nonlinear dispersive internal long waves at the interface of a two-layer system of fluids, the lower one being infinitely deep. It was derived by Craig, Guyenne and Kalisch (see equation (5.38) in [5]), using a Hamiltonian perturbation theory. Here, v represents the dislocation of the interface around its position of equilibrium, the coefficients a, b, c and d are respectively given by (1.3) a = h

²₁

2 ρ

²

ρ

²₁

− 1

3 s

gh

1

(ρ − ρ

1

) ρ

1

, b = ρh

²₁

2ρ

²₁

s

gρ

1

(ρ − ρ

1

) h

1

,

(1.4) c = 3 √ 2 4ρ

1

4

s gρ

1

(ρ − ρ

1

) h

1

and d =

√ 2ρh

1

2ρ

²₁

4

s gρ

1

(ρ − ρ

1

) h

1

,

2010Mathematics Subject Classification. Primary 35Q53, 35A01; Secondary 76B55.

Key words and phrases. Initial value problem, Benjamin-Ono equation, gauge transformation.

‡Partially supported by CNPq/Brazil, grant 200001/2011-6.

1

where h

1

represents the depth of the upper layer when the fluid is at rest, ρ

1

is the density of the upper fluid and ρ is the density of the lower fluid. Moreover, the system is assumed to be in a stable configuration, which is to say that ρ > ρ

1

, so that the coefficients a, b, c and d are positive.

It is worth noting that the equation obtained at the first order approximation of the above physical model is the well-known Benjamin-Ono equation

(1.5) ∂

t

v − b H ∂

_x²

v = cv∂

x

v,

and therefore equation (1.1) can be seen as an higher-order perturbation of equation (1.5). Moreover, the quantities

(1.6) M (v) =

Z

R

v

²

dx

and

(1.7) H (v) = Z

R

aǫ(∂

x

v)

²

− bv H ∂

x

v − c

3 v

³

+ dǫv

²

H ∂

x

v dx are conserved by the flow associated to (1.1).

The initial value problem (IVP) associated to the Benjamin-Ono equation on the line has been extensively studied in the recent years and has been proved to be globally well-posed in L

²

( R ) by Ionescu and Kenig in [9] (see [23] for another proof and [1, 4, 11, 12, 17, 26, 27] for former results). The IVP associated to (1.1) presents the same mathematical difficulties as for the Benjamin-Ono equation. Indeed, it has been shown in [25] that the flow map data-solution cannot be C

²

in any L

²

- based Sobolev space H

^s

( R ), s ∈ R , by using the same counter-example as for the Benjamin-Ono equation in [24]. On the other hand, the Cauchy problem associated to (1.1) was proved in [18] to be locally well-posed in H

^s

( R ), for s ≥ 2 (and also in weighted Sobolev spaces H

^k

( R ) ∩ L

²

( R ; x

²

dx), for k ∈ Z

₊

, k ≥ 2). However, there are no conserved quantities at the H

²

level and thus it is not known wether these local solutions extend globally in time or not. Therefore, as commented in [18], the question of the local well-posedness in H

¹

( R ), which would directly imply global well-posedness by using (1.6) and (1.7), arises naturally.

The first aim of this paper is to give a positive answer to this issue. The result states as follows.

Theorem 1.1. Fix ǫ > 0 and let s ≥ 1 be given. Then, for all v

0

∈ H

^s

( R ) and all T > 0, there exists a unique solution v to equation (1.1) in the space

(1.8) C([0, T ]; H

^s

( R )) ∩ L

⁴_T

W

_x^s,4

∩ L

²_x

L

^∞_T

∩ X

_ǫ,T^s−2θ,θ

, for all 0 ≤ θ ≤ 1.

satisfying

(1.9) v( · , 0) = v

0

and

(1.10) w = ∂

x

P

+hi

e

^−iF[v]

∈ X

_ǫ,T^s,12,1

, where F [v] is a spatial primitive of v defined in Section 3.

Moreover, v ∈ C

b

( R ; H

¹

( R )) and the flow map data-solution S

ǫ

: v

0

7→ v is

continuous from H

^s

( R ) into C([0, T ]; H

^s

( R )).

Note that above H

^s

( R ) denotes the space of all real-valued functions with the usual norm, while X

_ǫ,T^s,b

and X

_ǫ,T^s,b,q

are Bourgain spaces defined in Subsection 2.2.

Since it follows from the result of ill-posedness in [25] that the Cauchy problem associated to (1.1) cannot be solved by using a fixed point theorem on the integral equation, we use a compactness argument based on the smooth solutions obtained in [18]. To derive a priori estimates at the H

¹

level, we introduce a gauge trans- formation which weakens the high-low frequency interactions in the nonlinearity of (1.1), as it was done by Tao in [27] for the Benjamin-Ono equation. Note that the same kind of gauge transformation was already introduced in [18] to obtain the solutions in H

²

( R ). However, to lower the regularity till H

¹

( R ), we will need to combine this transformation with the use of Bourgain’s spaces (as it was already done in [4, 9, 23] for BO). More precisely, we need to work in a Besov version of Bourgain’s spaces (introcuded in [29] in the context of waves maps). Indeed, on one hand we have to work in Bourgain’ spaces of conormal regularity 1/2 to establish the main bilinear estimate (see Proposition 4.2 below). On the other hand, to con- trol some remaining terms appearing in the transformation, we need the full Kato smoothing effect for functions that are localized in space frequencies (see Proposi- tion 4.4). The rest of the proof follows closely the one in [23] for the Benjamin-Ono equation (see also [20]).

In the second part of this article, we investigate the limit behavior of the solutions v

ǫ

to (1.1), obtained in Theorem 1.1, as ǫ tends to zero. First, it is interesting to observe that a direct argument based on compactness methods (see for example [22]

in the case of the Benjamin-Ono-Burgers equation) does not seem to work. Indeed, the leading terms in the energy H , which is to say aǫ(∂

x

v)

²

and bv H v, have opposite signs, so that (1.6) and (1.7) do not provide a priori bounds, uniformly in ǫ, on ǫ k v

ǫ

k

²H¹

+ k v

ǫ

k

²

H¹²

. Therefore, the problem of studying the limit of v

ǫ

, as ǫ goes to zero, turns out to be far from trivial.

Nevertheless, we are able to prove the convergence of solutions of (1.1) toward a solution of the Benjamin-Ono equation in the special case where the ratio of the densities is equal to √

3.

Theorem 1.2. Assume that

^3ac_4d

= b ⇔ ρ

²

= 3ρ

²₁

. Let v

0

∈ H

¹

( R ) and for any ε > 0 denote by S

ε

(t)v

0

∈ C( R ; H

¹

( R )) the solution to (1.1) emanating from v

0

. Then for any T > 0 it holds

(1.11) k S

ε

(t)v

0

− S(t)v

0

k

L^∞(0,T;H¹(R))

−→ 0 as ε → 0

where S(t)v

0

is the solution to the Benjamin-Ono equation emanating from v

0

. In the case where

_ρ^ρ

1

= √

3, the spatial primitive chosen to perform the gauge

transformation for equation (1.1) corresponds to the one chosen for the Benjamin-

Ono equation. Then, we can show that the Cauchy problem associated to (1.1) is

uniformly in ǫ well-posed in H

¹

( R ), which will in a classical way (see for example [7])

lead to Theorem 1.2. The main difficulty here arises from the fact that the dispersive

linear terms ǫ∂

_x³

and H ∂

_x²

compete together as in the Benjamin equation (see the

introduction in [2]). Therefore, we are only allowed to use the dispersive smoothing

effects associated to (1.1) in some well behaved regions in spatial frequency and we

need to refine the bilinear estimates obtained in the proof of Theorem 1.1 in the

other regions.

It would be interesting to derive a class of higher-order equation for internal long waves from the first order one derived by Bona, Lannes and Saut in [3]. Among those equations, which would be formally equivalent to (1.1), one might find some with better behaved linear parts, which would avoid to deal with those technical difficulties.

Finally, we observe that the techniques introduced here would likely lead to similar results for the following intermediate long wave equation

(1.12) ∂

t

u − b F

_h

∂

²_x

u + (a

1

F

²_h

+ a

2

)ǫ∂

_x³

u = cu∂

x

u − dǫ∂

x

(u F

_h

∂

x

u + F

_h

(u∂

x

u)) where F

_h

is the Fourier multiplier − i coth(hξ), u is a real-valued solution, and a

1

, a

2

, b, c, d and h are positive constants, and which was also derived in [5].

Note that the same ill-posedness results as for equation (1.1) also hold for this equation (see [25]).

The paper is organized as follows: in the next section, we introduce the notations, define the functions spaces and recall some classical estimates. Sections 3 and 4 are devoted the key nonlinear estimates, which are used in Section 5 to prove Theorem 1.1. Finally, in Section 6, we prove Theorem 1.2.

2. Notations, function spaces and preliminary estimates

2.1. Notation. For any positive numbers a and b, the notation a . b means that there exists a positive constant c such that a ≤ cb. We also denote a ∼ b when a . b and b . a. Moreover, if α ∈ R , α

+

, respectively α

₋

, will denote a number slightly greater, respectively lesser, than α.

For u = u(x, t) ∈ S ( R

²

), F u = u b will denote its space-time Fourier transform, whereas F

_x

u = (u)

^∧x

, respectively F

_t

u = (u)

^∧t

, will denote its Fourier transform in space, respectively in time. For s ∈ R , we define the Bessel and Riesz potentials of order − s, J

_x^s

and D

_x^s

, by

J

_x^s

u = F

⁻_x¹

(1 + | ξ |

²

)

^s2

F

_x

u

and D

^s_x

u = F

⁻_x¹

| ξ |

^s

F

_x

u . Throughout the paper, we fix a smooth cutoff function η such that

η ∈ C

₀^∞

( R ), 0 ≤ η ≤ 1, η

_|[−1,1]

= 1 and supp(η) ⊂ [ − 2, 2].

Then if A is a positive number, P

_.A

denote the Fourier multiplier whose symbol is given by η(

_cA^·

) and P

_&A

is defined by P

_&A

= 1 − P

_.A

. For l ∈ Z

₊

, we define

φ(ξ) := η(ξ) − η(2ξ), φ

2^l

(ξ) := φ(2

^−l

ξ), and

ψ

2^l

(ξ, τ ) = φ

2^l

(τ − b | ξ | ξ + aǫξ

³

).

By convention, we also denote

φ

0

(ξ) := η(2ξ), and ψ

0

(ξ) := (ξ, τ ) = φ

0

(2(τ − b | ξ | ξ + aǫξ

³

)),

Any summations over capitalized variables such as N, L, K or M are presumed to be dyadic with N, L, K or M ≥ 0, i.e., these variables range over numbers of the form { 2

ⁿ

: n ∈ Z

₊

} ∪ { 0 } . Then, we have that

X

N

φ

N

(ξ) = 1, supp (φ

N

) ⊂ { N

2 ≤ | ξ | ≤ 2N } , N ≥ 1, and supp (φ

0

) ⊂ {| ξ | ≤ 1 } . Let us define the Littlewood-Paley multipliers by

P

N

u = F

⁻¹

x

φ

N

F

_x

u

, Q

L

u = F

⁻¹

ψ

L

F u

,

and P

_≥N

:= P

K≥N

P

K

. Moreover, we also define the operators P

hi

, P

HI

, P

lo

and P

LO

by

P

hi

= X

N≥2

P

N

, P

HI

= X

N≥2⁴

P

N

, P

lo

= 1 − P

hi

, and P

LO

= 1 − P

HI

. Let P

+

and P

₋

denote the projection on respectively the positive and the nega- tive Fourier frequencies. Then

P

_±

u = F

⁻_x¹

χ

_R±

F

_x

u ,

and we also denote P

_±hi

= P

_±

P

hi

, P

_±HI

= P

_±

P

HI

, P

_±lo

= P

_±

P

lo

, P

_±LO

= P

_±

P

LO

and P

_±N

= P

_±

P

N

. Observe that P

hi

, P

HI

, P

lo

, P

LO

, P

N

and P

_±N

are bounded (uniformly in N) operators on L

^p

( R ) for 1 ≤ p ≤ ∞ , while P

_±

are only bounded on L

^p

( R ) for 1 < p < ∞ . We also note that

H = − iP

+

+ iP

₋

.

Finally, we denote by V

ǫ

(t) = e

^{t(bH∂x2+aǫ∂x3)}

the free group associated with the linearized part of equation (1.1), which is to say,

(2.1) F

_x

V

ǫ

(t)f

(ξ) = e

^{it(b|ξ|ξ−aǫξ3)}

F

_x

f (ξ).

2.2. Function spaces. For 1 ≤ p ≤ ∞ , L

^p

( R ) is the usual Lebesgue space with the norm k·k

^Lp

, and for s ∈ R , the real-valued Sobolev spaces H

^s

( R ) and W

^s,p

( R ) denote the spaces of all real-valued functions with the usual norms

k φ k

^Hs

= k J

_x^s

φ k

L²

and k φ k

^Ws,p

= k J

_x^s

φ k

^Lp

.

If f = f (x, t) is a function defined for x ∈ R and t in the time interval [0, T ], with T > 0, if B is one of the spaces defined above, 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞ , we will define the mixed space-time spaces L

^p_T

B

x

, L

^p_t

B

x

, L

^q_x

L

^p_T

by the norms

k f k

L^p_TBx

= Z

^T

0

k f ( · , t) k

^pB

dt

¹_p

k f k

L^p_tBx

= Z

R

k f ( · , t) k

^pB

dt

_p¹

,

and

k f k

L^qxL^p_T

= Z

R

Z

^T

0

| f (x, t) |

^p

dt

_p^q

dx

!

_q¹

.

Moreover, if s ∈ R , 1 ≤ q ≤ ∞ and X denotes one of the mixed space-time spaces defined above, we define its dyadic version B

^s,q

(X ) as

k f k

^Bs,q(X)

= X

N

h N i

^sq

k P

N

f k

^qX

!

¹q

.

In the special case (s, q) = (0, 2), the space B

^s,q

(X ) will be simply denoted by X e . For s, b ∈ R , we introduce the Bourgain spaces X

_ǫ^s,b

related to the linear part of (1.1) as the completion of the Schwartz space S ( R

²

) under the norm

(2.2) k v k

Xǫ^s,b

:=

Z

R²

h τ − b | ξ | ξ + aǫξ

³

i

^2b

h ξ i

^2s

|b v(ξ, τ ) |

²

dξdτ

¹₂

,

where h x i := 1 + | x | . We will also use a dyadic version of those spaces introduced in [29] in the context of wave maps. For s, b ∈ R , 1 ≤ q ≤ ∞ , X

_ǫ^s,b,q

will denote the completion of the Schwartz space S ( R

²

) under the norm

(2.3) k v k

Xǫ^s,b,q

:= X

N

X

L

h N i

^sq

h L i

^bq

k P

N

Q

L

v k

^q_L²_x,t

²q

!

¹2

.

Moreover, we define a localized (in time) version of these spaces. Let T > 0 be a positive time and Y = X

_ǫ^s,b

or Y = X

_ǫ^s,b,q

. Then, if v : R × [0, T ] → C , we have that

k v k

^YT

:= inf {k ˜ v k

^Y

| v ˜ : R × R → C , ˜ v |

R×[0,T]

= v } .

When ǫ = 1, we will denote X

^s,b

= X

₁^s,b

, X

_T^s,b

= X

_1,T^s,b

, X

^s,b,q

= X

₁^s,b,q

and X

_T^s,b,q

= X

_1,T^s,b,q

.

Finally we list some useful properties of the Bourgain spaces defined above.

Proposition 2.1. Fix δ > 0, s ∈ R and ǫ > 0. Then it holds that

(2.4) k v k

X^s,

1 ǫ 2

. k v k

X^s,

1 2,1 ǫ

. k v k

X^s,

1 2+δ ǫ

, (2.5) k v k

L^∞_t H^s_x

. k J d

_x^s

v k

L²_ξL¹_τ

. k v k

X^s,

1 2,1 ǫ

,

and

(2.6) k f k

X^s,−

1 2+δ ǫ

. k f k

_L^1+δ′_{t Hx}^s

,

for δ

^′

> 0 satisfying 1 + δ

^′

=

₁₋¹_δ

. In other words, the injections X

ǫ^s,12+δ

֒ → X

ǫ^s,12,1

֒ → X

ǫ^s,12

, X

ǫ^s,12,1

֒ → L

^∞_t

H

_x^s

, and

L

^1+δ_t ^′

H

_x^s

֒ → X

^s,−

1 2+δ ǫ

are continuous.

2.3. Linear estimates. First, we recall some linear estimates in Bourgain’s spaces which will be needed later (see for instance [29]).

Lemma 2.2 (Homogeneous linear estimate) . Let s ∈ R and ǫ > 0. Then

(2.7) k η(t)V

ǫ

(t)φ k

X^s,

1 2,1 ǫ

. k φ k

^Hs

.

Lemma 2.3 (Non-homogeneous linear estimate). Let s ∈ R and ǫ > 0. Then, it holds that

(2.8) η(t) Z

t

0

V

ǫ

(t − t

^′

)g(t

^′

)dt

^′

X^s,

1 2,1 ǫ

. k g k

X^s,−

1 2,1 ǫ

.

Next, we derive local and global smoothing effects associated to the group { V

ǫ

(t) } , for the KdV scaling, in the context of Bourgain’s spaces. We begin with the Strichartz estimates.

Lemma 2.4. For all 0 < ǫ < 1, T > 0 and 0 ≤ θ ≤ 1, we have that (2.9) k v k

L^pθ_x,t

. k v k

Lg^pθ_x,t

. ǫ

^−θ8

k v k

Xǫ^{0, θ2+}

,

and

(2.10) k v k

L^pθ_x,T

. ǫ

^−θ8

k v k

X_ǫ,T^{0, θ2+}

, where

_p¹

θ

=

^θ₈

+

¹⁻₂^θ

.

Proof. First, we observe, arguing as in Lemma 2.1 in [18], that w is a solution to the linear equation

(2.11) ∂

t

w − aǫ∂

_x³

w ± ib∂

_x²

w = 0, if and only if

(2.12) u(x, t) = e

^{±i 2b}

3

27a2ǫ2t

e

^{∓i3aǫb x}

w(x − b

²

t 3aǫ , t) is a solution to

(2.13) ∂

t

u − aǫ∂

_x³

u = 0.

Let us denote by { W

_ǫ^±

(t) } and { U

ǫ

(t) } the groups associated to (2.11) and (2.13).

Since U

ǫ

(t) = U

1

(ǫt), we deduce from the classical Strichartz estimate for the KdV equation (cf. for example [19], chapter 4) that

(2.14) k U

ǫ

(t)φ k

L⁸_x,t

. ǫ

⁻¹⁸

k φ k

L²

. Then, it follows gathering (2.11)–(2.14) with the identity (2.15) V

ǫ

(t) = W

_ǫ⁺

(t)P

+

+ W

_ǫ⁻

(t)P

₋

, that

(2.16) k V

ǫ

(t)φ k

L⁸_x,t

. ǫ

⁻¹⁸

k φ k

L²

.

Next, we use Lemma 3.3 in [6] to rewrite estimate (2.16) in the context of Bourgain’s spaces. We get that

(2.17) k v k

L⁸_x,t

. ǫ

⁻¹⁸

k v k

_X^0,1₂⁺

.

Therefore, we deduce by using Stein’s theorem to interpolate estimate (2.17) with Plancherel’s identity k v k

L²_x,t

= k v k

Xǫ^0,0

, that

(2.18) k v k

L^pθ_x,t

. ǫ

^−θ8

k v k

Xǫ^{0, θ2+}

.

Finally, estimate (2.9) follows directly by applying estimate (2.18) to each dyadic

block of k v k

Lg^pθ_x,t

.

Next, we turn to the local Kato type smoothing effect.

Lemma 2.5. Let 0 < ǫ ≤ 1 and T > 0 and N &

¹_ǫ

. Then, it holds that (2.19) k ∂

x

P

N

v k

L^∞_xL²_t

. ǫ

⁻¹²

k P

N

v k

X^0,

1 2,1 ǫ

,

and

(2.20) k ∂

x

v k

_L^∞

xL²_T

. T

¹²

ǫ

⁻³²⁺

k v k

X^0,

1 2,1 ǫ

.

Proof. Since N &

¹_ǫ

, we obtain applying estimate (4.3) in Theorem 4.1 of [13] that k ∂

x

V

ǫ

(t)P

N

v

0

k

L^∞_xL²_t

. Z

|ξ|&¹ǫ

| ξ |

²

| 2bξ − 3aǫξ

²

| | P

N

v

0

∧

(ξ) |

²

dξ

¹2

. ǫ

⁻¹²

k P

N

v

0

k

L²_x

. (2.21)

Moreover, by applying the Fourier inverse formula, it follows that

∂

x

P

N

v(x, t) = Z

R

∂

x

V

ǫ

(t) V

ǫ

( −· )P

N

v

∧t

(x, τ)e

^itτ

dτ.

Therefore, Minkowski’s inequality, estimate (2.21), Plancherel’s identity and the Cauchy-Schwarz inequality imply that

k ∂

x

P

N

v k

L^∞_xL²_t

. Z

R

V

ǫ

( −· )P

N

v

∧

( · , τ)

_L2 ξ

dτ . X

L

h L i

¹²

φ

L

(τ) V

ǫ

( −· )P

N

v

∧

_L2 ξ,τ

, (2.22)

which leads to estimate (2.19) since V

ǫ

( −· )P

N

v

∧

(ξ, τ ) = P

N

v

∧

(ξ, τ + b | ξ | ξ − aǫξ

³

).

On the other hand, if N .

¹_ǫ

, we deduce from the Sobolev embedding H

^s

( R ) ֒ → L

^∞

( R ), whenever s >

¹₂

, that

k ∂

x

V

ǫ

(t)P

N

v

0

k

L^∞_xL²_T

. T

¹²

k ∂

x

V

ǫ

(t)P

N

v

0

k

L^∞_x,T

. T

¹²

k ∂

x

V

ǫ

(t)P

N

v

0

k

^L∞TH_x^s

. T

¹²

ǫ

⁻¹

(1 + ǫ

^−s

) k P

N

v

0

k

L²_x

. Therefore, we deduce arguing as above that

(2.23) k ∂

x

P

N

v k

L^∞_xL²_T

. T

¹²

ǫ

⁻¹

(1 + ǫ

^−s

) k P

N

v k

X^0,

1 2,1 ǫ,T

, whenever N .

¹_ǫ

.

Estimate (2.20) follows gathering estimates (2.19) and (2.23) and by squaring

and summing over N .

Finally, we derive the maximal function estimate.

Lemma 2.6. Let s >

³₄

, 0 < ǫ ≤ 1, and T > 0 be such that 0 < ǫT ≤ 1. Then, we have that

(2.24) k v k

_L^2

xL^∞_T

. ǫ

^−s

k v k

X^s,

1 2,1 ǫ,T

.

Proof. The L

²_x

-maximal function for the KdV group { U

1

(t) } derived in Theorem 2.7 of [15] implies that

(2.25) Z

R

sup

|t|≤1

| U

1

(t)u

0

(x) |

²

dx

¹2

. k u

0

k

H^s

, if s >

³₄

. Then, a scaling argument and estimate (2.25) yield

k U

ǫ

(t)u

0

k

L²_xL^∞_T

= k U

1

(ǫt)u

0

k

L²_xL^∞_T

= Z

R

sup

|s|≤ǫT

| U

1

(s)u

0

(x) |

²

dx

¹₂

. k u

0

k

²H^s

,

(2.26)

since s >

³₄

and 0 < ǫT ≤ 1.

Thus, if w and u are the solutions associated to (2.11) and (2.13) with respective initial data w

0

and u

0

, it follows from (2.12) and (2.26) that

(2.27) k w k

L²_xL^∞_T

= k u k

L²_xL^∞_T

. k u

0

k

H_x^s

. ǫ

^−s

k w

0

k

H_x^s

. Therefore, we conclude gathering (2.15) and (2.27) that

(2.28) k V

ǫ

(t)v

0

k

L²_xL^∞_T

. ǫ

^−s

k v

0

k

H_x^s

,

whenever s >

³₄

and ǫ, T satisfying 0 < ǫT ≤ 1. This implies estimate arguing as in (2.22) that

k P

N

v k

L²_xL^∞_T

. ǫ

^−s

h N i

^s

k P

N

v k

X^0,

1 2,1 ǫ,T

.

for any N ≥ 0 and s >

³₄

, which leads to (2.24) by squaring and summing over

N .

2.4. Fractional Leibniz’s rules. First we state the classical fractional Leibniz rule estimate derived by Kenig, Ponce and Vega (See Theorems A.8 and A.12 in [15]).

Proposition 2.7. Let 0 < α < 1, p, p

1

, p

2

∈ (1, + ∞ ) with

_p¹₁

+

_p¹₂

=

¹_p

and α

1

, α

2

∈ [0, α] with α = α

1

+ α

2

. Then,

(2.29) D

_x^α

(f g) − f D

_x^α

g − gD

^α_x

f

_Lp

. k D

_x^α1

g k

^Lp1

k D

^α_x²

f k

^Lp2

. Moreover, for α

1

= 0, the value p

1

= + ∞ is allowed.

The next estimate is a frequency localized version of estimate (2.29), proved in [20], in the same spirit as Lemma 3.2 in [27].

Lemma 2.8. Let α ≥ 0 and 1 < q < ∞ . Then, (2.30) D

^α_x

P

+

f P

₋

∂

x

g

L^q

. k D

^α_x¹

f k

^Lq1

k D

_x^α2

g k

^Lq2

,

with 1 < q

i

< ∞ ,

_q¹₁

+

_q¹₂

=

¹_q

and α

1

≥ α, α

2

≥ 0 and α

1

+ α

2

= 1 + α.

We also state an estimate to handle the multiplication by a term on the form e

^±2iF

, where F is a real-valued function, in fractional Sobolev spaces.

Lemma 2.9. Let 2 ≤ q < ∞ and 1 ≤ s ≤

³2

. Consider F and F

1

two real-valued functions such that v = ∂

x

F and v

1

= ∂

x

F

1

belong to L

²

( R ). Then, it holds that (2.31) k J

x^s

e

^±iF

g

k

^Lq

. (1 + k v k

²H¹

) k J

x^s

g k

^Lq

.

Remark 2.10. The proof follows the lines of Lemma 2.7 in [23] (see also [20, 21]).

A version of Lemma 2.9 could also be stated for s >

³₂

. 3. The gauge transformation

The gauge transform we will use is the one introduced by Tao in [27]. First we define an antiderivative F = F [v] of v. We determine F on the time axis x = 0 by solving the ODE

∂

t

F (0, t) = b H v

x

+ aǫv

xx

+

^c₂

A

⁻¹

v

²

−

³2

aǫ(v H v

x

+ H (vv

x

)) (0, t), F(0, 0) = 0,

Then we extend F on the whole plan by setting

∂

x

F = Av, where A = 2d

3a .

Clearly, it holds

∂

x

∂

t

F − b H ∂

_x²

F − aǫ∂

_x³

F

= ∂

x

c

2 A

⁻¹

F

_x²

− 3

2 aǫ(F

x

H F

xx

+ H (F

x

F

xx

)) . and, according to the choice of F on the time axis, it satisfies the equation (3.1) ∂

t

F − b H ∂

_x²

F − aǫ∂

_x³

F = c

2 A

⁻¹

F

_x²

− 3

2 aǫ(F

x

H F

xx

+ H (F

x

F

xx

)).

Now, we perform the following nonlinear transformation (3.2) W = P

+hi

(e

^iF

) and w = W

x

= iAP

+hi

(e

^iF

v).

First, using the identity H P

+

= − iP

+

, we compute

∂

t

W + ib∂

_x²

W − aǫ∂

_x³

W

= iP

+hi

e

^iF

(∂

t

F + ib∂

_x²

F − aǫ∂

_x³

F − bF

_x²

− 3iaǫF

x

F

xx

− aǫF

_x³

) . Then using (3.1) and the identity H + i = 2iP

₋

it follows that

∂

t

W + ib∂

_x²

W − aǫ∂

_x³

W

= P

+hi

e

^iF

(i( c

2 A

⁻¹

− b)F

_x²

− iaǫF

_x³

− 2bP

₋

F

xx

+ 3aǫF

x

P

₋

F

xx

+ 3aǫP

₋

(F

x

F

xx

))

= P

+hi

(e

^iF

α

1

v

²

+ α

2

ǫv

³

)

+ α

3

P

+hi

(W P

₋

v

x

) + α

3

P

+hi

(P

lo

(e

^iF

)P

₋

v

x

) +α

4

ǫP

+hi

(wP

₋

v

x

) + α

5

ǫP

+hi

(P

lo

(e

^iF

v)P

₋

v

x

) + α

6

ǫP

+hi

(W P

₋

(vv

x

)) +α

6

ǫP

+hi

(P

lo

(e

^iF

)P

₋

(vv

x

)),

where α

j

, j = 1 · · · 6 are complex constants depending on a, b, c and d.

Remark 3.1. We observe from the definition of the coefficients a, b, c and d in (1.3) and (1.4) that

(3.3) α

1

= 0 ⇔ 3ac

4d = b ⇔ ρ

²

= 3ρ

²₁

.

In the following, we will fix α

1

= · · · α

6

= 1 for sake of simplicity. Therefore, we deduce by differentiating the above equation that w is a solution to

∂

t

w + ib∂

_x²

w − aǫ∂

_x³

w = ∂

x

P

+hi

(e

^iF

(v

²

+ ǫv

³

))

+ ∂

x

P

+hi

(W P

₋

v

x

) + ∂

x

P

+hi

(P

lo

(e

^iF

)P

₋

v

x

) + ǫ∂

x

P

+hi

(wP

₋

v

x

) + ǫ∂

x

P

+hi

(P

lo

(e

^iF

v)P

₋

v

x

) + ǫ∂

x

P

+hi

(W P

₋

(vv

x

)) + ǫ∂

x

P

+hi

(P

lo

(e

^iF

)P

₋

(vv

x

))

:= N (e

^iF

, v, W, w).

(3.4)

On the other hand, we can recover v as a function of w by writing

(3.5) iAv = e

^−iF

∂

x

(e

^iF

) = e

^−iF

w + e

^−iF

∂

x

P

lo

(e

^iF

) + e

^−iF

∂

x

P

₋hi

(e

^iF

), so that it follows from the frequency localization

iAP

+HI

v = P

+HI

(e

^−iF

w) + P

+HI

(P

+hi

e

^−iF

∂

x

P

lo

(e

^iF

)) + P

+HI

(P

+HI

e

^−iF

∂

x

P

₋hi

(e

^iF

)).

(3.6)

Then, we have the following a priori estimates on v in terms of w.

Proposition 3.2. Let s ≥ 1, 0 < T ≤ 1, 0 ≤ θ ≤ 1, 0 < ǫ ≤ 1 and v be a solution to (1.1) in the time interval [0, T ]. Then, it holds that

(3.7) k v k

X_ǫ,T^s−2θ,θ

. k v k

^L∞TH_x^s

+ k v k

²L^∞_TH^s_x

+ ǫ k J

_x^s

v k

²L⁴_{T ,x}

. Moreover, if 1 ≤ s ≤

³2

, it holds that

(3.8) k J

_x^s

v k

L^∞_TL²_x

. k v

0

k

H¹

+ 1 + k v k

²L^∞_TH¹_x

k w k

X^s,

1 2,1 ǫ,T

+ k v k

²L^∞_TH_x¹

,

(3.9) k J

_x^s

v k

L⁴_x,t

. k v

0

k

H¹

+ 1 + k v k

²L^∞_TH¹_x

ǫ

⁻¹²¹

k w k

X^s,

1 3+ ǫ,T

+ k v k

²L^∞_TH_x¹

,

(3.10) k v k

L²_xL^∞_T

. ε

⁻¹

k v

0

k

H¹

+ k v k

L^∞_TH¹_x

( k w k

_X^1,1/2,1

ǫ,T

+ k v k

L^∞_TH¹_x

+ k v k

L²_xL^∞_T

) ,

and

k J

_x^s

∂

x

v k

_L^∞

xL²_T

.

ǫ

⁻³²⁺

k v

0

k

H^s

+ ǫ

⁻³²⁺

k w k

X^s,

1 2,1 ǫ,T

+ k v k

L^∞_TH_x¹

k w k

X^1,

1 2,1 ǫ,T

+ k J

_x^s

∂

x

v k

_L^∞

xL²_T

+ k v k

L^∞_TH_x¹

. (3.11)

Remark 3.3. It is worth noticing that estimates (3.8) and (3.9) could be rewritten in a convenient form for s >

³₂

.

Proof. (3.8) and (3.9) follow from (3.6) for the high frequencies and from (1.1) for the low frequencies (see for instance [23]). To prove (3.7) we proceed as in [23], noticing that according to (1.1),

k ∂

t

(V

ε

( − t)u(t)) k

L²_THx^s−2

. k J

_x^s

u k

²L⁴_{T x}

.

To prove estimate (3.10), we also split v between its high and low Fourier modes

(3.12) k v k

_L^2

xL^∞_T

. k P

LO

v k

_L^2

xL^∞_T

+ k P

HI

v k

_L^2 xL^∞_T

.

The low frequency term on the right-hand side of (3.12) can be treated by using (1.1) and the maximal function estimate (2.28) to get

(3.13) k P

LO

v k

_L^2

xL^∞_T

. ǫ

⁻⁽³⁴⁺⁾

k v

0

k

H¹

+ k v k

²L^∞_TH¹_x

.

To treat the high frequency term we use that v is real-valued to first notice that k P

HI

v k

L²_xL^∞_T

. 2 k P

+HI

v k

L²_xL^∞_T

so that we are reduced to estimate each terms on the right-hand side of (3.6).

Now the problem is that P

+HI

is not continuous in L

²_x

L

^∞_T

. We will overcome this difficulty by noticing that P

+HI

= P

k≥4

P

+2^k

and that the family of operators P

+2k

is bounded in L

²_x

L

^∞_T

. To treat the first term of the right-hand side of (3.6) we first notice that for k ≥ 4,

(3.14)

P

₊₂^k

(e

^−iF

w) = P

₊₂^k

X

j≥k−3

P

2^j

wP

_≤2^j−1

(e

^−iF

)

+ P

₊₂^k

X

j≥k−3

P

_≤2^j

wP

2^j

(e

^−iF

)

so that

k P

+HI

(e

^−iF

w) k

L²_xL^∞_T

. X

k≥4

k P

₊₂^k

(e

^−iF

w) k

L²_xL^∞_T

. X

k≥4

h X

j≥k−3

k P

2^j

w k

L²_xL^∞_T

+ X

j≥k−3

k P

2^j

(e

^−iF

) k

^L∞T ,x

k P

_≤2^j

w k

L²_xL^∞_T

i .

But on one hand, for s ≥ 1 we deduce from (2.24) and Bernstein inequalities that for α ∈ ]0, 1/4[,

X

k≥4

X

j≥k−3

k P

+2^j

w k

L²_xL^∞_T

. X

k≥4

X

j≥k−3

2

^−αj

k D

_x^α

P

+2^j

w k

L²_xL^∞_T

. sup

k≥3

k D

^α_x

P

2^k

w k

L²_xL^∞_T

. ε

⁻¹

k w k

_X^1,1/2,1

T

and on the other hand, X

k≥4

X

j≥k−3

k P

2^j

(e

^−iF

) k

L^∞_{T x}

k P

_≤2^j

w k

L²_xL^∞_T

. X

k≥4

X

j≥k−3

2

^−j

k v k

L^∞_{T x}

k P

_≤2^j

w k

L²_xL^∞_T

. k v k

L^∞_TH_x¹

sup

k≥1

k P

2^k

w k

L²_xL^∞_T

. ε

⁻¹

k w k

_X^1,1/2,1

T

k v k

L^∞_TH¹_x

,

which completes the estimate of the term P

+HI

(e

^−iF

w). To treat the second term and third terms of the right-hand side of (3.6) we proceed as above, using the frequency localization due to the projections, to obtain

P

+HI

e

^−iF

P

₋hi

(e

^iF

v)

L²_xL^∞_T

. X

N≥2⁴

X

K≥N/4

k P

K

e

^iF

k

^L∞x,T

k P

_≤K

P

₋hi

(e

^iF

v) k

L²_xL^∞_T

. X

N≥2⁴

X

K≥N/4

K

⁻¹

k v k

^L∞x,T

X

2≤Q≤K

k P

+Q

v k

L²_xL^∞_T

. k v k

L^∞_TH_x¹

k v k

L²_xL^∞_T

. (3.15)

which completes the proof of (3.10).

Finally, to prove (3.11) we proceed similarly. First we use (1.1) and Sobolev inequality to get

k P

LO

v k

_L^∞

xL²_T

.

k v

0

k

H¹

+ k v k

²L^∞_TH_x¹

.

Second, from (3.14) we deduce that for any integer k ≥ 4, k J

_x^s

∂

x

P

+2^k

(e

^−iF

w) k

L^∞_xL²_T

. 2

^k(s+1)

X

j≥k−3

k P

2^j

w k

L^∞_xL²_T

+ X

j≥k−3

k P

_≤2^j

w k

^L∞T ,x

k P

2^j

e

^−iF

k

L^∞_xL²_T

. X

j≥k−3

2

^(k−j)(s+1)

k J

_x^s

∂

x

P

2^j

w k

L^∞_xL²_T

+ k w k

L²_TH¹_x

X

j≥k−3

2

^−j

k J

_x^s

∂

x

P

2^j

v k

L^∞_xL²_T

.

Therefore, noticing that the first term on the right-hand side of the above inequality

is a discrete convolutions between { 2

^q(s+1)

}

^q≤1

∈ l

¹

( Z ) and {k J

_x^s

∂

x

P

2^j

w k

L^∞_xL²_T

}

^j∈Z+

,

we deduce from Young’s inequality that k J

_x^s

∂

x

P

+HI

(e

^−iF

w) k

_L^∞

xL²_T

. k J

_x^s

∂

x

w k

_L^∞

xL²_T

+ k w k

L²_TH_x¹

sup

j≥3

k J

_x^s

∂

x

v k

L^∞_xL²_T

. ǫ

⁻³²⁺

k w k

_X^s,1/2,1

ǫ,T

+ k w k

_X^1,1/2,1

ǫ,T

k J

_x^s

∂

x

v k

_L^∞

xL²_T

, where we made use of (2.20) in the last step. Third, we proceed similarly to estimate the second and third term of the right-hand side of (3.6) by

J

_x^s

∂

x

P

+HI

e

^−iF

P

lo

(e

^iF

v)

_^

L^∞_xL²_T

+ J

_x^s

∂

x

P

+HI

e

^−iF

P

₋hi

(e

^iF

v)

_^

L^∞_xL²_T

. k v k

L^∞_TH_x¹

sup

j≥3

k J

_x^s

∂

x

P

₂^j

v k

L^∞_xL²_T

.

This completes the proof of (3.11) and of the proposition.

4. Bilinear estimates

In this section, we fix ǫ = 1. The aim of this section is to derive an estimate on k w k

X^s,

1 2,1 T

.

Proposition 4.1. Let 0 < T ≤ 1, 1 ≤ s ≤

³₂

, v be a solution to (1.1) on the time interval [0, T ] and w defined in (3.2). Then it holds that

k w k

X^s,

1 2,1 T

. 1 + k v

0

k

²H¹

k v

0

k

^Hs

+ p k w k

_X^s,1₂^,1

, k v k

L^∞_TH_x¹

, k v k

L⁴_TWx^1,4

, k v k

L²_xL^∞_T

, k J

_x^s

∂

x

v k

_L^∞

xL²_T

, sup

0≤θ≤1

k v k

_XT^1−2θ,θ

, (4.1)

where p is a polynomial function at least quadratic in its arguments.

The main tools to prove Proposition 4.1 are the following crucial bilinear esti- mates.

Proposition 4.2. For any s ≥ 1, we have that (4.2) k ∂

x

P

+hi

wP

₋

∂

x

v

k

_X^s,−1

2,1

. k w k

_X^s,1₂^,1

sup

0≤θ≤1

k v k

X^1−2θ,θ

.

Proof. We only prove estimate (4.2) in the case s = 1, since the case s > 1 follows by similar arguments due to the frequency localization on the functions w and v.

By duality, estimate (4.2) is equivalent to (4.3) I . X

N

sup

L

k h

N,L

k

²L²_ξ,τ

¹₂

k w k

_X^1,1₂^,1

sup

0≤θ≤1

k v k

X^1−2θ,θ

, where

(4.4) I = X

N,L

h N ih L i

⁻¹²

Z

D

ξh

N,L

(ξ, τ )φ

N

(ξ)ψ

L

(ξ, τ ) w(ξ b

1

, τ

1

)ξ

2

v(ξ b

2

, τ

2

)dν, (4.5) dν = dξdξ

1

dτ dτ

1

, ξ

2

= ξ − ξ

1

, τ

2

= τ − τ

1

,

(4.6) σ = τ − | ξ | ξ + ξ

³

, σ

i

= τ

i

− ξ

i

| ξ

i

| + ξ

³_i

, i = 1, 2, and

(4.7) D =

(ξ, ξ

1

, τ, τ

1

) ∈ R

⁴

| ξ ≥ 1, ξ

1

≥ 1 and ξ

2

≤ 0 .

Observe that we always have in D that

(4.8) ξ

1

≥ ξ ≥ 1 and ξ

1

≥ | ξ

2

| .

Then, we obtain by performing dyadic decompositions in ξ

1

, ξ

2

, σ

1

and σ

2

that

I = X

N,N,N2

X

L,L1,L2

h N ih L i

⁻¹²

Z

D

ξh

N,L

(ξ, τ )φ

N

(ξ)ψ

L

(ξ, τ )

× (P

N1

Q

L1

w)

^∧

(ξ

1

, τ

1

)ξ

2

(P

N2

Q

L2

v)

^∧

(ξ

2

, τ

2

)dν.

(4.9)

Due to the second identity in (4.5) and (4.8), we can always assume that one of the following cases holds:

(1) high-low interaction: N

1

∼ N and N

2

≤ N

1

(2) high-high interaction: N

1

∼ N

2

and N ≤ N

1

. Moreover, the resonance identity

(4.10) σ − σ

1

− σ

2

= 3ξξ

2

(ξ

1

− 2 3 )

holds in D , so that for fixed N, N

1

and N

2

, we can always assume that (4.11) L

max

∼ max { L

med

, N N

1

N

2

} ,

where L

max

, L

med

and L

min

denote respectively the maximum, median and mini- mum of L, L

1

and L

2

.

To estimate I, we will divide the sum in (4.9) depending on the high-low or high-high interactions regime and on wether L

max

= L, L

1

or L

2

.

Case high-low interaction and L

max

= L

2

. In this case, we can estimate I as

| I | . X

N1,N2≤N1

X

L2,L≤L2

h N

1

ih L i

^−δ

h L

2

i

Z

D

| ξξ

2

|h ξ

2

i h ξ

1

i

| h

N1,L

(ξ, τ ) |

h σ i

^12−δ

φ

N1

(ξ)ψ

L

(ξ, τ )

× | (P

N1

J

_x¹

w)

^∧

(ξ

1

, τ

1

) | h σ

2

i

h ξ

2

i | (P

N2

Q

L2

v)

^∧

(ξ

2

, τ

2

) | dν.

Since L

2

= L

max

, we deduce from (4.10) that L

2

≥ N

₁²

N

2

. Therefore, L

2

∼ 2

^k

N

₁²

N

2

for k ∈ Z

₊

, so that, we obtain by using Plancherel’s identity and H¨ older’s inequality

| I | . X

N1,N2≤N1

N

2

N

1

sup

L

| h

N1,L

| h σ i

^12−δ

∨

_L4 x,t

(P

_N1

J

_x¹

w)

^∧

^∨

_L4 x,t

× X

k∈Z+

2

^−k

k h σ i

h ξ i (P

N2

Q

2^kN₁²N2

v)

^∧

k

L²_ξ,τ

. . X

N1

sup

L

k h

N1,L

k

L²_ξ,τ

(P

N1

J

_x¹

w)

^∧

^∨

_L4

x,t

k v k

X^−1,1

. X

N1

sup

L

k h

N1,L

k

²L²_ξ,τ

¹₂

X

N1

(P

N1

J

_x¹

w)

^∧

^∨

²_L4 x,t

¹₂

k v k

X^−1,1

, which, combined to estimates (2.9) and (2.4) leads to estimate (4.3) in this case.

Note that we have used here any 0 < δ <

¹₆

− , since X

^0,13+

֒ → L

⁴_x,t

due to estimate

(2.9).