Scaling-sharp dispersive estimates for the Korteweg-de Vries group

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Scaling-sharp dispersive estimates for the Korteweg-de

Vries group

Raphaël Côte, Luis Vega

To cite this version:

Raphaël Côte, Luis Vega. Scaling-sharp dispersive estimates for the Korteweg-de Vries group. Comptes Rendus Mathématique, Elsevier Masson, 2008, 346, pp.845-848. �10.1016/j.crma.2008.05.003�. �hal-00273050�

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hal-00273050, version 1 - 14 Apr 2008

Scaling-sharp dispersive estimates for the Korteweg-de Vries

group

Rapha¨el Cˆote Luis Vega

Abstract

We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations.

The purpose of this short note is to give a simple proof of two dispersive estimates which are heavily used in the proof of small data scattering for the generalized Korteweg-de Vries equations [2].

The proof of these estimates can be easily extended to other dispersive equations. Denote U (t) the linear Korteweg-de Vries group, i.e v = U (t)φ is the solution to vt+ vxxx= 0, v(t = 0) = φ, i.e. U (t)φ = e\ itξ3 ˆ φ or (U (t)φ)(x) = 1 t1/3 Z Ai x − y t1/3 φ(y)dy, where Ai is the Airy function

Ai(z) = 1 π Z ∞ 0 cos ξ 3 3 + ξz dξ. Theorem 1. _{Let φ, ψ ∈ L}2_{, such that xφ, xψ ∈ L}2. Then

kU(t)φk2L∞ ≤ 2k Ai k2_L∞t−2/3kφk_L2kxφk L2, (1) kU(t)φU(−t)ψxkL∞ ≤ Ct−1(kφk_L2kxψk L2 + kψk L2kxφk L2). (2)

Furthermore, the constant 2k Ai k2L∞ in the first estimate is optimal.

Remark _1. _{These estimates are often used with φ replaced by U (−t)φ : denoting J(t) =} U (t)xU (−t), they take the form

kφk2L∞ ≤ Ct−2/3kφk_L2kJ(t)φk L2, (3) kφψxkL∞ ≤ Ct−1(kφk_L2kJ(t)ψk L2 + kψk L2kJ(t)φk L2). (4)

Proof. Due to a scaling argument (and representation in term of the Airy function), we are reduced to show that

kU(1)φkL∞ ≤ Ckφk_L2kxφk

L2,

and similarly for the second inequality. Hence we consider (U (1)φ)(x) =

Z

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and we recall that the Airy function satisfies | Ai(x)| ≤ C(1 + |x|)−1/4 and | Ai′(x)| ≤ C(1 + |x|)1/4_{. Then}

|U(1)φ|2(x) = Z Z

Ai(x − y)φ(y) Ai(x − z) ¯φ(z)y − z y − zdydz =

Z

Ai(x − y)yφ(y)Z Ai(x − z) y − z φ(z)dz¯ | {z } Hz7→y(Ai(x−z)φ(z))(y) dy − Z Ai(x − z)z ¯φ(z)Z Ai(x − y) y − z φ(y)dy | {z } −Hy7→z(Ai(x−y)φ(y))(z) dz = 2ℜ Z

Ai(x − y)yφ(y)Hz7→y(Ai(x − z)φ(z))(y)dy,

where H denotes the Hilbert transform (and with the slight abuse of notation 1x for

vp 1_x_{). As H : L}2_{→ L}2 _{is isometric and hence continuous (with norm 1), and Ai ∈ L}∞, we get

|U(1)φ|2(x) ≤ 2k Ai(x − y)yφ(y)kL2_(dy)kH(Ai(x − ·)φ)(y)k_L2_(dy) (5)

≤ 2k Ai k2L∞kyφk_L2kφk_L2. (6)

This is the first inequality. Let us now prove that the constant is sharp. First consider the minimizers in the following Cauchy-Schwarz inequality :

Z yψ(y)H(ψ)(y)dy ≤ kyψ(y)kL2(dy)kψkL2. (7) There is equality if yψ(y) = λH(ψ)(y) for some λ ∈C. Then a Fourier Transform shows

that ∂ξψ(ξ) = λ sgn ξ ˆˆ ψ(ξ), hence ˆψ(ξ) = C exp(−λ|x|), or equivalenty, one has equality

in (7) as soon as

ψ(y) = C

1 + (y/λ)2 for some λ, C ∈R. (8)

(Notice that all the functions involved lie in L2)

We now go back to (6). Let x0 ∈ R where | Ai | reaches its maximum. Now as

Ai(x0) 6= 0, let ε > 0 such that for all y ∈ [−ε, ε], | Ai(x0 − y)| ≥ | Ai(x0)|/2, and

consider the sequence of functions φn(x) = √ n 1 + (ny)2 1 |y|≤ε Ai(x0− y) . Denote ψn(y) = 1|y|≤nε

1+y2 . As Ai(x0− y)φn(ny) =

√

nψn(ny),

|U(1)φn|2(x0) = 2

Z

y√nψn(ny)Hz→y(√nψn(nz))(y)dy =

2 n

Z

yψn(ny)H(ψn)(ny)dy.

One easily sees that ψn(y) → _1+|y|1 2 in L2 and yψn(y) →

y

1+|y|2 in L2, and hence, in view

of (8), as H is homogeneous of degree 0 and L2 isometric, we have |U(1)φn|2(x0) ∼ 2 n Z _y 1 + |y|2H( 1 1 + | · |2)(y)dy ∼ 2 nk y 1 + |y|2kL2k 1 1 + |y|2kL2

∼ 2ky√nψn(ny)kL2k√nψ_n(ny)k

L2

∼ 2ky Ai(x0− y)φn(y)kL2k Ai(x₀− y)φ_n(y)k

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As φn concentrates at point 0, we deduce

|U(1)φn|2(x0) ∼ 2| Ai(x0)|2kyφn(y)kL2kφ_n(y)k_L2 as n → ∞, (9)

which proves that the sharp constant in the first inequality is 2k Ai k2 L∞.

For the second inequality (estimate of the derivative), we have as for the first in-equality : (U (1)φU (1) ¯ψx)(x) = Z Z Ai(x − y)φ(y) Ai′(x − z) ¯ψ(z)y − z y − zdydz = Z Ai′_{(x − y)yφ(y)}Z Ai(x − z)φ(z) y − z dz dy − Z z Ai′_{(x − y)φ(y)}Z Ai(x − z)z ¯ψ(z) z − y dz dy = Z

Ai′_{(x − y)yφ(y)H}z7→y(Ai(x − z)φ(z))(y)dy

+ Z

Ai′_{(x − y)φ(y)H}z→y(Ai(x − z)z ¯ψ(z))(y)dy.

Denote ωx(y) = √ 1

1+|x−y| : ω −1

x ∈ A2 (with the notation of [4]), so that there exists C

not depending on x such that ∀v,

Z

|Hv|2ω−1_x _{≤ C} Z

|v|2ω−1_x . Recall the well-know asymptotic | Ai′(x)| ≤ C(1 + |x|1/4_{). Then}

Z Ai′_{(x − y)yφ(y)H(Ai(x − ·) ¯}ψ)(y)dy ≤ Z | Ai′(x − y)yφ(y)|2ωxdy 1/2Z

|H(Ai(x − ·) ¯ψ)(y)|2ω_x−1(y)dy 1/2

≤ Ckyφ(x)kL2

Z

| Ai(x − y) ¯ψ(y)|2ω_y−1(y)dy 1/2

≤ CkyφkL2kψk

L2.

In the same way,

Z

Ai′_{(x − y)φ(y)H}z→y(Ai(x − z)z ¯φ(z))(y)dy

≤ Z | Ai′(x − y)φ(y)|2ωx(y) 1/2Z

|Hz→y(Ai(x − z)zψ(z))(y)|2ωx−1(y)dy

1/2

≤ CkφkL2

Z

| Ai(x − y)yψ(y)|2ω−1_x (y)dy 1/2 ≤ CkφkL2kyψk_L2. So that : kU(1)φU(1) ¯ψxkL∞ ≤ 2C(kφk_L2kxψk L2 + kψk L2kxφk L2).

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Remark _2. _{This proof (especially (5)) is reminiscent of that in [3] (see also [1])} kφk2L∞ ≤ kφk_L2kφ′k

L2,

where the constant is sharp and minimizers are Ce−λ|x|_{. This has application to the}

Schr¨_{odinger group U(t) (i.e \}_{U(t)φ = e}itξ2

ˆ

φ). We have the following Schr¨odinger version of estimate (3) (notice that U(t)xU(−t) = ei|x|

2 4t it 2∂xe− i|x|2 4t ) : kψk2L∞ = ke i|x|2 4t ψk2 L∞ ≤ kei |x|2 4t ψk L2kei |x|2 4t ∂_xei |x|2 4t ψk L2 ≤ 2 tkψkL2kU(t)xU(−t)ψkL2. From Theorem 1, we can easily obtain the optimal decay in a scaling sharp Besov like space. Let ϕ ∈ D(R) be non-negative with support in ]−2, 2[ and such that ϕ equals

1 in a neighbourhood of [−1.5, 1.5]. Denote ψ(x) = ϕ(2x) − ϕ(x) and ψj(x) = ψ(x/2j).

Finally introduce kφkNt = ∞ X j=−∞ 2j/2_kψjU (−t)φkL2. Corollary 1. We have : kφkL∞ ≤ Ct−1/3kφk_N t.

Proof. Notice that |xψj(x)| ≤ 2j+1ψj(x). As φ =P_jU (t)ψjU (−t)φ, we have :

kφkL∞ ≤ X j kU(t)ψjU (−t)φkL∞ ≤ Ct−1/3X j kU(t)ψjU (−t)φk1/2_L2 kU(t)xU(−t)U(t)ψjU (−t)φk 1/2 L2 ≤ Ct−1/3X j kU(t)ψjU (−t)φk1/2_L2 kxψjU (−t)φk 1/2 L2 ≤ Ct−1/3X j kU(t)ψjU (−t)φk1/2_L2 2j/2kU(t)ψjU (−t)φk 1/2 L2 ≤ Ct−1/3kφkNt.

References

[1] J. Duoandikoetxea, L. Vega, Some weighted Gagliardo-Nirenberg inequalities and applications, Proc. Amer. Math. Soc. 135 (2007) 2795–2802 (electronic).

[2] N. Hayashi, P. I. Naumkin, Large time asymptotics of solutions to the generalized Korteweg-De Vries equation, J. Funct. Anal. 159 (1998) 110–136.

[3] E. H. Lieb, M. Loss, Analysis, American Mathematical Society, Providence, RI, 2001.

[4] E. M. Stein, Singular integrals and differentiability properties of functions, Prince-ton University Press, PrincePrince-ton, N.J., 1970.

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Rapha¨el Cˆote

Centre de Math´ematiques Laurent Schwartz, ´Ecole polytechnique 91128 Palaiseau Cedex, France

cote@math.polytechnique.fr Luis Vega

Departamento de Matem´aticas, Universidad del Pa´ıs Vasco Aptdo. 644, 48080 Bilbao, Espa˜na