On KP-II type equations on cylinders

0.34 MB·English

by  Axel Grünrock

ON KP-II TYPE EQUATIONS ON CYLINDERS
AXEL GRU¨NROCK,MAHENDRAPANTHEE,AND JORGEDRUMONDSILVA
9
0
0
2

n
Abstract. InthisarticlewestudythegeneralizeddispersionversionoftheKadomtsev-Petviashvili
a
J II equation, on T×R and T×R2. We start by proving bilinear Strichartz type estimates, de-

4 pendent only on the dimension of the domain but not on the dispersion. Their analogues in
1
terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of

] the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy
P
problem.
A
.
h
t
a
m
[

1
Contents
v
4
0 1. Introduction 2
0
2. Strichartz Estimates 4
2
. 2.1. Proof of the Strichartz estimate in the T R case 6
1
×
0 2.2. Proof of the Strichartz estimate in the T R2 case 11
9 ×
0 2.3. Counterexample for global Strichartz estimate in T R 13
: ×
v 3. Bilinear Estimates 14
i
X 3.1. Proof of the bilinear estimates in the T R2 case 16
×
ar 3.2. Proof of the bilinear estimates in the T R case 22
×
4. Local Well-posedness 26
Appendix A. Failure of regularity of the flow map in T R 27
×
References 31
1991 Mathematics Subject Classification. 35Q53.
A. Gru¨nrock was partially supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 611.
M. Panthee was partially supported through the program POCI 2010/FEDER. J. Drumond Silva was partially
supported through theprogram POCI 2010/FEDER and by theproject POCI/FEDER/MAT/55745/2004.
1
2 A.GRU¨NROCK,M.PANTHEE,ANDJ.DRUMONDSILVA
1. Introduction
In this paper, we consider the initial value problem (IVP) for generalized dispersion versions
of the Kadomtsev-Petviashvili-II (defocusing) equation on T R
x y
×
∂ u D α∂ u+∂ 1∂2u+u∂ u= 0 u:R T R R,
(1) t −| x| x x− y x t× x× y →
( u(0,x,y) = u0(x,y),
and on T R2
x× y
∂ u D α∂ u+∂ 1∆ u+u∂ u= 0 u: R T R2 R,
(2) t −| x| x x− y x t× x× y →
( u(0,x,y) = u0(x,y).
We consider the dispersion parameter α 2. The operators D α∂ and ∂ 1 are defined by
≥ | x| x x−
their Fourier multipliers ik αk and (ik) 1, respectively.
−
| |
The classical Kadomtsev-Petviashvili (KP-I and KP-II) equations, when α= 2,
∂ u+∂3u ∂ 1∂2u+u∂ u= 0
t x ± x− y x
are thenaturaltwo-dimensional generalizations of the Korteweg-de Vries (KdV)equation. They
occur as models for the propagation of essentially one-dimensional weakly nonlinear dispersive
waves, with weak transverse effects. The focusing KP-I equation corresponds to the minus (-)
sign in the previous equation, whereas the defocusing KP-II is the one with the plus (+) sign.
The well-posedness of the Cauchy problem for the KP-II equation has been extensively stud-
ied, in recent years. J. Bourgain [1] made a major breakthrough in the field by introducing
Fourier restriction norm spaces, enabling a better control of the norms in the Picard iteration
method applied to Duhamel’s formula, and achieving a proof of local well-posedness in L2(T2)
(and consequently also global well-posedness, due to the conservation of the L2 norm in time).
Since then, a combination of Strichartz estimates and specific techniques in the framework of
Bourgain spaces has been used by several authors to study KP-II type equations in several set-
tings (see [9], [10], [13], [14], [15], [16], [17] and references therein). Recently, an optimal result
was obtained by M. Hadac [6] for the generalized dispersion KP-II equation on R2, in which
local well-posedness for the range of dispersions 4 < α 6 was established for the anisotropic
3 ≤
Sobolev spaces Hs1,s2(R2), provided s > max(1 3α, 1 3α), s 0, thus reaching the scal-
1 − 4 4 − 8 2 ≥
ing critical indices for 4 < α 2. This includes the particular case α = 2 corresponding to the
3 ≤
classical KP-II equation. In this case the analysis was pushed further to the critical regularity
by M. Hadac, S. Herr, and H. Koch in [8], where a new type of basic function spaces - the so
called Up-spaces introduced by H. Koch and D. Tataru - was used. Concerning the generalized
dispersion KP-II equation on R3, a general result was also shown by M. Hadac in [7], which is
KP-II TYPE EQUATIONS 3
optimal in the range 2 α 30 by scaling considerations. For the particular case α = 2, he
≤ ≤ 7
obtained local well-posedness in Hs1,s2(R3) for s > 1 and s > 0.
1 2 2
In this article, we aim to study the local well-posedness of the initial value problem for the
general dispersion KP-II type equations (1) and (2), on the cylinders T R and T R2
x × y x × y
respectively. We will show that the initial value problem (1) is locally well-posed for data
u Hs1,s2(T R) satisfying the mean zero condition 2πu(x,y)dx = 0, provided α 2,
0 ∈ × 0 ≥
s > max(3 α, 1 α), and s 0. Combined with the conservation of the L2 -norm this
1 4 − 2 8 − 4 2 ≥ R xy
local result implies global (in time) well-posedness, whenever s 0 and s = 0. Concerning
1 2
≥
(2) we will obtain local well-posedness for u Hs1,s2(T R2), satisfying again the mean zero
0
∈ ×
condition, in the following cases:
α = 2, s 1, s > 0,
• 1 ≥ 2 2
2 < α 5, s > 3 α, s 0,
• ≤ 1 −2 2 ≥
5 < α, s > 1 α, s 0.
• 1 −4 2 ≥
Forα > 3ourresulthereisin,andbelow,L2 . Inthiscaseweagainobtainglobalwell-posedness,
xy
whenever s 0 and s = 0.
1 2
≥
We proceed in three steps. First, in Section 2, we will establish bilinear Strichartz estimates
for the linear versions of (1) and (2), depending only on the domain dimension but not on the
dispersion parameter. We believe, these estimates are of interest on their own, independently of
their application here1. Inthesecond step, inSection 3, wewillusethese Strichartzestimates to
provebilinear estimates forthenonlineartermof theequations, inBourgain’s Fourier restriction
norm spaces. Finally, in Section 4, a precise statement will be given of our local well-posedness
results for the associated initial value problems, with data in Sobolev spaces of low regularity.
TheirproofsfollowastandardfixedpointPicarditerationmethodappliedtoDuhamel’sformula,
using the bilinear estimates obtained in the previous section. In the appendix we provide a
counterexample, due to H. Takaoka and N. Tzvetkov [18], concerning the two-dimensional case.
Thisexampleshowsthenecessityofthelowerbounds 3 α andhencetheoptimality(except
1 ≥ 4−2
for the endpoint) of our two-dimensional result in the range 2 α 5. For higher dispersion
≤ ≤ 2
(α > 5) we unfortunately lose optimality as a consequence of the case when an interaction of
2
two high frequency factors produces a very low resulting frequency. The same problem occurs
in three space dimensions, but the effect is much weaker. Here, by scaling considerations, our
1Forexampleourtwo-dimensionalspacetimeestimate,whichisequallyvalidforthelinearizedKP-Iequation,
together with the counterexamples presented later on gives a definite answer to a question raised by J. C. Saut
and N.Tzvetkov in [16, remark on top of p. 460].
4 A.GRU¨NROCK,M.PANTHEE,ANDJ.DRUMONDSILVA
result is optimal for 2 α 5, and we leave the line of optimality only for very high dispersion,
≤ ≤
when α> 5.
2. Strichartz Estimates
Strichartz estimates have, in recent years, been playing a fundamental role in the proofs of
local well-posedness results for the KP-II equation. Their use has been a crucial ingredient for
establishing the bilinear estimates associated to the nonlinear terms of the equations, in the
Fourier restriction spaces developed by J. Bourgain, the proof of which is the central issue in
the Picard iteration argument in these spaces. Bourgain [1] proved an L4 L2 Strichartz-type
−
estimate, localized in frequency space, as the main tool for obtaining the local well-posedness
of the Cauchy problem in L2, in the fully periodic two-dimensional case, (x,y) T2. J.C. Saut
∈
and N. Tzvetkov [15] proceeded similarly, for the fifth order KP-II equation, also in T2 as well
as T3. Strichartz estimates for the fully nonperiodicversions of the (linearized) KP-II equations
have also been extensively studied and used, both in the two and in the three-dimensional cases.
In these continuous domains, R2 and R3, the results follow typically by establishing time decay
estimates for the spatial L norms of the solutions, which in turn are usually obtained from the
∞
analysisoftheiroscillatoryintegralrepresentations,asin[3],[11]or[13]. TheStrichartzestimates
obtained this way also exhibit a certain level of global smoothing effect for the solutions, which
naturally depends on the dispersion factor present in the equation.
As for our case, we prove bilinear versions of Strichartz type inequalities for the generalized
KP-II equations on the cylinders T R and T R2. The main idea behind the proofs that
× ×
we present below is to use the Fourier transform in the periodic x variable only. And
x
F
then, for the remaining y variables, to apply the well known Strichartz inequalities for the
Schro¨dinger equation in R or R2. This way, we obtain estimates with a small loss of derivatives,
but independent of the dispersion parameter.
So, consider the linear equations corresponding to (1) and (2),
(3) ∂ u D α∂ u+∂ 1∂2u= 0,
t −| x| x x− y
respectively
(4) ∂ u D α∂ u+∂ 1∆ u= 0.
t −| x| x x− y
The phase function for both of these two equations is given by
η 2
φ(ξ) =φ (k) | | ,
0
− k
KP-II TYPE EQUATIONS 5
where φ (k) = k αk is the dispersion term and ξ = (k,η) Z R, respectively ξ = (k,η)
0 ∗
| | ∈ × ∈
Z R2, is the dual variable to (x,y) T R, respectively (x,y) T R2, so that the unitary
∗
× ∈ × ∈ ×
evolution group for these linear equations is eitφ(D), where D = i . For the initial data
− ∇
functions u , v that we will consider below it is assumed that u (0,η) = v (0,η) = 0 (mean
0 0 0 0
zero condition).
c b
The two central results of this section are the following.
Theorem 1. Let ψ C (R) be a time cutoff function with ψ = 1 and supp(ψ) ( 2,2),
∈ 0∞ [−1,1] ⊂ −
and let u ,v :T R R satisfy the mean zero condition in the x variable. Then, for s 0
0 0 x y (cid:12) 1,2
× → (cid:12) ≥
such that s +s = 1, the following inequality holds:
1 2 4
(5) kψ eitφ(D)u0eitφ(D)v0kL2txy . ku0kHxs1L2ykv0kHxs2L2y.
Theorem 2. Let u ,v :T R2 R satisfy the mean zero condition in the x variable. Then,
0 0 x× y →
for s 0 such that s +s > 1, the following inequality holds:
1,2 1 2
≥
(6) keitφ(D)u0eitφ(D)v0kL2txy . ku0kHxs1L2ykv0kHxs2L2y.
Choosing u = v and s = s = 1+, we have in particular
0 0 1 2 2
eitφ(D)u . u .
k 0kL4txy k 0kHx12+L2y
Note that in the case of Theorem 1, in the T R domain, the Strichartz estimate is valid
x y
×
only locally in time. A proof of this fact is presented in the last result of this section
Proposition 1. There is no s R such that the estimate
∈
eitφ(D)u 2 . Dsu u ,
k 0 kL2txy k x 0kL2xyk 0kL2xy
holds in general. (cid:0) (cid:1)
The use of a cutoff function in time is therefore required in T R, whose presence will be
×
fully exploited in the proof of Theorem 1. In the case of Theorem 2, where y R2, the result is
∈
valid globally in time and no such cutoff is needed to obtain the analogous Strichartz estimate2.
As a matter of fact, in the three-dimensional case T R2, the proof that we present is equally
×
valid for the fully nonperiodic three-dimensional domain, R3. As pointed out above, Strichartz
2In any case, for our purposes of proving local well-posedness in time for the Cauchy problems (1) and (2),
further on in this paper, this issue of whether the Strichartz estimates are valid only locally or globally will not
be relevant there.
6 A.GRU¨NROCK,M.PANTHEE,ANDJ.DRUMONDSILVA
estimates have been proved and used for the linear KP-II equation, in R2 and R3. But being
usually derived through oscillatory integral estimates and decay in time, they normally exhibit
dependence on the particular dispersion under consideration, leading to different smoothing
properties of the solutions. For estimates independent of the dispersion term φ one can easily
0
apply a dimensional analysis argument to determine - at least for homogeneous Sobolev spaces
H˙s - the indices that should be expected. So, for λ R, if u(t,x,y) is a solution to the linear
∈
equation (4) on R3, then uλ = Cu(λ3t,λx,λ2y), C R, is also a solution of the same equation,
∈
with initial data uλ = Cu (λx,λ2y). An L4 H˙sL2 estimate for this family of scaled solutions
0 0 txy− x y
then becomes
λ12−skukL4txy . ku0kH˙xsL2y,
leading to the necessary condition s = 1. Theorem 2, for nonhomogeneous Sobolev spaces,
2
touches this endpoint (not including it, though).
2.1. Proof of the Strichartz estimate in the T R case.
×
Proof of Theorem 1. It is enough to prove the estimate (5) when s = 1/4 and s = 0.
1 2
We have, forthespace-time Fourier transformoftheproductof thetwosolutions to thelinear
equation 3
(7) (eitφ(D)u eitφ(D)v )(τ,ξ) = δ(τ φ(ξ ) φ(ξ ))u (ξ )v (ξ )µ(dξ ),
0 0 1 2 0 1 0 2 1
F − −
Z
∗
where µ(dξ ) = dη , and
∗ 1 kk=1,kk12+6=k02 η1+η2=η 1 c b
R P R
1
φ(ξ )+φ(ξ )= φ (k )+φ (k ) (kη2 2ηk η +k η2).
1 2 0 1 0 2 − k k 1 − 1 1 1
1 2
Thus the argument of δ, as a function of η , becomes
1
1
g(η ) := τ φ(ξ ) φ(ξ ) = (kη2 2ηk η +k η2)+τ φ (k ) φ (k ).
1 − 1 − 2 k k 1 − 1 1 1 − 0 1 − 0 2
1 2
The zeros of g are
ηk
1
η = ω,
1± k ±
with
k k η2
ω2 = 1 2 φ (k )+φ (k ) τ ,
0 1 0 2
k − k −
(cid:18) (cid:19)
whenever the right hand side is positive, and we have
2k ω
|g′(η1±)|= k| k| .
1 2
| |
3Throughout the text we will disregard multiplicative constants, typically powers of 2π, which are irrelevant
for thefinal estimates.
KP-II TYPE EQUATIONS 7
There are therefore two contributions I to (7), which are given by
±
k k ηk ηk
I±(τ,ξ) = k −1 | 1 2|u0 k1, 1 ω v0 k2, 2 ω ,
| | ω k ± k ∓
Xk1 (cid:18) (cid:19) (cid:18) (cid:19)
k1,k2=0
6 c b
and the space-time Fourier transform of ψ eitφ(D)u eitφ(D)v then becomes
0 0
(ψeitφ(D)u eitφ(D)v )(τ,ξ) = ψ I+(τ,ξ)+I (τ,ξ) =
0 0 τ −
F ∗
k1k2 (cid:0) ηk1 (cid:1) ηk2
ψ(τ τ ) | b | u k , +ω(τ ) v k , ω(τ )
1 0 1 1 0 2 1
− ω(τ )k k k −
Z Xk1 1 | | (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
b k1,k26=0 c b
ηk ηk
1 2
+u k , ω(τ ) v k , +ω(τ ) dτ .
0 1 1 0 2 1 1
k − k
(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
For the L2 estimate of this quantity we mcay assume, without lobss of generality, that k and
1
k are both positive (cf. pg. 460 in [16]), so that 0< k ,k < k.
2 1 2
We will now prove the result, by breaking up the sum into two cases which are estimated
separately.
Case I (ω(τ )2 > k k ):
1 1 2
In this case we start by using the elementary convolution estimate,
ψ (I+(,ξ)+I (,ξ)) . ψ (I+(,ξ)+I (,ξ)) .
k ∗τ · − · ω(τ1)2>k1k2 kL2τ k kL1τk · − · ω2>k1k2 kL2τ
(cid:12) (cid:12)
Now, tobestimate the L2 norm (cid:12)of the sum, we fix abny small 0 < ǫ < 1/4 an(cid:12)d Cauchy-Schwarz
gives
1
2
k k
(8) I (τ,ξ) . k 2ε 1 2
| ± ||ω(τ)2>k1k2  1− kω(τ)
k1X,k2>0
  1
k k ηk ηk 2 2
k2ε 1 2 u (k , 1 ω(τ))v (k , 2 ω(τ)) .
× 1 kω(τ) 0 1 k ± 0 2 k ∓
!
k1X,k2>0 (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)c b (cid:12)
The condition ω(τ)2 > k k implies ω(τ)2 = 1(φ (k )+φ (k )) η2 τ > 1, so that
1 2 k1k2 |k 0 1 0 2 − k2 − k|
√k k 1
1 2
.
ω(τ) ∼ hk1(φ0(k1)+φ0(k2))− ηk22 − kτi21
8 A.GRU¨NROCK,M.PANTHEE,ANDJ.DRUMONDSILVA
We also have √k1k2 1, from which we conclude then that the first factor in (8) is bounded by
k ≤
a constant C independent of k,τ,η, for
ǫ
k k 1
k 2ε 1 2 . k 2ε
k1X,k2>0 1− kω(τ) k1X,k2>0 1− hk1(φ0(k1)+φ0(k2))− kη22 − kτi21
1 1
p q
1
. Xk1 k1−2εp k1X,k2>0 hk1(φ0(k1)+φ0(k2))− kη22 − τkiq2 ,
   
which, using H¨older conjugate exponents p > 1/2ǫ > 2 and q = p/(p 1) < 2, as well as the
−
easy calculus fact that
1
sup (φ (k )+φ (k )) a δ C ,
0 1 0 2 − δ
a R hk − i ≤
k∈N kX1>0
∈
valid for any fixed α 2 and δ > 1/2, implies
≥
k k
k 2ε 1 2 C .
1− kω(τ) ≤ ǫ
k1X,k2>0
We thus have
I (,ξ) 2
k ± · ω2>k1k2 kL2τ
. (cid:12)(cid:12) k12ε kk1kω2 u0(k1,ηkk1 ±ω)v0(k2, ηkk2 ∓ω) 2dτ
k1X,k2>0 Z (cid:12) (cid:12)
(cid:12) (cid:12)
. k2ε u (k(cid:12)c,ηk1 ω)v (k ,bηk2 ω) 2dω.(cid:12)
1 0 1 k ± 0 2 k ∓
k1X,k2>0 Z (cid:12) (cid:12)
(cid:12) (cid:12)
Herewehaveuseddτ = 2ωkdω. Int(cid:12)ecgratingwithresbpecttodη and(cid:12)usingthechange ofvariables
k1k2
η+ = ηkk1 ±ω, η− = η(k−kk1) ∓ω with Jacobian ∓1 we arrive at
I (,ξ) 2 . k2ε u (k , ) 2 v (k , ) 2 .
k ± · ω2>k1k2 kL2τ 1 k 0 1 · kL2ηk 0 2 · kL2η
(cid:12) k1X,k2>0
Finally summing up over k(cid:12)= 0 we obtain c b
6
(I+ +I ) 2 . Dεu 2 v 2 .
k − ω2>k1k2 kL2τkη k x 0kL2xyk 0kL2xy
(cid:12)
(cid:12)
Case II (ω(τ )2 k k ):
1 1 2
≤
In this case 1(φ (k )+φ (k )) η2 τ1 1. Here we make the further subdivision
|k 0 1 0 2 − k2 − k |≤
1 = χ τ−τ1 1 +χ τ−τ1 >1 .
{| k |≤ } {| k | }
KP-II TYPE EQUATIONS 9
When τ τ1 1 we have
| −k | ≤
1 η2 τ τ τ
1
(φ (k )+φ (k )) 1+ − 2,
k 0 1 0 2 − k2 − k ≤ k ≤
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
andforeveryfixedτ,k,(cid:12)η wehaveonlyafinitenumber(cid:12)ofk ’ss(cid:12)atisfyin(cid:12)gthiscondition. Therefore,
1
k k ηk ηk
kX1>0Z |ψˆ(τ −τ1)|kω1(τ21)(cid:12)u0(k1, k1 ±ω(τ1))v0(k2, k2 ∓ω(τ1))(cid:12)χ{ω(τ1)2≤k1k2}χ{|τ−kτ1|≤1}dτ1
(cid:12) (cid:12)
(cid:12)c b (cid:12) 1
2 2
k k ηk ηk
. ψˆ(τ τ ) 1 2 u (k , 1 ω(τ ))v (k , 2 ω(τ )) dτ .
1 0 1 1 0 2 1 1
 | − |kω(τ ) k ± k ∓ 
kX1>0(cid:18)Z 1 (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12)
 (cid:12)c b (cid:12) 
Now, the L2 norm of this quantity is bounded by
τη
1
2 2
k k ηk ηk
ψˆ 1 2u (k , 1 ω)v (k , 2 ω)
 ∗τ1 kω 0 1 k ± 0 2 k ∓ 
kX1>0(cid:13)(cid:13) (cid:16) (cid:17)(cid:13)(cid:13)L2τη
 (cid:13) c b (cid:13)  1
(cid:13) (cid:13) 2
= ψ eit(φ0(k1)+φ0(k2))eikt1∂y2 xu0(k1, ) eikt2∂y2 xv0(k2, ) 2 ,
 F · F · L2 
kX1>0(cid:13) (cid:13) ty
(cid:13) (cid:13)
 (cid:13) (cid:13) 
where the equality is due to Plancherel’s theorem, applied to the t,y variables only. By H¨older
(9) ψ eit(φ0(k1)+φ0(k2))eikt1∂y2 xu0(k1, ) eikt2∂y2 xv0(k2, )
F · F · L2
ty
(cid:13) (cid:13)
(cid:13) i t ∂2 (cid:13) i t ∂2
(cid:13) ≤ kψkL4tke k1 yFxu0(k(cid:13)1,·)kL4tL∞y ke k2 yFxv0(k2,·)kL∞t L2y.
The partial Fourier transform of a free solution with respect to the periodic x variable only
x
F
x eitφ(D)u0 (k,y) = eitφ0(k)eikt∂y2 xu0(k,y),
F F
(cid:0) (cid:1)
is, for every fixed k, a solution of the homogeneous linear Schro¨dinger equation with respect
to the nonperiodic y variable and the rescaled time variable s := t, multiplied by a phase
k
factor of absolute value one. So, for the second factor on the right hand side of (9) we use the
endpoint Strichartz inequality for the one-dimensional Schro¨dinger equation, thus producing
|k1|14kFxu0(k1,·)kL2y, where the k1 factor comes from dt = k1ds in L4t. By conservation of the
10 A.GRU¨NROCK,M.PANTHEE,ANDJ.DRUMONDSILVA
L2 norm, the last factor is nothing but v (k , ) . We thus get
y kFx 0 2 · kL2y
1
2
ψ eit(φ0(k1)+φ0(k2))eikt1∂y2 xu0(k1, ) eikt2∂y2 xv0(k2, ) 2
 F · F · L2 
kX1>0(cid:13) (cid:13) ty
(cid:13) (cid:13)
 (cid:13) 1 (cid:13) 
2
. |k1|21kFxu0(k1,·)k2L2ykFxv0(k2,·)k2L2y . kDx14u0kL2xykv0kL2xy.
kX1>0
 
Finally, when τ τ1 > 1 τ τ > k, we exploit the useof thecutoff function; the estimate
| −k | ⇒ | − 1|
1
ψˆ(τ τ ) . ,
| − 1 | τ τ kβ
1
h − i
is valid, for arbitrarily large β, because ψ (R) (with the inequality constant depending only
∈ S
on ψ and β). Fixing any such β > 1, we can write
k k ηk ηk
Z kX1>0|ψˆ(τ −τ1)|kω1(τ21)(cid:12)u0(k1, k1 ±ω(τ1))v0(k2, k2 ∓ω(τ1))(cid:12)χ{ω(τ1)2≤k1k2}χ{|τ−kτ1|>1}dτ1
(cid:12) (cid:12)
1 1 (cid:12)ck1k2 ηk1 b ηk2 (cid:12)
. u (k , ω(τ ))v (k , ω(τ )) χ dτ .
Z hτ −τ1i kX1>0(k1k2)β/2kω(τ1)(cid:12) 0 1 k ± 1 0 2 k ∓ 1 (cid:12) {ω(τ1)2≤k1k2} 1
(cid:12) (cid:12)
(cid:12)c b (cid:12)
The L2 norm of this quantity is bounded, using the same convolution estimate as before, by
τ
1 k k ηk ηk
1 1 2 u (k , 1 ω(τ))v (k , 2 ω(τ)) χ dτ
kh·i− kL2τ Z kX1>0(k1k2)β/2kω(τ)(cid:12) 0 1 k ± 0 2 k ∓ (cid:12) {ω(τ)2≤k1k2}
(cid:12) (cid:12)
(cid:12)c 1 b ηk1 (cid:12) ηk2
. u (k , ω)v (k , ω) dω,
0 1 0 2
(k k )β/2 k ± k ∓
kX1>0 1 2 Zω≤√k1k2(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)c b (cid:12)
wherewe have doneagain thechange of variables of integration dτ = 2ωkdω. ApplyingH¨older’s
k1k2
inequality to the integral, we then get
1/2
1 ηk ηk 2
k k 1/4 u (k , 1 ω)v (k , 2 ω) dω
1 2 0 1 0 2
(k k )β/2| | k ± k ∓
kX1>0 1 2 (cid:18)Z (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12)
(cid:12)c b ηk(cid:12) ηk 2 1/2
1 2
. u (k , ω)v (k , ω) dω ,
0 1 0 2
k ± k ∓
(cid:18)kX1>0Z (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12)
(cid:12)c b (cid:12)
valid for our initial choice of β. The proof is complete, once we take the L2 norm of this last
kη
formula, which is obviously bounded by u v . (cid:3)
0 L2 0 L2
k k xyk k xy