Well-posedness of the Fifth Order Kadomtsev-Petviashvili I Equation in Anisotropic Sobolev Spaces with Nonnegative Indices

0.24 MB·English

by  Junfeng Li

Well-posedness of the Fifth Order
Kadomtsev-Petviashvili I Equation in
8
0
0 Anisotropic Sobolev Spaces with
2
Nonnegative Indices∗
n
a
J
Junfeng Li
4
1 School of Mathematical Sciences

Laboratory of Math and Complex Systems, Ministry of Education
]
P Beijing Normal University, Beijing 100875, P. R. China
A
Email: junfl[email protected]
.
h
t
a Jie Xiao
m Department of Mathematics and Statistics
[ Memorial University of Newfoundland, St John’s, NL AIC 5S7, Canada

1 Email: [email protected]
v
9
2
1
Abstract
2
1. Inthispaperweestablish thelocal andglobal well-posednessof therealvalued
0 fifth order Kadomstev-Petviashvili I equation in the anisotropic Sobolev spaces
8 with nonnegative indices. In particular, our local well-posedness improves Saut-
0 Tzvetkov’s one and our global well-posedness gives an affirmative answer to Saut-
: Tzvetkov’sL2-dataconjecture.
v
i
X
Key Words: Fifth KP-I equation, anisotropic Sobolev space, Bourgain space, dyadic
r decomposed Strichartz estimate, smoothing effect.
a
2000 Mathematics Subject Classification: 35Q53, 35G25.
1 Introduction
In their J. Math. Pures Appl. (2000) paper on the initial value problem (IVP) of the
real valued fifth order Kadomtsev-PetviashviliI (KP-I) equation (for (α,t,x,y)∈R4):
∂ u+α∂3u+∂5u+∂−1∂2u+u∂ u=0,
t x x x y x (1)
u(0,x,y)=φ(x,y),
(cid:26)
J.C. Saut and N. Tzvzetkov obtained the following result (cf. [16, Theorems 1 & 2]):
∗Thisprojectwascompletedwhenthefirst-namedauthorvisitedMemorialUniversityofNewfound-
land under the financial support from the NNSF of China No.10626008 as well as the second-named
author’sNSERC(Canada) grantandDeanofScience(MUN,Canada)Start-upfund.
1
Saut-Tzvzetkov’s Theorem (i) The IVP (1) is locally well-posed for initial data φ
satisfying
φˆ(ξ,η)
kφkL2(R2)+ |−i∂x|sφ L2(R2)+ |−i∂y|kφ L2(R2) <∞withs−1, k ≥0; |ξ| ∈S′(R2).
(cid:13) (cid:13) (cid:13) (cid:13) (2)
(cid:13) (cid:13) (cid:13) (cid:13)
(ii) The IVP (1) is globaly well-posed for initial data φ satisfying
1 α 1 1
kφkL2(R2) <∞; 2ZR2|∂x2φ|2+ 2 ZR2|∂xφ|2+ 2ZR2|∂x−1∂yφ|2− 6ZR2φ3 <∞. (3)
Here and henceforth, |−i∂ |s and |−i∂ |s are defined via the Fourier transform:
x y
|−\i∂ |sφ(ξ,η)=|ξ|sφˆ(ξ,η) and |−\i∂ |sφ(ξ,η)=|η|sφˆ(ξ,η).
x y
Since they simultaneously found in [16, Theorem 3] that the condition
kφkL2(R2) <∞; |ξ|−1φˆ(ξ,η)∈S′(R2) (4)
ensures the gobal well-posedness for the real valued fifth order Kadomtsev-Petviashvili
II (KP-II) equation (for (α,t,x,y)∈R4):
∂ u+α∂3u−∂5u+∂−1∂2u+u∂ u=0,
t x x x y x (5)
u(0,x,y)=φ(x,y),
(cid:26)
they made immediately a conjecture in [16, Remarks,p. 310]whichis now reformulated
in the following form:
Saut-Tzvzetkov’s L2-data Conjecture The IVP (1) is globally well-posed for initial
data φ satisfying (4).
In the above and below, as “localwell-posedness”we refer to finding a Banachspace
(X,k·k ) – when the initial data φ ∈ X there exists a time T depending on kφk
X X
such that (1) has a unique solution u in C([−T,T];X)∩Y (where Y is one of the Bour-
gain spaces defined in Section 2) and u depends continuously on φ (in some reasonable
topology). If this existing time T can be extended to the positive infinity, then “local
well-posedness” is said to be “global well-posedness”. Of course, the choice of a Banach
space relies upon the boundedness of the fundamental solution to the corresponding
homogenous equation or the conservation law for equation itself.
In our current paper, we settle this conjecture through improving the above-cited
Saut-Tzvzetkov’s theorem. More precisely, we have the following:
Theorem 1.1 The IVP(1)is not onlylocally butalsoglobally well-posed forinitial data
φ satisfying
φ∈Hs1,s2(R2) with s ,s ≥0; |ξ|−1φˆ(ξ,η)∈S′(R2). (6)
1 2
Here and henceafter, the symbol
Hs1,s2(R2)= f ∈S′(R2): kfkHs1,s2(R2) = (1+|ξ|2)s21(1+|η|2)s22fˆ(ξ,η) L2(R2) <∞
n (cid:13) (cid:13) o
stands for the anisotropic Sobolev space with(cid:13) nonnegative indices s1,s2 ∈(cid:13)[0,∞). Obvi-
ously,ifs1 =s2 =0thenHs1,s2(R2)=L2(R2)andhence(6)goesbackto(4)whichmay
be regarded as the appropriate constraint on the initial data φ deriving the global well-
posednedness of the IVP for the fifth order KP-I equation. And yet the understanding
of Theorem 1.1 is not deep enough without making three more observations below:
2
• Observation 1 The classificationof the fifth order KP equations is determined by
the dispersive function:
µ2
ω(ξ, µ)=±ξ5−αξ3+ , (7)
ξ
where the signs ± in (7) produce the fifth order KP-I and KP-II equations respectively.
The forthcoming estimates play an important role in the analysis of the fifth order KP
equations – for the fifth order KP-I equation, we have
µ2 µ
|ξ|2 >|α|⇒|∇ω(ξ,µ)|= 5ξ4+3αξ2− , 2 &|ξ|2; (8)
ξ2 ξ
(cid:12)(cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
and for the fifth order KP-II equation, w(cid:12)e have (cid:12)
µ2 µ
|ξ|2 >|α|⇒|∇ω(ξ,µ)|= 5ξ4+3αξ2+ , 2 &|ξ|4. (9)
ξ2 ξ
(cid:12)(cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
By (9), we can get more smooth effect(cid:12)estimates than by (8).(cid:12)These imply that we
can get a well-posedness (in other words, a lower regularity) for the fifth order KP-II
equation better than that for the fifth order KP-I equation. Another crucial concept is
the resonance function:
R(ξ ,ξ ,µ ,µ )
1 2 1 2
=ω(ξ +ξ ,µ +µ )−ω(ξ ,µ )−ω(ξ ,µ )
1 2 1 2 1 1 2 2 (10)
ξ ξ µ µ 2
= 1 2 (ξ +ξ )2 5(ξ2+ξ ξ +ξ2)−3α ∓ 1 − 2 .
(ξ +ξ ) 1 2 1 1 2 2 ξ ξ
1 2 (cid:18) h i (cid:16) 1 2(cid:17) (cid:19)
Evidently, the fifth order KP-II equation (corresponding to “+” in (10)) always enjoys
4
|R(ξ ,ξ ,µ ,µ )|& max{|ξ |,|ξ |,|ξ +ξ |} min{|ξ |,|ξ |,|ξ +ξ |}. (11)
1 2 1 2 1 2 1 2 1 2 1 2
Nevertheless,this lastinequa(cid:0)lity (11) is no longer true(cid:1)for the fifth order KP-I equation.
In the foregoing and following the notation A.B (i.e., B &A) means: there exists
a constant C >0 independent of A and B such that A≤CB. In addition, if there exist
two positive constants c and C such that 10−3 < c < C < 103 and cA ≤ B ≤ CB then
the notation A∼B will be used.
• Observation 2 Perhaps it worths pointing out that the well-posedness of the fifth
order KP-II equation is relatively easier to establish but also its result is much better
than that of the fifth orderKP-I equation. Although the study of the well-posedness for
the fifth order KP-II equation (without the third order partial derivative term) usually
focuses on the critical cases (which means s +2s = −2 by a scaling argument), in
1 2
[15] Saut and Tzvetkov only obtained the local well-posedness for the fifth order KP-
II equation in the anisotropic Sobolev space Hs1,s2(R2) with s1 > −41,s2 ≥ 0 with a
modificationofthelowfrequency,andfurthermorein[16]theyremovedthismodification
andobtainedtheglobalwell-posednessinL2(R2). Ontheotherhand,in[8]Isaza-L´opez-
Mej´ıa established the local well-posedness for Hs1,s2(R2) with s1 > −45,s2 ≥ 0 and the
globalwell-posednessfor Hs1,s2(R2)withs1 >−47,s2 ≥0. Morerecently,Hadac [4]also
gained the same local well-posedness in a broader context. Meanwhile in the fifth order
KP-Iequationcase,the attentionis mainly paidonthosespacespossessingconservation
law such as L2(R2) and the energy space
E1(R2)= f ∈L2(R2): (1+|ξ|2+|ξ|−1|µ|)fˆ(ξ,µ) <∞ .
L2(R2)
n (cid:13) (cid:13) o
(cid:13) (cid:13)
3
Toobtainthelocalwell-posednessofKP-IinE1(R2),in[16],besidestheabove-mentioned
results SautandTzvekovalsogotthe localwell-posednessinH˜s,k(R2) withs−1,k ≥0.
H˜s,k(R2)= f ∈L2(R2): (1+|ξ|s+|ξ|−1|η|k)fˆ(ξ,η) <∞ .
L2(R2)
For the energy case nH˜2,1(R2)=E1(R(cid:13)(cid:13)2), they obtained the global(cid:13)(cid:13)well-posednoess of (1).
In [5], Ionescu and Kenig got the global well-posedness for the fifth order periodic KP-I
equation(without the thirdorderdispersiveterm)inthe standardenergyspaceE1(R2).
Recently, in [3] Chen-Li-Miao obtained the local well-posedness in
Es(R2)= f ∈L2(R2): k(1+|ξ|2+|ξ|−1|µ|)sfˆ(ξ,µ)kL2(R2) <∞ , 0<s≤1.
n o
• Observation 3 The well-posednessfor the IVP of the third order KP equations in
R3:
∂ u∓∂3u+∂−1∂2u+u∂ u=0,
t x x y x (12)
u(0,x,y)=φ(x,y),
(cid:26)
inwhichthe sign∓givethethirdorderKP-IandKP-IIequationsrespectively,isanim-
portantbackgroundmaterialofthe investigationofthe well-posednessforthe fifthorder
KP equations. Molinet, Saut and Tzvetkov showed in [13, 14] that, for the third order
KP-Iequationonecannotobtainthelocalwell-posednessinanytypeofnonisotropicL2-
based Sobolev space or in the energy space using Picard’s iteration – see also [12]; while
I´orio and Nunes [7] applied a compactness method to deduce the local well-posednes for
the third KP-I equation with data being in the normal Sobolev space Hs(R2), s > 2
and obeying a “zero-mass” condition. On the other hand, the global well-posedness for
the thirdorderKP-Iequationwasdiscussedby using the classicalenergymethodin [10]
where Kenig established the global well-posedness in
f ∈L2(R2): kfkL2(R2)+k∂x−1∂yfkL2(R2)+k∂x2fkL2(R2)+k∂x−2∂y2fkL2(R2) <∞ .
n o
As far as we know,the best well-posedresult on the third order KP-I equation is due to
Ionescu, Kenig and Tataru [6] which gives the global well-posedness for the third order
KP-I equation in the energy space
f ∈L2(R2): kfkL2R2 +k∂x−1∂yfkL2(R2)+k∂xfkL2(R2) <∞ .
n o
Relatively speaking, the results on the third order KP-II equation are nearly perfect.
In [2], Bourgain proved the global well-posedness of the third order KP-II equation in
L2(R2)–theassertionwasthenextendedbyTakaokaandTzvetkov[18]andIsaza-Mej´ıa
[9] fromL2(R2) to Hs1,s2(R2) with s1 >−31, s2 ≥0. In [17], Takaoka obtainedthe local
well-posedness for the third order KP-II equation in Hs1,s2(R2) with s1 > −21, s2 = 0
under an additionallow frequency condition |−i∂x|−21+εφ∈L2(R2) whichwas removed
successfully inHadac’srecentpaper[4]. These resultsareverycloseto the criticalindex
s +2s =−1 which follows from the scaling argument.
1 2 2
The rest of this paper is devoted to an argument for Theorem 1.1. In Section 2 we
collectsomeusefulandbasicallyknownlinearestimatesforthefifthorderKP-Iequation.
InSection3wepresentthenecessaryandcrucialbilinearestimatesinordertosetupthe
local (and hence global) well-posedness – this part is partially motivated by [16] though
– the main difference between their treatmentand ours is how to dispose the “high-high
interaction” – their method exhausts no geometric structure of the resonant set of the
fifth order KP-I equation while ours does fairly enough. In Section 4 we complete the
argument through applying the facts verified in Sections 2 and 3 and Picard’s iteration
principle to the integral equation corresponding to (1).
4
2 Linear Estimates
We begin with the IVP of linear fifth order KP-I equation:
∂ u+α∂3u+∂5u+∂−1∂2u=0,
t x x x y (13)
u(0,x,y)=φ(x,y).
(cid:26)
By the Fourier transform (·), the solution of (13) can be defined as
u(t)(x,y)=cS(t)φ (x,y)= ei(xξ+yµ+tω(ξ,µ))φ(ξ,µ)dξdµ.
ZR2
(cid:0) (cid:1)
By Duhamel’s formula, (1) can be reduced to the integral reprbesentation below:
1 t
u(t)=S(t)φ− S(t−t′)∂ (u2(t′))dt′. (14)
x
2
Z0
So, in order to get the locall well-posedenss we will apply a Picardfixed point argument
in a suitable function space to the following integral equation:
ψ (t) t
u(t)=ψ(t)S(t)φ− T S(t−t′)∂ (u2(t′))dt′, (15)
x
2
Z0
where t belongs to R, ψ is a time cut-off function satisfying
ψ ∈C∞(R); suppψ ⊂[−2, 2]; ψ =1 on [−1, 1],
0
and ψ (·) represents ψ(·/T) for a given time T ∈ (0,1). Consequently, we need to
T
define an appropriate Bourgain type space, which is associated with the fifth order KP-
I equation. To this end, for s ,s ≥ 0 and b ∈ R the notation Xs1,s2 is used as the
1 2 b
Bourgain space with norm:
kukXs1,s2 =k<τ −ω(ξ,µ)>b<ξ >s1<µ>s2 uˆ(τ,ξ,µ)kL2(R3),
b
where < · > stands for (1+|·|2)12 ∼ 1+|·|. Furthermore, for an interval I ⊂ R the
localized Bourgain space Xs1,s2(I) can be defined via requiring
b
kukXbs1,s2(I) =w∈Xinsf1,s2 kwkXbs1,s2 : w(t)=u(t) on interval I .
b
(cid:8) (cid:9)
The following two results are known.
Proposition 2.1 [16] If
1
T ∈(0,∞); s ,s ≥0; − <b′ ≤0≤b≤b′+1,
1 2
2
then
kψS(t)φkXs1,s2 .kφkHs1,s2(R2). (16)
b
t
(cid:13)ψ(t/T)Z0 S(t−t′)h(t′)dt′(cid:13)Xbs1,s2 .T1−b+b′khkXbs′1,s2. (17)
for any khkXbs′1,s(cid:13)(cid:13)2 <∞. (cid:13)(cid:13)
Proposition 2.2 [1] If r ∈ [2,∞), then there exists a constant c > 0 independent of
T ∈(0,1) such that
|−i∂x|21−r1 S(t)φ (x,y) Lr2−r2Lr(R2) ≤ckφkL2(R2), (18)
T
(cid:13) (cid:0) (cid:1) (cid:13)
(cid:13) (cid:13)
5
where
r−2
T r−22 2r
kfk = |f(x,y,t)|rdxdy dt .
2r
LTr−2Lr(R2) Z−T (cid:18)ZR2 (cid:19) !
Toreachourbilinear inequalities inSection 3,we will use (·)∨ for the inverseFourier
transform, and take the dyadic decomposed Strichartz estimates below into account.
Proposition 2.3 Let η be a bump function with compact support in [−2,2] ⊂ R and
η = 1 on (−1,1) ⊂ R. For each integer j ≥ 1 set η (x) = η(2−jx)−η(21−jx), η (x) =
j 0
η(x), η (ξ,µ,τ)=η (τ−ω(ξ,µ)), and f (ξ,µ,τ)=(η (ξ,µ,τ)|fˆ|(ξ,µ,τ))∨ for any given
j j j j
f ∈L2(R3). Then for given r∈[2,∞) and any T ∈(0,1) we have
|−i∂x|12−r1fj Lr2−r2Lr(R2) .22jkfjkL2(R3). (19)
T
(cid:13) (cid:13)
(cid:13) (cid:13)
In particular,
|−i∂x|41fj L4L4(R2) .22jkfjkL2(R3). (20)
T
(cid:13) (cid:13)
Proof : Note first that (cid:13) (cid:13)
f (x,y,t)= ei(xξ+yµ+tτ)|fˆ|η (ξ,µ,τ)dξdµdτ.
j j
ZR3
So, changing variables and using f (ξ,µ)=|fˆ|(ξ,µ,λ+ω) we can write
λ
f (x,y,t)= ei(xξ+cyµ+t(λ+ω))|fˆ|(ξ,µ,λ+ω)η (λ)dξdµdλ
j j
ZR3
= eitλη (λ) ei(xξ+yµ+tω)|fˆ|(ξ,µ,λ+ω)dξdµ dλ
j
ZR hZR2 i
= eitλη (λ)S(t)f (x,y)dλ.
j λ
R
Z
Now the estimate (19) follows from Minkowski’s inequality, the Strichartz estimate (18)
and the Cauchy-Schwarz inequality.
The following well-known elementary inequalities are also useful – see for example
[16, Proposition 2.2].
Proposition 2.4 Let γ >1. Then
dt
.<a>−γ (21)
R <t>γ<t−a>γ
Z
and
dt
.<a>−12 (22)
ZR <t>γ |t−a|21
hold for any a∈R
6
3 Bilinear Estimates
Although there weremany worksonthe so-calledbilinear estimates, we havefound that
the Kenig-Ponce-Vega’sbilinear estimation approachintroduced in [11] is quite suitable
for our purpose. With the convention: when a ∈ R the number a± equals a±ǫ for
arbitrarily small number ǫ>0, we can state our bilinear estimate as follows.
Theorem 3.1 If s ,s ≥ 0 and functions u,v have compact time support on [−T,T]
1 2
with 0<T <1, then
k∂x(uv)kXs1,s2 .kukXs1,s2kvkXs1,s2. (23)
−21+ 21+ 12+
Proof In what follows, we derive (23) using the duality; that is, we are required to
dominate the integral
|ξ|<ξ >s1<µ>s2
ZA∗ <τ −ω(ξ,µ)>12−g(ξ,µ,τ)|uˆ|(ξ1,µ1,τ1)|vˆ|(ξ2,µ2,τ2)dξ1dµ1dτ1dξ2dµ2dτ2, (24)
where g ≥0, kgkL2(R2) ≤1 and
A∗ = (ξ ,µ ,τ ,ξ ,µ ,τ )∈R6 : ξ +ξ =ξ, µ +µ =µ, τ +τ =τ .
1 1 1 2 2 2 1 2 1 2 1 2
Let (cid:8) (cid:9)
σ =τ −ω(ξ,µ); σ =τ −ω(ξ ,µ ); σ =τ −σ(ξ ,µ ).
1 1 1 1 2 2 2 2
Define two functions below:
f1(ξ1,µ1,τ1)=<ξ1 >s1<µ1 >s2<σ1 >21+ |uˆ(ξ1,µ1,τ1)|
and
f2(ξ2,µ2,τ2)=<ξ2 >s1<µ2 >s2<σ2 >12+ |vˆ(ξ2,µ2,τ2)|.
Then we need to bound the integral
K(ξ ,µ ,τ ,ξ ,µ ,τ )g(ξ,µ,τ)f (ξ ,µ ,τ )f (ξ ,µ ,τ )dξ dµ dτ dξ dµ dτ (25)
1 1 1 2 2 2 1 1 1 1 2 2 2 2 1 1 1 2 2 2
ZA∗
from above by using a constant multiple of kf1kL2(R3)kf2kL2(R3). Here
|ξ +ξ |
1 2
K(ξ ,µ ,τ ,ξ ,µ ,τ ) =
1 1 1 2 2 2 (cid:18)<σ >12−<σ1 >21+<σ2 >21+(cid:19)
<ξ1+ξ2 >s1 <µ1+µ2 >s2
× .
(cid:18)<ξ1 >s1<ξ2 >s1(cid:19)(cid:18)<µ1 >s2<µ2 >s2(cid:19)
It is clear that for s ,s ≥0 we always have
1 2
|ξ +ξ |
K(ξ ,µ ,τ ,ξ ,µ ,τ ). 1 2 .
1 1 1 2 2 2 <σ >21−<σ1 >21+<σ2 >21+
Keeping a further assumption |ξ |≥|ξ | (which follows from symmetry) in mind, we
1 2
are about to fully control the integral in (25) through handling two situations.
• Situation 1 – Low Frequency |ξ +ξ |.max{10,|α|}.
1 2
◦ High+High→Low |ξ |,|ξ |&max{10,|α|}. We first deduce a dyadic decompo-
1 2
sition. Employing η in Proposition 2.3, we have η = 1, and consequently (25)
j j≥0 j
can be bounded from above by a constant multiple of
P
f (ξ ,µ ,τ ) f (ξ ,µ ,τ )
j≥02−j(21−)ZA∗ηj(σ)g(ξ,µ,τ)(cid:18) 1<σ11 >121+1 (cid:19)(cid:18) 2<σ22 >221+2 (cid:19)dξ1dµ1dτ1dξ2dµ2dτ2.
X
(26)
7
We may assume that for each natural number j,
G (x,y,t)=F−1 η (σ)g(ξ,µ,τ) (x,y,t),
j j
(cid:16) (cid:17)
has support compact in the interval [−T,T] whenever it acts as a time-dependent func-
tion, where F−1 also denotes the inverse Fourier transform. In fact, if we consider the
following functions generated by F−1:
f (ξ ,µ ,τ )
F (x,y,t)=F−1 l l l k (x,y,t) for l=1,2,
l <σl >12+ !
thentheintegralin(26)canbewrittenasanL2 innerproducthG ,F F i. Sinceuandv
j 1 2
(actingastime-dependentfunctions)havecompactsupportin[−T,T],sodoesF F . As
1 2
a result, the inner product hG ,F F i can be restricted on the interval [−T,T], namely,
j 1 2
we mayassumethat G has the same compactsupport(with respectto time) asF F ’s.
j 1 2
Now,anapplicationof(20)yieldsthatthesumin(26)isboundedbyaconstantmultiple
of
2−j(12−)hGj,F1F2i
j≥0
X
. 2−j(21−)2−j1(12+)2−j2(21+)
j,j1X,j2≥0(cid:16)
× |−i∂x|41(ηj1(σ1)f1)∨ L4L4(R2) |−i∂x|14(ηj2(σ2)f2)∨ L4L4(R2)kηj(σ)gkL2(R3)
T T
. (cid:13)(cid:13) 2−j(21−)2−j1[(12+(cid:13)(cid:13))−21]2−j2[(cid:13)(cid:13)(12+)−21] (cid:13)(cid:13) (cid:17)
j,j1X,j2≥0(cid:16)
×kηj1(σ1)f1kL2(R3)kηj2(σ2)f2kL2(R3)kηj(σ)gkL2(R3)
.kf1kL2(R3)kf2kL2(R3). (cid:17)
◦ Low+Low→Low |ξ |,|ξ |.max{15,|α|}. Viachangingvariablesandusingthe
1 2
Cauchy-Schwarzinequality we can bound (25) with
1
2
K |f (ξ ,µ ,τ )f (ξ−ξ ,µ−µ ,τ −τ )|2dτ dξ dµ g(ξ,µ,τ)dξdµdτ,
ll 1 1 1 1 2 1 1 1 1 1 1
Z (cid:18)Z (cid:19)
where
1
|ξ| dτ1dξ1dµ1 2
K = .
ll <σ >21− (cid:18)Z <τ1−ω(ξ1,µ1)>1+<τ −τ1−ω(ξ−ξ1,µ−µ1)>1+(cid:19)
We need only to control K using a constant independent of ξ,µ,τ. By (21) we have
ll
1
K . |ξ| dξ1dµ1 2 .
ll <σ >21− (cid:18)Z <τ −ω(ξ,µ)−ω(ξ−ξ1,µ−µ1)>1+(cid:19)
An elementary computation with the change of variables:
ν =τ −ω(ξ,µ)−ω(ξ−ξ ,µ−µ )
1 1
shows
dν
dµ &|ξ|21|σ+ξξ1(ξ−ξ1)(5ξ2−5ξξ1+5ξ12−3α)−ν|21
1
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8
and consequently,
1
K . |ξ|34 dξ1dν 2 .
ll <σ >21− Z <ν >1+ |σ+ξξ1(ξ−ξ1)(5ξ2−5ξξ1+5ξ12−3α)−ν|12!
By (22) we further get
1
dξ 2
K . 1 .1.
ll Z|ξ1|.max{15,|α|} <σ+ξξ1(ξ−ξ1)(5ξ2−5ξξ1+5ξ12−3α)>21!
• Situation 2 – High Frequency |ξ +ξ |&max{10,|α|}.
1 2
◦ High+Low→High |ξ | . max{10,|α|} . |ξ| ∼ |ξ |. As above, we apply the
2 1
Cauchy-Schwarz inequality to bound the integral in (25) from above with a constant
multiple of
1
2
K |f (ξ ,µ ,τ )f (ξ−ξ ,µ−µ ,τ −τ )|2dτ dξ dµ g(ξ,µ,τ)dξdµdτ
hl 1 1 1 1 2 1 1 1 1 1 1
Z (cid:18)Z (cid:19)
where
1
|ξ| dτ1dξ1dµ1 2
K = ,
hl <σ >21− (cid:18)Z <τ1−ω(ξ1,µ1)>1+<τ −τ1−ω(ξ−ξ1,µ−µ1)>1+(cid:19)
but also we have the following estimate
1
K . |ξ| dξ1dµ1 2 .
hl <σ >12− (cid:18)Z <τ −ω(ξ,µ)−ω(ξ−ξ1,µ−µ1)>1+(cid:19)
Under the change of variables
κ=ξξ (ξ−ξ )(5ξ2−5ξξ +5ξ2−3α); ν =τ −ω(ξ,µ)−ω(ξ−ξ ,µ−µ )
1 1 1 1 1 1
the Jacobian determinant J enjoys
|κ|12
J . .
|ξ|72|σ+κ−ν|12(|ξ|5−2|κ|)21
As a by-product of the last inequality and (22), we obtain
1
1 |κ|12 dκdν 2
K .
hl |ξ|34 <σ >21− Z |σ+κ−ν|21(|ξ|5−2|κ|)21 <ν >1+!
1
1 |κ|21 dκ 2
. .
|ξ|34 <σ >21− Z <σ+κ>21 (|ξ|5−2|κ|)12!
Since |ξ−ξ |.max{10,|α|}, we have |κ|.|ξ|4, whence getting
1
1
1 dκ 2
K . .1.
hl |ξ|2 <σ >12− Z|κ|.|ξ|4 <σ+κ>12!
◦ High+High→High |ξ |,|ξ | & max{10,|α|}. Since |ξ | ≥ |ξ |, we have |ξ | &
1 2 1 2 1
|ξ +ξ |. Under this circumstance, we will deal with two cases in the sequel.
1 2
9
⋄ Case (i) max{|σ|,|σ |} & |ξ |2. Decomposing the integral according to |ξ | ∼
2 1 1
2m where m=1,2,···, we can run the dyadic decomposition:
|σ|∼2j, |σ |∼2j1, |σ |∼2j2 for j,j ,j =0,1,2,....
1 2 1 2
If|σ|≥|σ |≥|ξ |2,thenanapplicationof(20)yieldsthattheintegralin(25)isbounded
2 1
from above by a constant multiple of
234m2−j(21−)2−j1(21+)2−j2(21+)kηj(σ)gkL2(R3)
mX≥1jX≥2mj1X,j2≥0(cid:16)
1 ∨ 1 ∨
× |−i∂x|4 ηm(ξ1)ηj1(σ1)f1 L4L4(R2) |−i∂x|4 ηj2(σ2)f2 L4L4(R2)
T T
. (cid:13)(cid:13) (cid:0) 2−j(12−)234m(cid:1)2−(cid:13)(cid:13)j1[(12+)−12(cid:13)(cid:13)]2−j2[(21+)−(cid:0)21] (cid:1) (cid:13)(cid:13) (cid:17)
mX≥1jX≥2mj1X,j2≥0(cid:16)
×kηj1(σ1)f1kL2(R3)kηj2(σ2)f2kL2(R3)kηj(σ)gkL2(R3)
.kf1kL2(R3)kf2kL2(R3). (cid:17)
If|σ |≥|σ|≥|ξ |2,thenafurtheruseof(20)derivesthattheintegralin(25)isbounded
2 1
from above by a constant multiple of
2m22−j(12−)2−j1(12+)2−j(21+)kηj2(σ2)f2kL2(R3)
mX≥1j2≥X2m,j≥0jX1≥0(cid:16)
× |−i∂x|41(ηm(ξ1)ηj1(σ1)f1)∨ L4L4(R2) |−i∂x|14(ηj(σ)g)∨ L4L4(R2)
T T
. (cid:13)(cid:13) 2−j2(14+)2m2 2(cid:13)(cid:13)−j1[(12+)−(cid:13)(cid:13)12]2−j[(43−)−21] (cid:13)(cid:13) (cid:17)
mX≥1j2X≥2mj1X,j2≥0(cid:16)
×kηj1(σ1)f1kL2(R3)kηj2(σ2)f2kL2(R3)kηj(σ)gkL2(R2)
.kf1kL2(R3)kf2kL2(R3). (cid:17)
⋄ Case (ii) max{|σ|,|σ |} . |ξ |2. In this case, we need to consider the size of
2 1
the resonancefunction evenmore carefully. This considerationwill be done via splitting
the estimate into two pieces according to the size of resonance function.
⊲ Subcase (i) max{|σ|,|σ |,|σ |} & |ξ |4. This means that the resonant interac-
1 2 1
tion does not happen and consequently |σ |&|ξ |4. The dyadic decomposition and (20)
1 1
are applied to deduce that the integral in (25) is bounded from above by a constant
multiple of
243m2−j(21−)2−j1(12+)2−j2(21+)kηm(ξ1)ηj1(σ1)f1kL2(R3)
mX≥1j1X≥4m2m≥Xj,j2≥0(cid:16)
× |−i∂x|14(ηj2(σ2)f2)∨ L4L4(R2) |−i∂x|14(ηj(σ)g)∨ L4L4(R2)
T T
. (cid:13)(cid:13) 234m(cid:13)(cid:13)2−j1(14+)(cid:13)(cid:13)2−j[(43−)−12]2−j2[(21+(cid:13)(cid:13))−21] (cid:17)
mX≥1j1X≥4m2m≥Xj,j2≥0(cid:16)
×kηj1(σ1)f1kL2(R3)kηj2(σ2)f2kL2(R3)kηj(σ)gkL2(R3)
.kf1kL2(R3)kf2kL2(R3). (cid:17)
⊲ Subcase (ii) max{|σ|,|σ |,|σ |} . |ξ |4. This means that the resonant interac-
1 2 1
tion does happen. By the definition of the resonant function we have
µ µ 2
1 − 2 >2−1|ξ +ξ |2|5(ξ2+ξ ξ +ξ2)−3α|.
ξ ξ 1 2 1 1 2 2
1 2
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
10