Global solutions with infinite energy for the 1-dimensional Zakharov system

0.21 MB·English

by  Hartmut Pecher

Global solutions with infinite energy for the
5 1-dimensional Zakharov system
0
0
2
Hartmut Pecher

n
Fachbereich Mathematik und Naturwissenschaften
a
J Bergische Universit¨at Wuppertal

1 Gaußstr. 20
1
D-42097 Wuppertal

] Germany
P
e-mail [email protected]
A
.
h
t
a Abstract
m
The 1-dimensional Zakharov system is shown to have a unique global
[
solution for data without finite energy. The proof uses the ” I-method ”

5
introduced by Colliander, Keel, Staffilani, Takaoka, and Tao in connection
v
with a refined bilinear Strichartz estimate.
4
0
2
0 Introduction
9
0
4
Consider the (1+1)-dimensional Cauchy problem for the Zakharov system
0
/
h iu +u = nu (1)
t xx
t
a n −n = (|u|2) (2)
m tt xx xx
u(0) = u , n(0) = n , n (0) = n (3)
: 0 0 t 1
v
i
X where u is a complex-valued und n a real-valued function defined for (x,t) ∈
r R×R+.
a
The Zakharov system was introduced in [Z] to describe Langmuir turbulence
in a plasma.
Our main result is the existence of a unique global solution for data without
finite energy, more precisely we assume u ∈ Hs(R) , n ∈ Hs 1(R) , A 1/2n ∈
0 0 − − 1
Hs 1(R) , where 1 > s > 5/6 , A:= − d2 .
− dx2
Thisresultcanbeprovenbyusingtheconservationlaws, namelyconservation
of ku(t)k and
1
E(u,n) := ku (t)k2+ (kn(t)k2 +kA 1/2n (t)k2)+ ∞ n(t)|u(t)|2dx
x − t
2 Z
−∞
although under our assumptions these quantities are not finite, in general.
Results of this type were given in various situations in the last years in the
framework of the Fourier restriction norm method in most of the applications.
One approach is to use Bourgain’s trick to split the data into high and low
1
frequency parts. He used it to prove global well-posedness for the (2+1)- and
(3+1)-dimensional Schro¨dinger equations with rough data without finite energy
[B1],[B2] and for the wave equation [B3]. Later it was also used for other model
equations [CST],[FLP],[KT],[KPV],[P2]. Concerning the problem at hand the
author had been able to show global well-posedness for data (u ,n ,n ) ∈ Hs ×
0 0 1
L2 × H˙ 1 for 1 > s > 9/10 [P1]. Remark here that no data n 6∈ L2 were
− 0
admissible because in such a case the nonlinear part of n(t) could not be shown
to belong to L2 which is necessary for this method. In contrast, the approach
here allows data n 6∈L2 and also less regular data u .
0 0
Another approach was initiated by Colliander, Keel, Staffilani, Takaoka and
Tao in [CKSTT3], called the I-method. The main idea is to use a modified en-
ergy functional which is also defined for less regular functions and not strictly
conserved. When one is able to control its growth in time explicitly this al-
lows to iterate a modified local existence theorem to continue the solution to
any time T and moreover to estimate its growth in time. This method was suc-
cessfully applied by these authors to several equations which have a scaling in-
variance with sometimes even optimal global well-posedness results. It was used
in [CKSTT3] to improve Bourgain’s global well-posedness results [B1],[B2] for
the (2+1)- and (3+1)-dimensional Schro¨dinger equation with a further improve-
ment in [CKSTT6]. Later it was applied to the (1+1)-dimensional derivative
Schro¨dinger equation [CKSTT1] with an (almost) optimal result in [CKSTT2]
and to the KdV and modified KdV equation with also optimal results in some
cases [CKSTT4],[CKSTT5].
Although in our situation such a scaling argument does not work we are able
to suitably modify the method to prove the above mentioned global existence
result for the Zakharov system.
The paper is organized as follows. We transform the system in the usual way
into a first order system. Then we apply the multiplier I for given s < 1 and
N
N >> 1 to it, where I f(ξ) := m (ξ)f(ξ). Here m (ξ) is a smooth, radially
N N N
symmetric, and nonincreasing function of |ξ|, defined by m (ξ) = 1 for |ξ| ≤ N
d b N
and m (ξ) = (N)1 s for |ξ| ≥ 2N. We drop N from the notation for short and
N ξ −
remark that I :|H| s → H1 is a smoothing operator in the following sense:
kuk ≤ ckIuk ≤ cN1 skuk
Xϕm,b Xϕm+1−s,b − Xϕm,b
Here we used the Xm,b - spaces which are defined as follows: for an equation of
ϕ
the form iu −ϕ(−i∂ )u = 0, where ϕ is a measurable function, let Xm,b be the
t x ϕ
completion of S(R×R) with respect to
kfk := khξimhτibF(eitϕ( i∂x)f(x,t))k
Xϕm,b − L2ξτ
= khξimhτ +ϕ(ξ)ibf(ξ,τ)k
L2
ξτ
b
For ϕ(ξ) = ±|ξ| we use the notation Xm,b and for ϕ(ξ) = |ξ|2 simply Xm,b.
For a given time interval I we define kfk± = inf kf˜k and similarly
Xm,b(I) f˜ =f Xm,b
|I
kfk .
Xm,b(I)
±
For the modified (by I multiplied) Zakharov system we then prove a local
existence theorem by usingtheprecise estimates given by [GTV]for thestandard
2
Zakharov system in connection with an interpolation type lemma in [CKSTT5].
Our aim is to extract a factor Tδ with maximal δ from the nonlinear estimates
in order to give an optimal lower bound for the local existence time T in terms
of the norms of the data. Because the difference of the differentiability classes of
the data is maximal (=1), one is forced also to use here the auxiliary spaces Ym
ϕ
(cf. [GTV]), defined by
kfkYϕm : = khξimhτi−1F(e−itϕ(−i∂x)f(x,t))kL2ξL1τ
= khξimhτ +ϕ(ξ)i 1f(ξ,τ)k
− L2L1
ξ τ
b
As is typical for the I-method one then has to consider in detail the modified
energy functional E(Iu,In) and to control its growth in time in dependence of
the time interval and the parameter N (cf. the definition of I above). The
increment of the energy has to be small for small time intervals and large N.
Because the modified energy functional is somehow close to the original one here
some sort of cancellation helps. An important tool is also a refined Strichartz
estimate for the product of a wave and a Schro¨dinger part along the lines of
Bourgain’s improvements for the simpler pure Schro¨dinger case (cf. Lemma 2.2).
This estimate for the modified energy functional can also control the growth of
the correspondingnorms of the solution of the problem duringits time evolution.
One iterates the local existence theorem with time steps of equal length in order
to reach any given fixed time T. To achieve this one has to make the process
uniform which can be done if s is close enough to 1 (namely s> 5/6).
We collect some elementary facts about the spaces Xm,b and Ym.
ϕ ϕ
The following interpolation property is well-known:
Xϕ(1−Θ)m0+Θm1,(1−Θ)b0+Θb1 = (Xϕm0,b0,Xϕm1,b1)[Θ] for Θ ∈[0,1].
If u is a solution of iu +ϕ(−i∂ )u = 0 with u(0) = f and ψ is a cutoff function
t x
in C (R) with suppψ ⊂ (−2,2) , ψ ≡ 1 on [−1,1] , ψ(t) = ψ(−t) , ψ(t) ≥ 0 ,
0∞
ψ (t) := ψ(t), 0 < δ ≤ 1, we have for b >0:
δ δ
kψ uk ≤ ckfk
1 Xϕm,b Hm
If v is a solution of the problem iv + ϕ(−i∂ )v = F , v(0) = 0 , we have for
t x
b +1≥ b ≥ 0≥ b > −1/2
′ ′
kψ vk ≤ cδ1+b′ bkFk
δ Xϕm,b − Xϕm,b′
and, if b +1 ≥ b ≥ 0 ≥ b, we have
′ ′
kψδvkXϕm,b ≤ c(δ1+b′−bkFkXϕm,b′ +δ12−bkFkYϕm)
(for a proof cf. [GTV], Lemma 2.1). Moreover, if w(t) = tei(t s)ϕ( i∂x)F(s)ds
0 − −
we have by [GTV], Lemma 2.2, especially (2.35), for δ ≤ 1R
kwk ≤ ckFk (4)
C0([0,δ],H1) Y1[0,δ]
x ϕ
Finally, if 1/2 > b >b ≥ 0 , m ∈ R , we have the embedding
′
kfk ≤ cδb b′kfk (5)
Xϕm,b′[0,δ] − Xϕm,b[0,δ]
3
For the convenience of the reader we repeat the proof of [G], Lemma 1.10. The
claimed estimate is an immediate consequence of the following
Lemma 0.1 For 1/2 > b > b ≥ 0 , 0 < δ ≤ 1 , m ∈ R the following estimate
′
holds:
kψ fk ≤ cδb b′kfk
δ Xϕm,b′ − Xϕm,b
Proof: The following Sobolev multiplication rule holds:
kfgk ≤ ckfk kgk
Htb′ H12−(b−b′) Htb
t
This rule follows easily by the Leibniz rule for fractional derivatives, using Js :=
F 1hτisF:
−
kfgk ≤ c(k(Jb′f)gk +kf(Jb′g)k )
Htb′ L2t L2t
≤ c(kJb′fkLptkgkLpt′ +kfkLqt′kJb′gkLqt)
with 1 = b , 1 = 1 −b , 1 = b−b , 1 = 1 −(b −b). Sobolev’s embedding
p p′ 2 q′ ′ q 2 ′
theorem gives the claimed result. Consequently we get
kψ gk ≤ ckψ k kgk ≤ cδb b′kgk
δ Htb′ δ H12−(b−b′) Htb − Htb
t
and thus
kψ fk = keitϕ( i∂x)ψ fk ≤ cδb b′keitϕ( i∂x)fk
δ Xϕm,b′ − δ Hxm⊗Htb′ − − Hxm⊗Htb
= cδb b′kfk
− Xϕm,b
FundamentalarethefollowinglinearStrichartztypeestimatesfortheSchro¨dinger
equation (cf. e.g. [GTV], Lemma 2.4):
keit∂x2ψkLq(I,Lr(R)) ≤ ckψkL2(R)
t x x
and
kukLqt(I,Lrx(R)) ≤ ckukX0,21+(I)
if 0≤ 2 = 1 − 1 , especially
q 2 r
kuk ≤ ckuk
L6xt X0,12+
which by interpolation with the trivial case kuk = kuk gives:
L2 X0,0
xt
kukLpxt ≤ ckukX0,32(12−p1)+
if 2< p ≤ 6. For the wave equation we only use kn k ≤ ckn k .
± L∞t L2x ± X0,21+
±
We use the notation hλi := (1+λ2)1/2. Let a± denote a number slightly larger
(resp., smaller) than a.
Acknowledgement: I thank Axel Gru¨nrock for very helpful discussions.
4
1 Local existence
The system (1),(2),(3) has the following conserved quantities:
ku(t)k =:M
and
+
E(u,n) := kA1/2u(t)k2+1/2(kn(t)k2 +kV(t)k2)+ ∞n(t)|u(t)|2dx
Z
−∞
where V := −n and A:= − d2 .
x t dx2
Thesystem(1),(2),(3)is nowtransformedinto afirstordersystem int as follows:
withn := n±iA 1/2n ,i.e. n= 1(n +n ),2iA 1/2n = n −n ,andn = n
− t 2 + − t + +
± − − −
this gives
1
iu +u = (n +n )u (6)
t xx +
2 −
in ∓A1/2n = ±A1/2(|u|2) (7)
t
± ±
u(0) = u , n (0) = n := n ±iA 1/2n (8)
0 0 0 − 1
± ±
The energy is given by
1 1
E(u,n )= kA1/2uk2+ kn k2+ (n +n )|u|2dx
+ + + +
2 2 Z
By Gagliardo-Nirenberg
1 1
n|u|2dx ≤ n2dx+c |u|4dx ≤ knk2+cku kkuk3
x
Z 4Z Z 4
1
≤ (knk2 +ku k2)+c kuk6
x 0
4
This easily implies
kA1/2uk2 +knk2 +kVk2 ≤ c(E +kuk6)= c (E +M6) (9)
0
and also
E ≤ c (kA1/2uk2+knk2+kVk2+M6) (10)
0
We want to apply the I-method (for the definition of I see the introduction). A
crucial role is played by the modified energy E(Iu,In ) for the system
+
1
iIu +Iu = I[(n +n )u] (11)
t xx +
2 −
iIn ∓A1/2In = ±IA1/2(|u|2) (12)
t
± ±
Iu(0) = Iu , In (0) = In = I(n ±iA 1/2n ) (13)
0 0 0 − 1
± ±
namely
1 1
E(Iu,In ):= kIu k2+ kIn k2+ I(n +n )|Iu|2dx
+ x + + +
2 2 Z
5
which is not conserved but its growth is controllable.
An elementary but lengthy calculation shows
d
E(Iu,In ) = RehI(n +n )Iu−I((n +n )u),Iu i
+ + + + + t
dt
+RehIn ,iA1/2(|Iu|2 −I(|u|2))i (14)
+
If I = id this again shows the conservation of E(u,n ).
+
Before considering this modified energy in detail we give a local existence
result for the system (11),(12),(13), which essentially uses the bilinear estimates
given by [GTV] for their local existence result of the Zakharov system.
Proposition 1.1 Assume s ≥ 1/2. Let (u ,n ,n ) ∈ Hs ×Hs 1 ×Hs 1 be
0 +0 0 − −
given. Thenthere existsapositive numberδ ∼ − 1 such
(kIu0kH1+kIn+0kL2+kIn−0kL2)4+
that the system (11), (12), (13) has a unique local solution in the time interval
[0,δ] with the property (dropping from now on [0,δ] from the notation):
kIuk +kIn k +kIn k ≤ c(kIu k +kIn k +kIn k )
X1,12 + X0,12+ − X0,21+ 0 H1 +0 L2 −0 L2
+ −
This solution also belongs to C0([0,δ],H1(R)) and
x
kIukC0([0,δ],H1(R)) ≤ c(kIu0kH1 +kIn+0kL2 +kIn 0kL2)
x −
Proof: We use the corresponding integral equations to define a mapping S =
(S ,S ) by
0 1
1 t
S0(Iu(t)) = Ieit∂x2u0+ ei(t−s)∂x2I(u(s)((n+(s)+n (s))ds
2Z0 −
t
S (In (t)) = IeitA1/2n ±i e i(t s)A1/2A1/2I(|u(s)|2)ds
1 0 ∓ −
± ± Z0
We use [GTV], Lemma 4.3 to conclude for s ≥ 1/2:
kn uk ≤ ckn k kuk
± Xs,−21 ± Xs−1,83+ Xs,83+
±
Similarly [GTV], Lemma 4.5 shows
kn uk ≤ ckn k kuk
± Ys ± Xs−1,38+ Xs,38+
±
Finally, [GTV], Lemma 4.4 shows for s ≥ 0:
kA1/2(|u|2)k ≤ ckuk2
Xs−1,−12+ Xs,14++
±
These estimates imply similar estimates including the I-operator by the interpo-
lation lemma of [CKSTT5], namely
kI(n u)k +kI(n u)k ≤ ckIn k kIuk
± X1,−12 ± Y1 ± X0,38+ X1,83+
±
6
and
kIA1/2(|u|2)k ≤ ckIuk2
X0,−12+ X1,41++
±
where c is independent of N.
The same estimates also hold true for functions defined on [0,δ] only, and for
such functions we can also use the embedding (5). This gives
1
kI(n±u)kX1,−21 +kI(n±u)kY1 ≤ ckIn±kX0,12+kIukX1,21δ4−
±
and
kIA1/2(|u|2)kX0,−12+ ≤ ckIuk2X1,21δ12−
±
Using these estimates the integral equations lead to (remark here that one needs
the space Y1, cf. [GTV], Lemma 2.1):
1
kS0(Iu)kX1,12 ≤ ckIu0kH1 +c(kIn+kX0,12+ +kIn−kX0,12+)kIukX1,21δ4−
+ −
kS1(In±)kX0,12+ ≤ ckIn±0kL2 +ckIuk2X1,12δ12−
±
The standard contraction argument gives the existence of a unique solution on
[0,δ] with
kIuk +kIn k +kIn k ≤ 2c(kIu k +kIn k +kIn k )
X1,12 + X0,21+ − X0,12+ 0 H1 +0 L2 −0 L2
+ −
provided
1
cδ4−(kIu0kH1 +kIn+0kL2 +kIn 0kL2) < 1
−
Concerning the property Iu ∈ C0([0,δ],H1) we refer to [GTV], Lemma 2.2 (use
x
the first integral equation and I(un )∈ Y1). Moreover (4) gives
±
kIuk ≤ kIu k +c(kI(n u)k +kI(n u)k )
C0([0,δ],H1) 0 H1 + Y1 Y1
−
1
≤ kIu0kH1 +cδ4−(kIn+kX0,21+ +kIn−kX0,12+)kIukX1,21
+ −
(kIu k +kIn k +kIn k )2
≤ kIu0kH1 +c kIu0kH1 +kIn+0kL2 +kIn−0kL2
0 H1 +0 L2 0 L2
−
≤ c(kIu k +kIn k +kIn k )
0 H1 +0 L2 0 L2
−
2 A bilinear Strichartz estimate
Lemma 2.1
k(D1/2u)n k ≤ ckn k kuk
x ± L2xt ± X0,21+ X0,12+
±
Proof: We split the domain of integration into the following parts
a) suppu⊂ {|ξ| ≥ 2}
b) suppu⊂ {|ξ| ≤ 2}
b
b
7
a) We assume suppm ⊂ {ξ ≥ 0} (the other part suppm ⊂ {ξ ≤ 0} can be
± ±
treated similarly) and suppv ⊂ {|ξ| ≥ 2}. In this region we conclude as follows:
d d
keit∂x2Dx1/2ve±it|∂x|mb k2L2 (15)
± xt
2
= dξdt(cid:12) e−itξ12±it|ξ2|v(ξ1)m (ξ2)|ξ1|21dξ1(cid:12)
Z (cid:12)(cid:12)Zξ=ξ1+ξ2,ξ2≥0,|ξ1|≥2 ± (cid:12)(cid:12)
(cid:12) b d (cid:12)
= dξdt(cid:12) e−it(ξ12±|ξ2|−η12∓|η2|)v((cid:12)ξ1)v(η1)·
Z Z
ξ=ξ1+ξ2=η1+η2,η2,ξ2 0, ξ1,η1 2
≥ | || |≥
1 b 1 b
·m (ξ2)m (η2)|ξ1|2|η1|2dξ1dη1
± ±
1 1
= dξ dξ1dη1δ(P(η1))v(ξ1)v(η1)m (dξ2)m (dη2)|ξ1|2|η1|2
Z Z ± ±
with b b d d
P(η ) := ξ2±|ξ |−η2∓|ξ −η | = ξ2±|ξ |−η2∓|η |
1 1 2 1 1 1 2 1 2
= ξ2±ξ −η2∓η = ξ2±(ξ −ξ )−η2∓(ξ −η )
1 2 1 2 1 1 1 1
= ξ2−η2∓(ξ −η ) =(ξ −η )[(ξ +η )∓1]
1 1 1 1 1 1 1 1
This function has the simplezeroes η = ξ and η = ±1−ξ . Moreover P (η ) =
1 1 1 1 ′ 1
−2η ± 1 , thus |P (η )| ∼ |η | in our region |η | ≥ 2. Using the well-known
1 ′ 1 1 1
identity
f(x )
k
δ(P(η ))f(η )dη =
1 1 1
Z |P (x )|
X ′ k
wherex denotesthesimplezeroesofP,weremarkthatinourcaseforthezeroes
k
1 1
wehave|η1| ∼ |ξ1|, andthereforethefactor |ξ1|2|η1|2 cancels with|P′(xk)|. Thus
we can estimate (15) using Schwarz’ inequality by
c dξ dξ |v(ξ )v(ξ )m (ξ−ξ )m (ξ−ξ )|
1 1 1 1 1
Z Z ± ±
b b d d
+ c dξ dξ |v(ξ )v(±1−ξ )m (ξ−ξ )m (ξ−(±1−ξ ))|
1 1 1 1 1
Z Z ± ±
≤ ckvk2 km k2b = cbkvk2 km dk2 d
L2 L2 L2 L2
± ±
and the claimedbestimdate follows directly in the region suppu ⊂ {|ξ| ≥ 2} (cf.
e.g. [P2], Lemma 1.4, [G], Lemma 2.1 or [KS], Section 3).
b
b) In the region suppu⊂ {|ξ| ≤ 2} we have
1 1 1
k(D2u)n k ≤b kD2uk kn k ≤ ckD2uk kn k
x ± L2xt x L2tL∞x ± L∞t L2x x L2tHx12+ ± X±0,21+
≤ ckuk kn k ≤ ckuk kn k
L2tL2x ± X0,12+ X0,12+ ± X0,21+
± ±
Lemma 2.2
k(D1/2u)n k ≤ ckn k kuk
x ± L2xt ± X0,12+ X0+,12
±
Proof: Interpolate the estimate of the previous lemma with
1 1
k(D2u)n k ≤ kn k kD2uk ≤ ckn k kuk
x ± L2xt ± L∞t L2x x L2tL∞x ± X0,21+ L2tHx1+
±
≤ ckn k kuk
± X0,21+ X1+,0
±
8
Lemma 2.3
k(D1/2u)n k ≤ ckn k kuk
x ± L2t+L2x ± X0,12 X0+,21
±
Proof: By Sobolev’s embedding theorem and Strichartz’ estimate we have
1 1 1
k(D2u)n k ≤ ckn k kD2uk ≤ ckn k kD2uk
x ± L4t−L2x ± L∞t −L2x x L4tL∞x ± X±0,12− x L4tHx41+,4
≤ ckn k kuk
± X0,12− X43+,38+
±
Interpolation with Lemma 2.1 gives the claimed result.
A variant of this lemma is given in the following
Lemma 2.4
1/2
k(D u)∗n k ≤ ckn k kuk
x ± L2ξL2τ− ± X0,12+ X0+,12
d ±
c
Proof: On one hand Lemma 2.1 gives
1/2
k(D u)∗n k ≤ ckn k kuk (16)
x ± L2ξτ ± X0,12+ X0,12+
d ±
c
On the other hand Young’s inequality shows
1/2 1/2
k(D u)∗n k ≤ ckn k kD uk (17)
x 4 L2L1 x 4
d ± L2ξLτ3 ± ξ τ d L1ξLτ3
c c
Now by Schwarz’ inequality
kn±kL2ξL1τ = kZ |n±(ξ,τ)|hτ ±|ξ|i12+hτ ±|ξ|i−12−dτkL2ξ
c ≤ ckn±(cξ,τ)hτ ±|ξ|i12+kL2ξτ = ckn±kX0,12+
c
and by H¨older’s inequality in τ and Schwarz’ inequality in ξ:
kDx1/2uk 4 = kDx1/2u(ξ,τ)hτ +ξ2i14+hτ +ξ2i−41−k 4
d L1ξLτ3 d L1ξLτ3
≤ ckDx1/2u(ξ,τ)hτ +ξ2i41+kL1L2
ξ τ
d
= ckDx1/2u(ξ,τ)hτ +ξ2i41+hξi12+hξi−12−kL1L2
ξ τ
d
≤ ckDx1/2u(ξ,τ)hτ +ξ2i41+hξi12+kL2
ξτ
d
≤ ckuk
X1+,41+
Interpolating (17) and (16) we get the result.
We also need a bilinear Strichartz’ refinement for the pure Schro¨dinger prob-
lem. We have the well-known
9
Lemma 2.5 If u ,u fulfill |ξ | >> |ξ | ≥ 1 for ξ ∈ suppu (i = 1,2), the
1 2 1 2 i i
following estimate holds:
b
1
k(D2u )u k ≤ cku k ku k
x 1 2 L2xt 1 X0,21+ 2 X0,21+
Proof: [CKSTT1], Lemma 7.1.
We also have the following variant:
Lemma 2.6 Under the assumptions of Lemma 2.5 we have:
1
k(D2u )u k ≤ cku k ku k
x 1 2 L2t+L2x 1 X0+,12 2 X0,21
Proof: similarly as the proof of Lemma 2.3.
Remark: All the estimates in this section remain true, if any of the functions
on the l.h.s. of the estimates are replaced by their complex conjugates.
3 Estimates for the modified energy
The main step towards global existence is an exact control of the increment of
the modified energy.
Proposition 3.1 Let (u,n ) be a solution of (6),(7),(8) on [0,δ] in the sense of
±
Prop. 1.1. Then the following estimate holds (for N ≥ 1 , s > 3/4):
|E(Iu(δ),In (δ))−E(Iu(0),In (0))|
+ +
≤ c((N−12+δ12−+N−23+δ0+)kIn+kX0,12+kIuk2X1,21
+
+(N−3++N−1+δ12−)kIn+k2X0,21+kIuk2X1,12)
+
Proof: Using (14) and replacing Iu by (11) we have to show
t
δ c
(cid:12)(cid:12)(cid:12)Z0 Z In+A1/2(|Iu|2 −I(|u|2))dxdt(cid:12)(cid:12)(cid:12) ≤ N1−δ12−kIn+kX+0,21+kIuk2X1,12 (18)
(cid:12) (cid:12)
(cid:12) (cid:12)
and
δ
(Iu) (I(n u)−In Iu)dxdt (19)
(cid:12) xx + + (cid:12)
(cid:12)Z0 Z (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) ≤ c(N−12+δ12− +N−23+δ0+)kIn+(cid:12)(cid:12)kX0,21+kIuk2X1,12
+
as well as
δ 1
(cid:12)(cid:12)Z0 Z I(n+u)(I(n+u)−In+Iu)dxdt(cid:12)(cid:12) ≤ c(N3−+N−1+δ12−)kIn+k2X0,21+kIuk2X1,21
(cid:12) (cid:12) +
(cid:12) (cid:12) (20)
(cid:12) (cid:12)
Here and in the sequel we assume w.l.o.g. the Fourier transforms of all these
functions to be nonnegative, ignore the appearance of complex conjugates, use
10