On an estimate for the wave equation and applications to nonlinear problems

24 Pages·0.29 MB·English

by  Sigmund Selberg

On an estimate for the wave equation and
1
0
applications to nonlinear problems
0
2

n Sigmund Selberg
a
J Department of Mathematics

3 Johns Hopkins University
1 Baltimore, MD 21218

]
P
A
.
h Abstract
t
a We prove estimates for solutions of the Cauchy problem for the in-
m homogeneous wave equation on R1+n in a class of Banach spaces whose
norms only depend on the size of the space-time Fourier transform. The
[
estimates arelocal intime,andthisallows one,essentially, toreplacethe

1 symbol of the wave operator, which vanishes on the light cone in Fourier
v space, withaninhomogeneoussymbol,whichcanbeinverted. Ourresult
9
improves earlier estimates of this type proved by Klainerman-Machedon
1
[4,5]. Asa corollary, oneobtains arather general result concerning local
1
well-posednessofnonlinearwaveequations,whichwasusedextensivelyin
1
the recent article [8].
0
1
0
1 Introduction
/
h
at Consider the Cauchy problem for the wave equation on R1+n,
m
(cid:3)u=F, u =f, ∂ u =g, (1)
v: t=0 t t=0
Xi where (cid:3)= ∂2+∆ is the wave o(cid:12)(cid:12)perator. (cid:12)(cid:12)
− t
The purpose of this note is to prove estimates for u in a certain class of
r
a Banach spaces whose norms only depend on the size of the space-time Fourier
transform. The estimates are local with respect to time t, and this allows
one, essentially, to replace the symbol of the wave operator, which vanishes on
the light cone in Fourier space, with an inhomogeneous symbol, which can be
inverted. ThisideaoriginatesintheworkofBourgain[1]ontheSchr¨odingerand
KdV equations, and was later simplified by Kenig-Ponce-Vega[3] in their work
on KdV. Following this, Klainerman-Machedon[4, Lemma 4.3], [5, Lemma 1.3]
provedestimatesofthistypeforthewaveequation;seealsoKlainerman-Tataru
[9].
AMSSubjectClassification: 35L.
1
The improvement in our result comparedto [3, 4, 5] lies mainly in showing,
assuggestedin[9,Remark1.8],thatforsufficientlysmallε>0,thenormofthe
inhomogeneous part of the solution, restricted to the time slab [0,T] Rn, is
×
O(Tε) as T 0, at the expense of a loss of essentially ε derivatives. As shown
→
inthe recentarticle[8],thisallowsonetoremovethe assumptionofsmall-norm
data in the well-posedness results proved in [4, 5, 6, 7, 9].
As an application of our estimate, we also prove a simple but useful re-
sult concerning local well-posedness of nonlinear wave equations, which is used
extensively in [8].
2 The main estimate
We are interested in finding complete subspaces s of
X
C (R,Hs) C1(R,Hs 1) (2)
b ∩ b −
such that solutions of (1) satisfy estimates of the type
u C (f,g) +C (cid:3) 1Fε (3)
k kXTs ≤ k k(s) T,ε − s
(cid:13) (cid:13)X
for all 0<T <1 and ε 0. We also want (cid:13)(cid:13)e (cid:13)(cid:13)
≥
lim C =0 (4)
T,ε
T 0
→
when ε is strictly positive.
Precise definitions will be supplied presently. For the moment suffice it to
say that s stands for the restriction to the time slab [0,T] Rn, (cid:3) may be
XT ×
thought of as an inhomogeneous and invertible versionof the wave operator (cid:3),
Fε is F with a certain operator of order ε applied to it, and e
(f,g) = f + g
k k(s) k kHs k kHs−1
with Hs the usual Sobolev space.
We use coordinates (t,x) on R1+n. The Fourier transform of f(x) [resp.
u(t,x)] is denoted f(ξ) = f(ξ) [resp. u(τ,ξ) = u(τ,ξ)]. For any α R we
F F ∈
define pseudodifferential operators Λα, Λα and Λα by
+
b b −
Λαf(ξ)= 1+ ξ 2 α/2f(ξ),
| |
Λαu(τ,ξ)=(cid:0)1+τ2+(cid:1) ξ 2 α/2u(τ,ξ),
+d | |b
α/2
(cid:0) (τ2 ξ(cid:1)2)2
Λdαu(τ,ξ)= 1+ −| | b u(τ,ξ),
− 1+τ2+ ξ 2!
| |
Observe that the Fdourier symbol of Λα is comparable tbo 1+ τ ξ α. The
− | |−| |
operator (cid:3) in (3) is just Λ Λ , and Fε =ΛεF.
+ (cid:0) (cid:12) (cid:12)(cid:1)
− − (cid:12) (cid:12)
e
2
The Sobolev and Wave Sobolev spaces Hs and Hs,θ are givenby the norms
f = Λsf ,
k kHs k kL2(Rn)
u = ΛsΛθu .
k kHs,θ − L2(R1+n)
(cid:13) (cid:13)
For the basic properties of the latte(cid:13)r, see, e(cid:13).g., [8]. In particular, we shall use
the fact that Hs,θ embeds in C (R,Hs) when θ > 1. Associated to Hs,θ is the
b 2
space s,θ with norm
H
u = u + ∂ u Λs 1Λ Λθu .
k kHs,θ k kHs,θ k t kHs−1,θ ∼ − + − L2
(cid:13) (cid:13)
By the above, s,θ embeds in (2) for θ > 1. (cid:13) (cid:13)
H 2
Ingeneral,ifaBanachspace s embedsin(2)thenitmakessensetorestrict
its elements to any interval I XR. The resulting restriction space is denoted
⊂
s. Itisalwayspossibletodefineanormonthisspacewhichmakesitcomplete.
XI
Indeed, s is the quotient space s/ , where is the equivalence relation
XI X ∼I ∼I
u v u(t)=v(t) for all t I.
I
∼ ⇐⇒ ∈
Since s embeds in (2), the equivalence classes are closed sets in s, so the
X X
quotient space is complete when equipped with the norm
u = inf v .
k kXIs v∼Iuk kXs
If I =[0,T], we always write s instead of s.
XT XI
We now state the precise result.
Theorem 1. Let s be a Banach space such that
X
(i) s embeds in s,θ for some θ > 1,
X H 2
(ii) u v = u v ,
| |≤| | ⇒ k k s ≤k k s
X X
(iii) there exists γ <2 such that2
b b
u . Λs 1Λ Λγu(τ,ξ)
k kXs F − + − L2ξ(L∞τ )
(cid:13) (cid:13)
for all u. (cid:13) (cid:13)
Let ε 0. Then for all (f,g) Hs Hs−1 and F (cid:3)Λ−ε( s), there is
a uniqu≥e u C(R,Hs) C1(R,H∈s 1)×which solves (1)∈, and−theXestimate (3)
−
∈ ∩
holds for all 0<T <1. Moreover, if ε>0, then (4) holds.e
Remarks. (1) Estimate (3) in the special case s = s,θ and ε=0 was proved
X H
in [4].
2Weusethenotation kv(τ,ξ)kL2ξ(L∞τ )= kv(·,ξ)k2L∞ dξ 1/2.
(cid:0)R (cid:1)
3
(2) The proofgivessomething stronger: for all T >0, there is a linear operator
W such that
T
(cid:3)W F =F on [0,T] Rn,
T
×
with vanishing initial data at t=0, and W is bounded from
T
s,ε =(cid:3)Λ−ε( s) (5)
Y − X
into s for all ε 0. Moreover,when ε>0, the operator norm
e
X ≥
W 0 as T 0.
k Tk s,ε s → →
Y →X
(3) The reason for the condition γ < 2 is as follows. The proof of the theorem
shows that
C CTmin αε,δ ,
T,ε { }
≤
where
δ =2 γ α/2+min 0,α(θ+ε 1)
− − −
and 0 α 1 can be chosen at will. The(cid:8)constant C is i(cid:9)ndependent of ε and
≤ ≤
α. In the typical applications γ is close to 1; see [8].
(4) s plays no role. Indeed, if s satisfies the hypotheses of the theorem, then
so does s′ =Λs s′ s for all Xs.
− ′
X X
3 Applications
For t R we denote by τ the time-translation operator τ u = u( +t, ), and
t t
for an∈y interval I R we denote restriction to the time-slab I Rn· by · .
I
⊆ × |
Suppose s satisfies the hypotheses of Theorem 1, is invariant under time-
translation3,Xand for all φ C (R), the multiplication map u φ(t)u(t,x) is
∈ c∞ 7→
bounded from s into itself.
Consider aXsystem of wave equations on R1+n of the form
(cid:3)u= (u), (6)
N
where u takes values in RN and : s is (i) time-translation invariant
′
N X → D
( commutes with τ ); (ii) local in time4; and (iii) satisfies (0)=0.
t
N N
Furthermore, we assume that for some ε > 0, satisfies, with notation as
N
in (5), a Lipschitz condition
(u) (v) A max u , v u v (7)
kN −N k s,ε ≤ {k k s k k s} k − k s
Y X X X
for all u,v s, where A is a cont(cid:0)inuous function. (cid:1)
∈X
Theorem 2. Under the above assumptions, the Cauchy problem for (6) is lo-
cally well-posed for initial data in Hs Hs 1, in the following sense:
−
×
3Thatistosay,τt isanisomorphismofXs forallt.
4Bythiswemeanthatifu|I =v|I,whereI isanopeninterval,thenN(u)|I =N(v)|I.
4
(I) (Local existence) For all (f,g) Hs Hs 1 there exist a T > 0 and
−
a u s which solves (6) on S ∈= (0,×T) Rn with initial data (f,g).
∈ XT T ×
Moreover, T can be chosen to depend continuously on f + g .
k kHs k kHs−1
(II) (Uniqueness) If T > 0 and u,u s are two solutions of (6) on S
′ ∈ XT T
with the same initial data (f,g), then u=u.
′
(III) (Continuous dependence on initial data)If, forsomeT >0,u s
∈XT
solves (6) on S with initial data (f,g), then for all (f ,g ) Hs Hs 1
T ′ ′ −
∈ ×
sufficiently close to (f,g), there exists a u s which solves (6) on S
′ ∈ XT T
with initial data (f ,g ), and
′ ′
u u C (f f ,g g ) .
k − ′kXTs ≤ k − ′ − ′ k(s)
If, moreover, is C as a map from s into s,ε, then:
∞
N X Y
(IV) (Smooth dependence on initial data) Suppose u s solves (6) on
∈ XT
S for some T >0, with initial data (f,g) Hs Hs 1, and that (f ,g )
T − δ δ
∈ ×
is a smooth perturbation of the initial data, i.e.,
δ (f ,g ), R Hs Hs 1
δ δ −
7→ → ×
is C and takes the value (f,g) at δ = 0. Let u be the corresponding
∞ δ
solution of (6) (by (III), u s for δ < δ ). Then the map δ u
δ ∈ XT | | 0 7→ δ
from [ δ ,δ ] into s is C .
− 0 0 XT ∞
We write σ =Λs σ s and σ,ε =Λs σ s,ε for σ R. If for all σ >s there
− −
X X Y Y ∈
is a continuous function A such that
σ
(u) A u u (8)
kN k σ,ε ≤ σ k k s k k σ
Y X X
for all u s σ, then: (cid:0) (cid:1)
∈X ∩X
(V) (Persistence of higher regularity) If σ >s and u s solves (6) on
∈XT
S with initial data (f,g) Hσ Hσ 1 for some T >0, then
T −
∈ ×
u C [0,T],Hσ C1 [0,T],Hσ−1 .
∈ ∩
(cid:0) (cid:1) (cid:0) (cid:1)
Remark. Typically,provingthat isC isnoharderthanprovingitislocally
∞
N
Lipschitz. As an example, relevant for wave maps, consider
(u)=Γ(u)Q (u,u),
0
N
where u is real-valued, Γ:R R is C and Q is the bilinear “null form”
∞ 0
→
Q (u,v)= ∂ u∂ v+ u v.
0 t t x x
− ∇ ·∇
cFaixnssh>own2, faonrdaspeptrXop=riaHtes,θθ,>X1′ =anHdsε,θ>an0,dtYha=t (cid:3)Λ−−ε(X)=Hs−1,θ+ε−1. One
2
e
5
(i) Q is bounded, hence C , from to ;
0 ∞
X ×X Y
(ii) Φ :u Γ(u) Γ(0) is a locally bounded map of to itself;
Γ ′
7→ − X
(iii) multiplication is bounded, hence C , from to .
∞ ′
X ×Y Y
It remains to prove that Φ is C as a map of , but this follows from (ii)
Γ ∞ ′
X
(which is valid for any C function Γ) and the fact that is an algebra.
∞ ′
X
Indeed, since
1
Γ(v) Γ(u)= Γ′ u+t(v u) (v u)dt,
− − −
Z0
(cid:0) (cid:1)
Φ is locally Lipschitz. Then, since
Γ
1
Γ(v) Γ(u) Γ′(u)(v u)= Γ′ u+t(v u) Γ′(u) dt (v u)
− − − − − · −
Z0
(cid:8) (cid:0) (cid:1) (cid:9)
we have
Γ(v) Γ(u) Γ(u)(v u) C(u,v) v u ,
′ ′ ′
k − − − k ≤ k − k
X X
where
1
C(u,v) C Γ u+t(v u) Γ(u) dt.
≤ ′ − − ′ ′
Z0 X
Butbythe above,ΦΓ′ islocally(cid:13)(cid:13)Lip(cid:0)schitz,soC((cid:1)u,v)=O(cid:13)(cid:13)( v u ′). ThusΦΓ
k − k
is C1, and by induction C . X
∞
4 Abstract local well-posedness
Theorem2is convenientlyprovedin anabstractsetting, whichwe discusshere.
Observe that if s is a space satisfying the assumptions of the last section,
and denotes the sXet of compactintervals I R, then the family s has
I ⊆ {XI}I∈I
the following properties:
(S1) s embeds in C(I,Hs) C1(I,Hs 1) for all I ;
XI ∩ − ∈I
(S2) the solution of (cid:3)u=0 with (u,∂ u) =(f,g) Hs Hs 1 satisfies
t t=0 ∈ × −
u C(cid:12) (f,g)
k kXTs ≤ (cid:12) k k(s)
for all 0<T <1 and all (f,g);
(S3) τ is an isometry from s onto s for all t R and I ;
t XI X−t+I ∈ ∈I
(S4) if I J, then : s s is norm decreasing;
⊆ |I XJ →XI
(S5) whenever I and J are two overlapping intervals (I J has nonempty
∩
interior) and u s, v s agree on the overlap (u(t) = v(t) for all
∈ XI ∈ XJ
t I J), then
∈ ∩
w C u + v ,
k kXIs∪J ≤ I,J k kXIs k kXJs
(cid:16) (cid:17)
where w(t)=u(t) for t I and w(t)=v(t) for t J.
∈ ∈
6
Note that (S2) holds by Theorem 1; we write s instead of s . (S5) holds
XT X[0,T]
by the assumption that u φ(t)u(t,x) is bounded on s for all φ C (R).
7→ X ∈ c∞
We now consider an abstract family of spaces with these properties.
Theorem 3. Let s R. Let s be a family of Banach spaces satisfying
∈ XI I
(S1–5). Consider the system (6), wh∈eIre is an operator which
(cid:8) (cid:9) N
(N1) maps s into (a,b) Rn for all <a<b< ;
X[a,b] D′ × −∞ ∞
(N2) is time-translation (cid:0)invariant: (cid:1) τ =τ ;
t t
N ◦ ◦N
(N3) is local in time: (u )= (u) whenever [a,b] I and u s;
N |[a,b] N |(a,b) ⊂ ∈XI
(N4) satisfies (0)=0.
N
Assume further that for all 0 < T < 1 and u s, there exists v s
(necessarilyunique)whichsolves(cid:3)v = (u)onS ∈=X(0T,T) Rnwithvani∈shXinTg
T
N ×
initial data at t=0; we write v =W (u) and assume that
N
W (u) C A u u (9)
k N kXTs ≤ T k kXTs k kXTs
and, more generally, (cid:0) (cid:1)
W (u) (v) C A max u , v u v (10)
N −N XTs ≤ T {k kXTs k kXTs} k − kXTs
for al(cid:13)(cid:13)l 0<(cid:0) T <1 and u(cid:1),(cid:13)(cid:13)v∈XTs, whe(cid:0)re (cid:1)
lim C =0 (11)
T
T 0+
→
and A is a continuous function.
Then the system (6) is locally well-posed for initial data in Hs Hs 1, in
−
×
the sense that properties (I–III) of Theorem 2 hold.
Remarks. (1) (Lifespan.) By local existence (I), and properties (S4) and (N3),
for given (f,g) the set E(f,g) consisting of all T > 0 for which there exists
u s solving (6) on S with data (f,g), is a nonempty interval. Then, by
∈ XT T
localexistence(I)anduniqueness(II),aswellasproperties(S3–5)and(N2,3),it
followsthatthisintervalisopen. Moreover,bycontinuousdependenceoninitial
data (III), the lifespan T = supE(f,g) is a lower semicontinuous function of
∗
(f,g).
(2) (Higher regularity.) Set σ = Λs σ s. If, for any σ > s, there is a
XT − XT
continuous A such that
σ
W (u) C A u u (12)
k N kXTσ ≤ T σ k kXTs k kXTσ
and (cid:0) (cid:1)
W (u) (v) C A max u , v u v
N −N XTσ ≤ T σ {k kXTs k kXTs} k − kXTσ
(cid:13) (cid:0) (cid:1)(cid:13) +C(cid:0)A max u , v (cid:1) u v (13)
(cid:13) (cid:13) T σ {k kXTσ k kXTσ} k − kXTs
(cid:0) (cid:1)
7
forallu,v s σ,whereC istheconstantappearingin(9)and(10),then
∈XT∩XT T
(V) of Theorem 2 holds.
(3) (Smooth dependence on data.) Suppose S = W is not just Lipschitz
N
but C as a map of s. Recall that the k-th derivative S(k)(u), u s, is a
∞ XT ∈ XT
k-linear map from s s into s; let S(k)(u) denote its operator
XT ×···×XT XT (T)
norm. In view of (13),
(cid:13) (cid:13)
(cid:13) (cid:13)
S (u) C A u . (14)
′ (T) ≤ T k kXTs
Suppose there exist, for k(cid:13)=2,3,(cid:13)..., B con(cid:0)tinuous(cid:1)and increasing so that
(cid:13) (cid:13) k
sup sup S(k)(u) B (R). (15)
(T) ≤ k
0<T<1kukXTs≤R(cid:13) (cid:13)
(cid:13) (cid:13)
Then (IV) of Theorem 2 holds.
(4) If is multilinear, then so is W , so if the latter is bounded on some
N N
Banach space, it is trivially C on that space. Thus, by the previous remark,
∞
the dependence on initial data is C .
∞
Inthisconnection,wementionaninterestingobservationduetoKeel-Tao[2,
Section8],concerningthe feasibilityofprovingwell-posednessforwavemapsin
the critical data space by an iteration argument. For simplicity we take n=2,
but this is not essential.
A wave map u:R1+2 S1 C satisfies the equation
→ ⊆
(cid:3)u+u(∂ u ∂µu)=0, (16a)
µ
·
where is the Euclidean inner product on R2 =C. Consider initial data
·
u =1, ∂ u =ig, (16b)
t=0 t t=0
where i is the imaginary un(cid:12)it and g L2 is(cid:12)real-valued.
(cid:12) ∈ (cid:12)
Observe that if (16) is well-posed for g L2, then the solutions stay on S1
∈
and hence are wave maps. This is certainly true for any smooth solution,5 and
therefore true in general by an approximation argument, using the continuous
dependence on initial data.
If (16) could be proved well-posed for g L2 by an iteration argument in
∈
some Banach space, the dependence on the initial data would necessarily be
C , since in this case the operator is trilinear.
∞
N
As it turns out, (16) is well-posed for g L2, but the dependence on the
dataisnotevenC2. Infact,sinceθ eiθ,R∈ S1 isageodesic,thesolutionof
7→ →
(16) is given by u = eiv, where (cid:3)v = 0 with initial data (0,g); clearly (u,∂ u)
t
belongs to C(R,H˙1 L2) and depends continuously on g.
Thus u = eiεv×solves (16) with g replaced by εg. But since H˙1 is not
ε
an algebra, one would not expect the map ε u , R H˙1 to be twice
ε t=1
→ | →
differentiable at ε =0 for all choices of g, and indeed it is not, as proved in [2,
Proposition 8.3].
5Ifu:R1+2→R2 isasmoothsolutionof (16)thenφ=u·u−1solvesthelinearequation
(cid:3)φ+φ(∂µu·∂µu)=0withvanishinginitialdata, sobyuniqueness,φmustvanish.
8
5 Proof of Theorem 1
In this section, (i–iii) refer to the hypotheses of the theorem. With notation as
in (5), observe that by (i),
s,ε Hs−1,θ+ε−1 (17)
Y ⊆
for all ε 0, and that s,ε s,0.
≥ Y ⊆Y
We shall use the fact that the symbol of Λ is comparable to 1+ τ ξ .
− | |−| |
More precisely, there is a constant c>0 such that
(cid:12) (cid:12)
(cid:12) (cid:12)
1
(τ2 ξ 2)2 2
c−1 1+ |τ|−|ξ| ≤ 1+ 1+ τ−2|+| ξ 2! ≤c 1+ |τ|−|ξ| . (18)
(cid:0) (cid:12) (cid:12)(cid:1) | | | | (cid:0) (cid:12) (cid:12)(cid:1)
(cid:12) (cid:12) (cid:12) (cid:12)
Fix two bump functions χ,φ C (R) such that 0 χ,φ 1, χ = 1 on
∈ c∞ ≤ ≤
[ 2,2],andφ=1 on[ 2c,2c]withsupportin[ 4c,4c],where cis the constant
− − −
in (18).
Fix 0 α 1. Given 0<T <1, write, for any F Hs 1,θ 1,
− −
≤ ≤ ∈
F =φ TαΛ F + I φ TαΛ F =F +F ,
1 2
− − −
where I denotes the i(cid:0)dentity(cid:1)opera(cid:8)tor. In(cid:0)view o(cid:1)f(cid:9)(18),
τ ξ 4c2T α for (τ,ξ) suppF , (19)
− 1
| |−| | ≤ ∈
(cid:12)τ ξ (cid:12) T−α for (τ,ξ) suppF2. (20)
(cid:12)| |−| |(cid:12)≥ ∈ c
Now define (cid:12) (cid:12)
(cid:12) (cid:12) c
u=χ(t)u +χ(t/T)u +u ,
0 1 2
where
sin(t√ ∆)
u =∂ W(t)f +W(t)g with W(t)= − ,
0 t
√ ∆
−
t
u = W(t t)F (t, )dt,
1 ′ 1 ′ ′
− − ·
Z0
u =(τ2 ξ 2) 1F .
2 − 2
−| |
Observe that F Hs 1,0 L1 (R,Hs 1), so u is well-defined.
b 1 ∈ − ⊆ lobc − 1
Lemma 1. u solves (1) on [0,T] Rn.
×
Proof. The only point which is not evident is that u = ∂ u = 0.
2 t=0 t 2 t=0
This is clearly true when F , since then F , so u is necessarily given
∈ S 2 ∈ S (cid:12)2 (cid:12)
by Duhamel’s formula. The general case then follows by(cid:12) density, sin(cid:12)ce clearly
F u is linear and bounded from Hs 1,θ 1 into s,θ, and the latter space
2 − −
7→ H
embeds in (2).
Using hypothesis (iii) only, we shall prove the following lemmas.
9
Lemma 2. For all (f,g) Hs Hs 1,
−
∈ ×
χ(t)u C (f,g) .
k 0k s ≤ k k(s)
X
Lemma 3. For all ε 0 and F Hs 1,θ+ε 1,
− −
≥ ∈
χ(t/T)u CTδ F ,
k 1k s ≤ k kHs−1,θ+ε−1
X
where
δ =2 γ α/2+min 0,α(θ+ε 1) , (21)
− − −
(cid:8) (cid:9)
γ is as in hypothesis (iii), and C is independent of ε and α.
Since by (i) we have (17), and since by (20) and (ii) we clearly have
u CTαε F , (22)
k 2k s ≤ k k s,ε
X Y
the theorem follows.
We turn to the proofs of Lemmas 2 and 3.
Some notation: for γ R, let γ be defined by
∈ D
γχ(τ)=(1+ τ 2)γ/2χ(τ).
D | |
In what follows, p . q medans that p ≤ Cq for sobme positive constant C inde-
pendent of α and ε.
5.1 Proof of Lemma 3
We write F = F +F , where F (τ,ξ) and F (τ,ξ) are supported in the
1 1,1 1,2 1,1 1,2
regions ξ T α and ξ T α respectively. Let u be defined as u , but
− − 1,j 1
| | ≤ | | ≥
with F replaced by F for j =1,d2. d
1 1,j
The following lemma, proved in section 5.3, characterizes u and u .
1,1 1,2
Lemma 4. Given 0<T <1, let u , j =1,2 be defined as above. There exist
1,j
sequences fj+,fj− ∈Hs and gj ∈C([0,1],Hs−1) such that
suppfj± ⊆{ξ :|ξ|≥T−α},
[
suppg (ρ) ξ : ξ T α ,
jc −
⊆{ | |≤ }
fj± Hs, sup kgj(ρ)kHs−1 .Tα(1/2−j)kF1kHs−1,0 (23)
0 ρ 1
≤ ≤
(cid:13) (cid:13)
(cid:13) (cid:13)
for j =1,2,..., and
u = , u = +E, (24)
1,1 1,2
1 2
X X
10