Remark on Well-posedness of Quadratic Schrödinger equation with nonlinearity $u\bar u$ in $H^{-1/4}(\R)$

0.12 MB·English

by  Yuzhao Wang

REMARK ON WELL-POSEDNESS OF QUADRATIC SCHRO¨DINGER
EQUATION WITH NONLINEARITY uu IN H−1/4(R)
0 YUZHAOWANG
1
0
2 Abstract. In this note, we give another approach to the local well-posedness of quadratic
Schr¨odinger equation with nonlinearity uu in H−1/4, which was already proved by Kishimoto
n [3]. Ourresolutionspaceisl1-analogueofXs,b spacewithlowfrequencypartinaweakerspace
a L∞L2. SuchtypespacesweredevelopedbyGuo. [2]todealtheKdVendpointH−3/4regularity.
J t x

5

]
P 1. Introduction
A
This paper is mainly concerned with the following equation
.
h iu +u =|u|2, u(x,t):R×R→C,
t t xx (1.1)
a (cid:26) u(x,0)=φ(x)∈Hs(R).
m
The low regularity for this equation was first studied by Kenig, Ponce, Vega in [4], they proved
[ the local well-posedness in Hs, for s > −1/4, by using Xs,b spaces. The local well-posedness in

3 H−1/4 was already proved by Kishimoto [3], where Kishimoto solved (1.1) in the spaces
v
Z =X−1/4,1/2+β +Y
9
3
and
9
.0 Y ={f ∈S′(R2);kfkY =khξi−1/4hτ −ξ2i3βfˆkL2ξLpτ +khξi1/4−2βhτ −ξ2i3βfˆkL2ξL2τ},
0 with 0<β ≤1/24, 2β <1/p′ <3β and 1/p+1/p′ =1.
1
WegiveanotherapproachbasedontheargumentdevelopedbyGuo. [2],whichsolvedtheglobal
9
well-posedness for KdV equation in H−3/4. Our resolution space is l1-analogue of Xs,b space with
0
: low frequency part in a weaker space L∞L2, so as a resolution space, it has simple form.
v t x
It is well known that Xs,b failed for (1.1) in H−1/4 because of the logarithmic divergences from
i
X high×high → low interactions, it is natural to use the weaker structure in low frequency. We
r use L∞L2 to measure the low frequency part, however in [2] Guo used L2L∞. The reason for this
a t x x t
is that in the KdV case, the high×low interactions has one derivative, and the smoothing effect
norm L∞L2 was needed to absorb it. This method can also be adapted to other similar problems
x t
where some logarithmic divergences appear in the high-high interactions.
Theorem 1.1. The initial value problem (1.1) is local well-posedness in H−1/4.
For f ∈ S′ we denote by f or F(f) the Fourier transform of f. We denote by F the Fourier
x
transformonspatialvariable. LetZandNbethesetsofintegersandnaturalnumbersrespectively,
b
Z =N∪{0}. For k ∈Z let I ={ξ :|ξ|∈[2k−1,2k+1]}, k ≥1; I ={ξ :|ξ|≤2}. Let η :R→
+ + k 0 0
[0,1] denote an even smooth function supported in [−8/5,8/5] and equal to 1 in [−5/4,5/4]. We
define ψ(t)=η (t). For k ∈Z let η (ξ)=η (ξ/2k)−η (ξ/2k−1) if k ≥1 and η (ξ)≡0 if k ≤−1.
0 k 0 0 k
For k ∈Z , define P by P u(ξ)=η (ξ)u(ξ). For l∈Z let P = P ,P = P .
+ k k k ≤l k≤l k ≥l k≥l k
P P
2000 Mathematics Subject Cdlassification. 35Qb53,35L30.
Key words and phrases. QuadraticSchr¨odingerequation,Localwell-posedness,Lowregularity.
1
2 YUZHAOWANG
For u0 ∈S′(R), we denote W(t)u0 =eit∂x2u0 defined by Fx(W(t)φ)(ξ)=exp[−iξ2t]φ(ξ).
For k ∈Z we define the dyadic Xs,b-type normed spaces X =X (R2),
+ k k b
f(ξ,τ) is supported in I ×R and
Xk =(cid:26)f ∈L2(R2): kfkXk = ∞j=02j/2kηj(τk+ξ2)·fkL2, (cid:27) (1.2)
P
thus we have kfkL1τL2ξ ≤kfkXk. For −3/4≤s≤0, we define our resolution spaces
b
F¯s ={u∈S′(R2):kuk2 = 22skkη (ξ)F(u)k2 +kP (u)k2 <∞}. (1.3)
F¯s kX≥1 k Xk ≤0 L∞t L2x
It is easy to see that for k ∈Z
+
kPk(u)kL∞t L2x.kF[Pk(u)]kXk, (1.4)
as Laectoans,eaqu,eance∈, wRe, dheafivneekaukL∞t H=sm.akxuk{Fa¯s.,a ,a }, same as a ,a . Usually we use k ,k ,k
1 2 3 max 1 2 2 min med 1 2 3
and j1,j2,j3 to denote integers, Ni =2ki and Li =2ji for i=1,2,3 to denote dyadic numbers.
2. Dyadic Bilinear Estimates
In this section we will give some dyadic bilinear estimates for next section. We define
D ={(ξ,τ):ξ ∈[2k−1,2k+1] and τ +ξ2 ∈I }, k ∈Z,j ∈Z .
k,j j +
Following the [k;Z] methods [5] the bilinear estimates in Xs,b space reduce to some dyadic sum-
mations: for any k ,k ,k ∈Z and j ,j ,j ∈Z
1 2 3 1 2 3 +
(uk2,j2,suvkp3,j3)∈Ek1Dk1,j1(ξ,τ)·uk2,j2 ∗vk3,j3(ξ,τ)kL2ξ,τ (2.1)
where E = {(u,v) : kuk , kvk ≤ 1 and supp(u) ⊂ D , supp(v) ⊂ D } and D =
2 2 k2,j2 k3,j3 k3,j3
{(ξ,τ);(−ξ,−τ) ∈ D }. By checking the support properties, we get that in order for (2.1) to
k3,j3 e e
be nonzero one must have
|k −k |≤3, and j ≥k +k −10 (2.2)
max med max max min
The following sharp estimates on (2.1) were obtained in [5].
Lemma2.1(Proposition11.1,[5](–++)case). Letk1,k2,k3 ∈Zandj1,j2,j3 ∈Z+. LetNi =2ki
and Li =2ji for i=1,2,3. Then
(i) If N ∼N and L ∼N N , then we have
max min max max min
(2.1).L1/2 L1/4 . (2.3)
min med
(ii) If N ∼ N ≫ N and N N ∼ L = L , and N ∼ N ≫ N and N N ∼
1 3 2 max min 2 max 1 2 3 max min
L =L , then
3 max
(2.1).L1/2 L1/2 N−1/2. (2.4)
min med min
(iii) In all other cases, we have
(2.1).L1/2 N−1/2min(N N ,L )1/2. (2.5)
min max max min med
QUADRATIC SCHRO¨DINGER EQUATION 3
3. Proof of Theorem 1.1
For u,v ∈F¯s we define the bilinear operator
t t
B(u,v)=ψ W(t−τ)∂ ψ2(τ)u(τ)·v(τ) dτ. (3.1)
x
4 Z
(cid:0) (cid:1) 0 (cid:0) (cid:1)
Asin[2],theproofforTheorem1.1reducetoshowingtheboundnessofB :F¯−1/4×F¯−1/4 →F¯−1/4.
Lemma 3.1 (Linear estimates). (a) Assume s∈R, φ∈Hs. Then there exists C >0 such that
kψ(t)W(t)φkF¯s ≤CkφkHs. (3.2)
(b) Assume s∈R,k ∈Z and (i+τ −ξ3)−1F(u)∈X . Then there exists C >0 such that
+ k
t
F ψ(t) W(t−s)(u(s))ds ≤Ck(i+τ −ξ3)−1F(u)k . (3.3)
(cid:13) (cid:20) Z (cid:21)(cid:13) Xk
(cid:13) 0 (cid:13)Xk
(cid:13) (cid:13)
Proof. Such lin(cid:13)ear estimates have appeared in(cid:13)many literatures, see for example [1]. (cid:3)
Lemma 3.2 (Bilinear estimates). Assume −1/4≤s≤0. Then there exists C >0 such that
kB(u,v)kF¯s ≤C(kukF¯skvkF¯−1/4 +kukF¯−1/4kvkF¯s) (3.4)
hold for any u,v ∈F¯s.
Proof. It is easy to see
kB(u,v)kF¯s.kP≥1B(P≥1u,P≥1v)kFs +kP≥1B(P≥1u,P0v)kF¯s
+kP≥1B(P0u,P≥1v)kF¯s +kP≥1B(P0u,P0v)kF¯s +kP0B(u,v)kF¯s
,A+B+C+D+E
We notice that there is no low frequency in part A, so the proof for part A do not involve the
special structure in low frequency, and standard Xs,b argument will suffice, we omit the proof.
The proof for part B, C and D are similar, we just consider part B for example. By definition
and Lemma 3.1 (b), let S ={(k ,k );k ,k ≥1,|k −k |≤5}, then
B 1 3 1 3 1 3
\ 2
B2.(k1,Xk3)∈SB22sk3(cid:16)jX3≥02−j3/2k1Dk3,j3ψ(t)Pk1u∗Pd0vkL2ξ,τ(cid:17)
. 22sk3kψ(t)Pk1uk2L2kP0vk2L∞. 22sk3kPk1uk2L∞L2kP0vk2L∞ (3.5)
t x
X X
(k1,k3)∈SB (k1,k3)∈SB
which is sufficient by Bernstein inequality and (1.4).
NowweturntopartD.DenoteQ(u,v)=P B(P u,P v¯). Bystraightforwardcomputations,
≤0 k1 k2
F[Q(u,v¯)](ξ,τ)=c ψ(τ −τ′)−ψ(τ +ξ2)η (ξ) P[u(ξ ,τ )P[v¯(ξ ,τ ) dτ′.
ZR b τ′+ξb2 0 ZZ k1 1 1 k2 2 2
where Z ={ξ =ξ +ξ ,τ′ =τ +τ }. Fixing ξ ∈R, we decomposing the hyperplane as following
1 2 1 2
A = {ξ =ξ +ξ ,τ′ =τ +τ :|ξ|.2−k1};
1 1 2 1 2
A = {ξ =ξ +ξ ,τ′ =τ +τ :|ξ|≫2−k1,
2 1 2 1 2
|τ +ξ2|≪2k1|ξ|,|τ −ξ2|≪2k1|ξ|};
1 1 2 2
A = {ξ =ξ +ξ ,τ′ =τ +τ :|ξ|≫2−k1,|τ +ξ2|&2k1|ξ|};
3 1 2 1 2 1 1
A = {ξ =ξ +ξ ,τ′ =τ +τ :|ξ|≫2−k1,|τ −ξ2|&2k1|ξ|}.
4 1 2 1 2 2 2
4 YUZHAOWANG
Then we get
F[Q(u,v¯)](ξ,τ)=I +II+III,
where
I =C ψ(τ −τ′)−ψ(τ +ξ2)η (ξ) P[u(ξ ,τ )P[v¯(ξ ,τ )dτ′,
ZR b τ′+ξb2 0 ZA1 k1 1 1 k2 2 2
II =C ψ(τ −τ′)−ψ(τ +ξ2)η (ξ) P[u(ξ ,τ )P[v¯(ξ ,τ )dτ′,
ZR b τ′+ξb2 0 ZA2 k1 1 1 k2 2 2
III =C ψ(τ −τ′)−ψ(τ +ξ2)η (ξ) P[u(ξ ,τ )P[v¯(ξ ,τ )dτ′.
ZR b τ′+ξb2 0 ZA3∪A4 k1 1 1 k2 2 2
We consider first the the term I. By (1.4) and Proposition 3.1 (b),
kF−1(I)kL∞t L2x.kIkX0.(cid:13)(cid:13)(i+τ′+ξ2)−1η0(ξ)ZA1P[k1u(ξ1,τ1)P[k2v¯(ξ2,τ2)(cid:13)(cid:13)X0,
since in the set A we have |ξ|.2−(cid:13)k1, thus we continue with (cid:13)
1
. 2−j3/2 k1Dk3,j3 ·uk1,j1 ∗vk2,j2kL2
k3≤X−k1+10jX3≥0 j1≥X0,j2≥0
where
u =η (ξ)η (τ +ξ2)u, v =η (ξ)η (τ −ξ2)v. (3.6)
k1,j1 k1 j1 k,j2 k j2
Using Proposition 2.1 (iii), then we get b b
kF−1(I)kL∞t L2x . 2−j3/22jmin/22k3/2kuk1,j1kL2kvk2,j2kL2
k3≤X−k1+10jXi≥0
. 2−k1/2kP[uk kP[vk ,
k1 Xk1 k2 Xk2
which suffices to give the bound for the term I since |k −k |≤5.
1 2
NextweconsiderthecontributionofthetermIII. AstermI, by (1.4)andProposition3.1(b),
kF−1(III)kL∞t L2x . (cid:13)(i+τ′+ξ2)−1η0(ξ)Z P[k1u(ξ1,τ1)P[k2v(ξ2,τ2)(cid:13)
(cid:13) A3∪A4 (cid:13)X0
(cid:13) (cid:13)
. (cid:13) 2−j3/2 k1Dk3,j3 ·uk1,j1 ∗vk2,j2k(cid:13)L2.
−k1X≤k3≤0jX3≥0 j1≥X0,j2≥0
Without loss of generality, we assume |τ +ξ2|&|ξξ |, applying Proposition 2.1 (iii), then we get
1 1 1
kF−1(III)kL∞t L2x . 2j2/22−k1/2kuk1,j1kL2kvk2,j2kL2
−k1X≤k3≤0j1≥k3+Xk1−10,j2≥0
. 2−k1/2kP[uk kP[uk ,
k1 Xk1 k2 Xk2
which suffices to give the bound for the term III since |k −k |≤5.
1 2
Now we consider the main contribution term: term II. By direct computation, we get
t
F−1(II)=ψ(t) e−i(t−s)ξ2η (ξ)iξ eis(τ1+τ2) u (ξ ,τ )v (ξ ,τ ) dτ dτ ds
t Z0 0 ZR2 Zξ=ξ1+ξ2 k1 1 1 k2 2 2 1 2
where
uk1(ξ1,τ1)=ηk1(ξ1)1{|τ1+ξ12|≪2k1|ξ|}u(ξ1,τ1), vk2(ξ2,τ2)=ηk2(ξ2)1{|τ2−ξ22|≪2k1|ξ|}v(ξ2,τ2).
b b
QUADRATIC SCHRO¨DINGER EQUATION 5
By a change of variable τ′ =τ +ξ2, τ′ =τ −ξ2, we get
1 1 1 2 2 2
t
F−1(II) = ψ(t)e−itξ2η (ξ) eisξ2 eis(τ1+τ2)
t 0 Z0 ZR2
× e−isξ12u (ξ ,τ −ξ2)eisξ22v (ξ ,τ +ξ2) dτ dτ ds
Z k1 1 1 1 k2 2 2 2 1 2
ξ=ξ1+ξ2
= ψ(t)e−itξ2η (ξ) eit(τ1+τ2) eit(−ξ12+ξ22+ξ2)−e−it(τ1+τ2)
0 ZR2 Zξ=ξ1+ξ2 τ1+τ2−ξ12+ξ22+ξ2
× u (ξ ,τ −ξ2)v (ξ ,τ +ξ2) dτ dτ .
k1 1 1 1 k2 2 2 2 1 2
Then by Plancherel Theorem and Ho¨lder inequality, we can bound kF−1(II)k by
t Lξ
η (ξ)
0 |u (ξ ,τ −ξ2)v (ξ ,τ +ξ2)| dτ dτ
ZR2(cid:13)(cid:13)Zξ=ξ1+ξ2 |τ1+τ2−ξ12+ξ22+ξ2| k1 1 1 1 k2 2 2 2 (cid:13)(cid:13)L2ξ 1 2
(cid:13) χ (ξ) (cid:13)
. 2k/2 k |u (ξ ,τ −ξ2)v (ξ ,τ +ξ2)| dτ dτ
ZR2−kX1≤k≤0 (cid:13)(cid:13)Zξ=ξ1+ξ2 |ξξ1| k1 1 1 1 k2 2 2 2 (cid:13)(cid:13)L∞ξ 1 2
.2−k1/2kuk1kL1τ2L2ξ(cid:13)2kvk2kL1τ3L2ξ3.2−k1/2kP[k1ukXk1kP[k2ukXk2. (cid:13)
where we use |τ +τ −ξ2+ξ2+ξ2|&|ξξ |, which completes the proof of the lemma. (cid:3)
1 2 1 2 1
Acknowledgment. TheauthorisverygratefultoProfessorZihuaGuoforencouragingtheauthor
to work on this problem and helpful conversations.
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LMAM, School of Mathematical Sciences, Peking University, Beijing 100871,China
E-mail address: [email protected]