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Sharp well-posedness of the cauchy problem for the higher-order dispersive equation
Acta Mathematica Scientia 2017,37B(4):1061–1082 http://actams.wipm.ac.cn
SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION∗
ö¯#)
Minjie JIANG (
A%)
Wei YAN (
†
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China E-mail :[email protected]; [email protected]
ܬ)
Yimin ZHANG (
School of Science, Wuhan University of Technology, Wuhan 430070, China E-mail : [email protected] This current paper is devoted to the Cauchy problem for higher order dispersive
Abstract equation
ut + ∂x2n+1 u = ∂x (u∂xn u) + ∂xn−1 (u2x ),
n ≥ 2, n ∈ N+ .
By using Besov-type spaces, we prove that the associated problem is locally well-posed in n 3 1 H (− 2 + 4 ,− 2n ) (R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H (s,a) (R) with s < − n2 + 34 and all a ∈ R. Key words
Cauchy problem; sharp well-posedness; modified Bourgain spaces
2010 MR Subject Classification
1
35K30; 35G25
Introduction
In this paper, we investigate the Cauchy problem for the following higher-order dispersive equation ut + ∂x2n+1 u = ∂x (u∂xn u) + ∂xn−1 (u2x ),
n ≥ 2, n ∈ N, x, t ∈ R,
u(x, 0) = u0 (x).
(1.1) (1.2)
In 1988, Constantin and Saut [5] studied the Cauchy problem for the following equations ut + iP (D)u = F, ∗ Received February 24, 2016; revised December 24, 2016. This work is supported by Natural Science Foundation of China NSFC (11401180 and 11471330). The second author is also supported by the Young Core Teachers Program of Henan Normal University (15A110033). The third author is also supported by the Fundamental Research Funds for the Central Universities (WUT: 2017 IVA 075). † Corresponding author: Wei Yan
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∂ , ···, ∂x∂ n ) and P (D)u = where D = 1i ( ∂x 1
R
Rn
Vol.37 Ser.B
e2πihx,ξi p(ξ)Fx u(ξ)dξ. Kenig et al. [14] considered
ut + ∂x2n+1 u + P (u, ∂x u, · · ·, ∂x2n u) = 0,
j ∈ N, x, t ∈ R,
u(x, 0) = u0 (x),
(1.3) (1.4)
where P : R2n+1 −→ R (or P : C2n+1 −→ C) is a polynomial having no constants or linear terms. They proved that the problem (1.3)–(1.4) is well-posed in some weighted Sobolev spaces for small initial data and for arbitrary initial data with the aid of a gauge transformation. By using the Fourier restriction norm method introduced by Bourgain [2, 3], Wang and Cui [20] 1 proved that the case of n = 2 in (1.1)–(1.2) is well-posed in modified Sobolev space H (s,− 4 ) with s > − 41 . Recently, the following problem has been investigated: P ut + ∂x2n+1 u = an1 ,n2 ∂xn1 u∂xn2 u, n ∈ N, x, t ∈ R, (1.5) 0≤n1 +n2 ≤2n
u(x, 0) = u0 (x).
(1.6)
Pilod [19] studied the well-posedness of problem (1.5)–(1.6) in Besov paces and proved that the problem (1.5)–(1.6) is well-posed in the space H k (R) ∩ H k−2n (R; x2 dx) for small initial data with k > 2n + 14 . Pilod also proved some ill-posedness results when a0, k 6= 0 for some k > n in the sense that (1.5)–(1.6) cannot have flow map C 2 at the origin in H s (R) for any s ∈ R. In [22], Yan and Li studied the ill-posedness of modified Kawahara equation and Kaup-Kupershmidt equation. Recently, Kato [13] considered the Cauchy problem for ut − ∂x5 u + c1 ∂x (u3 ) + c2 ∂x (u2x ) + c3 ∂x (uuxx ) = 0,
(1.7)
where cj ∈ R, j = 1, 2, 3 with c3 6= 0, in modified Sobolev space. Very recently, Guo et al. [8] and Kenig and Pilod [15] studied the Cauchy problem for (1.7) in Sobolev space with the aid of the short time Xs, b space which is defined below. In this paper, as in [1, 7, 8, 11–13, 16], we use the modified Bourgain space to establish some new dyadic bilinear estimates, then we prove (1.1)–(1.2) is locally well-posed in n 3 1 H (− 2 + 4 ,− 2n ) (R). When n is even, we also prove that the associated equation is ill-posed in H (s,a) (R) with s < − n2 + 34 and all a ∈ R. We give some notations before presenting the main results. In this paper, IΩ denotes the 1 characteristic function of a set Ω. 0 < ǫ < 10000 . j, k ∈ N+ . C is a positive constant which may vary from line to line. A ∼ B means that |B| ≤ |A| ≤ 4|B|. A ≫ B means that |A| ≥ 4|B|. Let ψ(t) be the smooth function supported in [−1, 2] and equals to 1 in [0, 1]. F u denotes the Fourier transformation of u with respect to its all variables. F −1 u denotes the Fourier inverse transformation of u with respect to its all variables. Fx u denotes the Fourier transformation of u with respect to its space variable. Fx−1 u denotes the Fourier inverse transformation of u with respect to its space variable. Let φ(t) is a smooth cut-off function φ(t) satisfying suppφ ⊂ [−2, 2] and φ(t) = 1 for |t| < 1, Z n+1 2n+1 −t∂x2n+1 W (t)u0 = Ce u0 = C ei(−1) tξ Fx u0 (ξ)dξ, R
and
n o 2n+1 D1 := (τ, ξ) ∈ R2 : |ξ| ≤ 1, |τ | ≥ |ξ|− 2n−1 , n o 2n+1 D2 := (τ, ξ) ∈ R2 : |ξ| ≤ 1, |τ | ≤ |ξ|− 2n−1 ,
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Aj = (τ, ξ) : 2j ≤ hξi < 2j+1 ,
Bk = (τ, ξ) : 2k ≤ τ + (−1)n ξ 2n+1 < 2k+1
for nonnegative integers j, k. The modified Sobolev space is defined by n o
H (s,a) := u ∈ S ′ (R) : kukH (s, a) = hξis−a |ξ|a Fx u(ξ) L2 < ∞ . ξ
Bourgain spaces is defined by n
Xs, b := u ∈ S ′ (R2 ) : kukXs, b = hξis hτ + (−1)n ξ 2n+1 ib F u L2
τξ
(R2 )
ˆ s, a, b is defined by The modified Bourgain space X n
ˆ s, a, b := u ∈ S ′ (R2 ) : kuk ˆ s, a, b = hξis−a |ξ|a hτ + (−1)n ξ 2n+1 ib u 2 X X L
τξ
1 − 2n , n+1 2n
ˆ We denote X L
o <∞ . (R2 )
1 − 2n , n+1 2n ,1
ˆ ,X L
by
n+1 1
kf k − 1 , n+1 := |ξ|− 2n hτ + (−1)n ξ 2n+1 i 2n f 2 , ˆ 2n 2n Lτ ξ (A0 ) X L
X n+1 n+1 1
2 2n k |ξ|− 2n hτ + (−1)n ξ 2n+1 i 2n f kf k − 1 , n+1 ,1 := 2n 2n ˆ X L
o <∞ .
L2τ ξ (A0 ∩Bk )
k≥0
,
ˆ s, b is then defined by the norm respectively. The space X (2,1)
b
s
kukXˆ s, b = hξi τ + (−1)n ξ 2n+1 u 2
(2,1)
Lτ ξ (Aj ∩Bk ) j, k≥0 2
ιj ι1k
X ∼ 2js j
X
2bk kukL2τ ξ(Aj ∩Bk )
k
!2 1/2
1
We define the function space Z (s,− 2n ) as follows:
1 = χ|ξ|≥1 u kuk ˆ s,− 2n
1
ˆ s, 2 X (2,1)
Z
+ kuk
1
ˆ − 2n X L
.
,
where
kuk
1
ˆ − 2n X L
= kχD1 uk
1
ˆ − 2n , X L
n+1 2n
+ kχD2 uk
1
ˆ − 2n , X L
n+1 ,1 2n
.
We define
n 3 1 1
kf kY = hξi(− 2 + 4 )+ 2n |ξ|− 2n F u
L2ξ L1τ
, kF f kXˆ = kf kX , kf k
1
Z s,− 2n
1 , = kF f k ˆ s,− 2n
Z
kF f kXˆ s, b = kf kXs, b , kF f kXˆ s, a, b = kf kX s, a , b . (s,−
1
)
We denote the function space ZT 2n by 1 := 1 : kuk s,− 2n kvk s,− 2n ZT
ZT
u(t) = v(t)
on t ∈ [0, T ] .
The main results of this paper are as follows: n 3 1 Theorem 1.1 Let u0 (x) ∈ H (− 2 + 4 ,− 2n ) (R). Then the Cauchy problem for (1.1) is locally well-posed in the sense of Hadamard. Theorem 1.2 Let n be even and s < − n2 + 34 and a ∈ R. Then there is no T such that the flow map for (1.1), u0 → u(t) is C 3 as a map from Br (H (s,a) ) to H (s,a) for any t ∈ [0, T ].
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Remark Theorem 1 is sharp in the sense of Theorem 2. The rest of paper is arranged as follows. In Section 2, we present some preliminaries. In Section 3, we give some crucial bilinear estimates. In Section 4, we prove Theorem 1.1. In Section 5, we prove Theorem 1.2. In Section 6, we give Appendix which contains the example where the bilinear estimates is invalid in modified Bourgain spaces.
2
Preliminaries In this section, we make some preparations for the proof of bilinear estimates of the Section
3. Lemma 2.1 Assume that n ∈ N + . Then, for ξ = ξ1 + ξ2 , τ = τ1 + τ2 , we have that 2n+1 nξ n 2n+1 n 2n+1 τ + (−1) − τ + (−1) ξ − τ + (−1) ξ 1 2 1 2 4n 2 ξ ξ 2n+1 2n+1 2n+1 n+1 n+1 = (−1) = (−1) ξ ξ1 − F1 (ξ, ξ1 ), (2.1) ξ1 + ξ2 − 4n 2
where
"
2 2n−2 2n−2 1 1 ξ 4 ξ 2n−2 + C2n+1 ξ 2n−4 ξ1 − 2 2 2 2n−2 # ξ 2n + · · · + C2n+1 . ξ1 − 2
2 F1 (ξ, ξ1 ) = C2n+1
We omit the process of proof since the proof of Lemma 2.1 is easy. Lemma 2.2 Assume that each f and g is supported on Aj1 , Aj2 , respectively. Then k|ξ|
2n−1 4
f ∗ gkL2 (R2 ) ≤ Ckf k
1
ˆ 0, 2 X (2,1)
kgk
1
ˆ 0, 2 X (2,1)
.
(2.2)
Moreover, if K := inf {|ξ1 − ξ2 | : τ1 , τ2 , s.t.(τ1 , ξ1 ) ∈ suppf, (τ2 , ξ2 ) ∈ suppg} > 0, then, we have that k|ξ|1/2 f ∗ gkL2 (R2 ) ≤ CK − Proof
2n−1 2
kf k
1
ˆ 0, 2 X (2,1)
kgk
1
ˆ 0, 2 X (2,1)
.
To obtain (2.2) and (2.3), we firstly prove that Z Z (χBk1 f )(τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2
(2.3)
τ =τ1 +τ2
≤ C2
k1 +k2 2
kf kL2τ ξ (Bk1 ) kgkL2τ ξ (Bk2 ) k|ξ|−
2n−1 4
hkL2τ ξ
(2.4)
and Z Z (χBk1 f )(τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
≤ CK
− 2n−1 2
2
k1 +k2 2
1
kf kL2τ ξ (Bk1 ) kgkL2τ ξ (Bk2 ) k|ξ|− 2 hkL2τ ξ ,
(2.5)
where f, g are restricted to Bk1 , BK2 for k1 , k2 ≥ 0. By using the Cauchy-Schwartz inequality twice and the Fubini’s Theorem, we have that Z Z (χBk1 f )(τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
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≤C
sup (τ,
ξ)∈R2
m(τ, ξ)1/2 kf kL2τ ξ (Bk1 ) kgkL2τ ξ (Bk2 ) k|ξ|−1/2 hkL2τ ξ ,
(2.6)
where Z m(τ, ξ) = χΛ1 (τ1 , ξ1 , τ, ξ)dτ1 dξ1 , τ = τ1 + τ2 , ξ = ξ1 + ξ2 , Λ1 := (τ1 , ξ1 , τ, ξ) ∈ R4 , (τ1 , ξ1 ) ∈ supp(χBk1 ∩Aj1 f ), (τ2 , ξ2 ) ∈ supp(χBk2 ∩Aj2 g) .
Hence, (2.4) and (2.5) are reduced to o n 2n−1 m(τ, ξ) ≤ Cmin |ξ|− 2 2k1 +k2 , K −(2n−1) |ξ|−1 2k1 +k2 .
(2.7)
Now we prove (2.7). We fix τ, ξ 6= 0 and assume that G1 and G2 be the projections of Λ1 onto the ξ1 -axis and τ1 -axis, respectively. From (2.1), we have that 4|M − C(2k1 + 2k2 )| 2 |M + C(2k1 + 2k2 )| max ,K , (2.8) ≤ |2ξ1 − ξ|2 ≤ 4 |ξ|F1 (ξ, ξ1 ) |ξ|F1 (ξ, ξ1 ) 2n+1 where M = τ + (−1)n ξ 4n and C is some positive constant. 1/2 k1 +2k2 )| When K ≥ 4|M−C(2 , then the variation of |2ξ1 − ξ| is bounded by |ξ|F1 (ξ,ξ1 )
4|M + C(2k1 + 2k2 )| |ξ|F1 (ξ, ξ1 )
≤ |ξ|F1 (ξ, ξ1 ) ≤
1/2
−K =
C(2k1 + 2k2 )
4|M+C(2k1 +2k2 )| |ξ|F1 (ξ,ξ1 )
1/2
4|M+C(2k1 +2k2 )| |ξ|F1 (ξ,ξ1 )
4|M+C(2k1 +2k2 )| |ξ|F1 (ξ,ξ1 )
+K
C(2k1 + 2k2 ) C(2k1 + 2k2 ) ≤ . |ξ|F1 (ξ, ξ1 )K |ξ|K 2n−1
− K2
1/2
+K (2.9)
(2.10)
Moreover, from (2.9), the length of the interval |2ξ1 − ξ| is bounded by C(2k1 + 2k2 ) 1/2
|ξ|1/2 F1 (ξ, ξ1 )1/2 (|M + C(2k1 + 2k2 ))
≤
C(2k1 /2 + 2k2 /2 ) |ξ|
2n−1 2
.
From (2.10) and (2.11), we have that n o 2n−1 mes G1 ≤ Cmin |ξ|− 2 (2k1 /2 + 2k2 /2 ), K −(2n−1) |ξ|−1 (2k1 + 2k2 ) . When K ≤ by
4|M−C(2k1 +2k2 )| |ξ|F1 (ξ,ξ1 )
1/2
(2.11)
(2.12)
, we have that the length of the interval of |2ξ1 − ξ| is bounded
1/2 1/2 4|M + C(2k1 + 2k2 )| 4|M − C(2k1 + 2k2 )| − |ξ|F1 (ξ, ξ1 ) |ξ|F1 (ξ, ξ1 ) n o 2n−1 ≤ Cmin |ξ|− 2 (2k1 /2 + 2k2 /2 ), K −(2n−1) |ξ|−1 (2k1 + 2k2 ) .
From the above inequality, we have that in this case o n 2n−1 mes G1 ≤ Cmin |ξ|− 2 (2k1 /2 + 2k2 /2 ), K −(2n−1) |ξ|−1 (2k1 + 2k2 ) .
From
|τ1 + (−1)n ξ12n+1 | ≤ C2k1 ,
|τ − τ1 + (−1)n (ξ − ξ1 )2n+1 +
1 | ≤ C2k2 , ξ − ξ1
(2.13)
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we have that mes G2 ≤ Cmin 2k1 , 2k2 .
(2.14)
Combining (2.11), (2.13) with (2.14), we have (2.7).
We have completed the proof of Lemma 2.2. Lemma 2.3 Assume that f ∈ S (R), g ∈ S ′ (R) with suppg ⊂ Aj with j ≥ 0. Then, we have kf ∗ gkL2 (Bk ) ≤ C2k/4 kgk ˆ 0, 12 k|ξ|− X(2,1)
2n−1 4
f kL2 (R2 ) .
(2.15)
Moreover, if a non-empty set Ω ∈ R2 satisfies K1 := inf {|ξ2 + ξ| : τ, τ2 , s.t.(τ, ξ) ∈ Ω, (τ2 , ξ2 ) ∈ suppg} > 0, then, we have that − 2n−1 2
kf ∗ gkL2 (Ω∩Bk ) ≤ C2k/2 K1 Proof
Ck|ξ|−1/2 f kL2 (R2 ) kgkXˆ
0, 1 ,1 2
To obtain (2.15) and (2.16), we firstly prove that Z Z f (τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2
.
(2.16)
τ =τ1 +τ2
≤ C2
k2 /2+k/2
k|ξ|−
2n−1 4
f kL2τ ξ kgkL2τ ξ (Bk2 ) khkL2τ ξ
(2.17)
for h ∈ L2τ ξ (Bk ) and Z Z f (τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
≤
− 2n−1 CK1 2 2k/2+k3 /2 kf kL2τ ξ kgkL2τ ξ (Bk2 ) k|ξ|−1/2 hkL2τ ξ
(2.18)
for h ∈ L2τ ξ (Bk ∩ Ω). By using the Cauchy-Schwartz inequality twice and the Fubini’s Theorem, we have that Z Z f (τ1 , ξ1 )(g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
≤C
sup
(τ, ξ)∈R2
where m1 (τ, ξ) =
m1 (τ, ξ)1/2 kf kL2τ ξ kgkL2τ ξ khkL2τ ξ ,
(2.19)
R
χΛ2 (τ1 , ξ1 , τ, ξ)dτ1 dξ1 , and Λ2 := (τ1 , ξ1 , τ, ξ) ∈ R4 , (τ2 , ξ2 ) ∈ suppf, (τ, ξ) ∈ supph .
Hence, (2.15) and (2.16) are reduced to o n 2n−1 m1 (τ, ξ) ≤ Cmin |ξ|− 2 2k+k2 , K −(2n−1) |ξ1 |−1 2k+k2 .
(2.20)
Now we prove (2.20). We fix τ, ξ 6= 0 and assume that H1 and H2 be the projections of Λ2 onto the ξ1 -axis and τ1 -axis, respectively. It is easily checked that 2n+1 n ξ1 τ1 + (−1) − τ + (−1)n ξ 2n+1 + τ2 + (−1)n ξ22n+1 4n 2 ξ 2n+1 ξ = (−1)n+1 ξ 2n+1 − ξ22n+1 − 1 n = (−1)n+1 ξ1 ξ1 − F2 (ξ, ξ1 ), (2.21) 4 2
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where F2 (ξ, ξ1 ) 2n−2 2n−2 2 2n−2 ξ 1 1 ξ 2n−2 2n−4 2 4 2n = C2n+1 ξ1 + C2n+1 ξ1 ξ1 − + · · · + C2n+1 ξ1 − . 2 2 2 2 From (2.21), we have that |M + C(2k + 2k2 )| 4|M − C(2k + 2k2 )| 2 max ≤ |ξ − 2ξ1 |2 ≤ 4 ,K , (2.22) |ξ|F2 (ξ, ξ1 ) |ξ|F2 (ξ, ξ1 ) ξ 2n+1 where M1 = τ1 + (−1)n 14n and C is some positive constant. 1/2 k +2k2 )| , we have that the variation of |2ξ − ξ1 | is bounded by When K ≥ 4|M−C(2 |ξ1 |F2 (ξ,ξ1 )
4|M + C(2k + 2k2 )| |ξ1 |F2 (ξ, ξ1 )
≤ |ξ1 |F2 (ξ, ξ1 ) ≤
1/2
−K =
C(2k + 2k2 )
4|M+C(2k +2k2 )| |ξ1 |F2 (ξ,ξ1 )
4|M+C(2k1 +2k2 )| |ξ1 |F2 (ξ,ξ1 )
4|M+C(2k +2k2 )| |ξ1 |F2 (ξ,ξ1 )
1/2
+K
C(2k + 2k2 ) C(2k + 2k2 ) ≤ . |ξ1 |F2 (ξ, ξ1 )K |ξ1 |K 2n−1
− K2
1/2
+K (2.23)
(2.24)
Moreover, from (2.23), the variation of |ξ − 2ξ1 | is also bounded by C(2k + 2k2 ) |ξ1 |1/2 F2 (ξ, ξ1 )1/2 (|M + C(2k + 2k2 ))
1/2
≤
C(2k/2 + 2k2 /2 ) |ξ1 |
2n−1 2
.
From (2.24) and (2.25), we have that o n 2n−1 mes H1 ≤ Cmin |ξ1 |− 2 2k/2 + 2k2 /2 ), K −(2n−1) |ξ1 |−1 (2k + 2k2 ) . When K ≤ by
4|M−C(2k +2k2 )| |ξ1 |F2 (ξ,ξ1 )
1/2
(2.25)
(2.26)
, we have that the length of the interval of |2ξ1 − ξ| is bounded
1/2 1/2 4|M − C(2k1 + 2k2 )| 4|M + C(2k1 + 2k2 )| − |ξ|F2 (ξ, ξ1 ) |ξ|F2 (ξ, ξ1 ) n o 2n−1 ≤ Cmin |ξ|− 2 2k1 /2 + 2k2 /2 ), K −(2n−1) |ξ|−1 (2k1 + 2k2 ) .
In this case,
From
o n 2n−1 mes H1 ≤ Cmin |ξ1 |− 2 2k/2 + 2k2 /2 ), K −(2n−1) |ξ1 |−1 (2k + 2k2 ) . n 2n+1
|τ + (−1) ξ
k
| ≤ C2 ,
we have that
τ − τ1 + (−1)n (ξ − ξ1 )2n+1 + 1 ≤ C2k2 , ξ − ξ1
mes H2 ≤ Cmin 2k , 2k2 .
By using the Cauchy-Schwartz inequality and the triangle inequality, we have that Z Z f (τ1 , ξ1 )g(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
(2.27)
(2.28)
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X Z Z ≤C f (τ1 , ξ1 )(χBk3 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ . R2 ξ=ξ1 +ξ2 k3
(2.29)
τ =τ1 +τ2
Combining (2.17)–(2.18), (2.26)–(2.27) with (2.28)–(2.29), we have (2.15)–(2.16). We have completed Lemma 2.3. 0,
1 2
ˆ Lemma 2.4 The space X (2,1) has the following properties. (i) For any b > 1/2, there exists C > 0 such that k|f k
≤ Ckf kXˆ 0, b .
1
ˆ 0, 2 X (2,1)
(2.30)
(ii) For 1 < p ≤ 2, there exists C > 0 such that kf kL2ξ Lpτ ≤ Ckf k ˆ 0, 21 ,
(2.31)
kf kL2ξ L1τ ≤ Ckf k
(2.32)
X(2,1) 1
ˆ 0, 2 X (2,1)
.
Lemma 2.4 can be proved similarly to Lemma 2.9 of [18]. 2 2 P P Lemma 2.5 Let τ = τj , ξ = ξj . Then, we have that j=1
j=1
τ + (−1)n ξ 2n+1 − τ1 + (−1)n ξ 2n+1 − τ2 + (−1)n ξ 2n+1 1 2 = ξ 2n+1 − ξ12n+1 − ξ22n+1 ∼ |ξmin ||ξmax |2n ,
where
|ξmin | := min {|ξ|, |ξ1 |, |ξ2 |} ,
|ξmax | := max {|ξ|, |ξ1 |, |ξ2 |} .
For the proof of Lemma 2.6, we refer the reader to Lemma 2.5 of [21]. 3 P Lemma 2.6 Let ξ = ξj . Then, we have that j=1
ξ
2n+1
−
ξ12n+1
− ξ22n+1 − ξ32n+1 = C(ξ1 + ξ2 )(ξ1 + ξ3 )(ξ2 + ξ3 )f (ξ1 , ξ2 , ξ3 ),
where f (ξ1 , ξ2 , ξ3 ) ≥ C (ξ1 + ξ2 )2n−2 + (ξ1 + ξ3 )2n−2 + (ξ2 + ξ3 )2n−2 .
For the proof of Lemma 2.6, we refer the reader to Lemma 5 of [10]. Lemma 2.7 Let ǫ > 0. Then, we have that ˆ s, a, X Proof
n+1 2n +ǫ
ˆ s, a, n+1 4n . ֒→ Zˆ s, a ֒→ X
From the definition, it is easily checked that Lemma 2.7 is true.
Lemma 2.8 Let s, a ∈ R and u(t) = φ(t)W (t)u0 . Then, we have that kukZ s, a + kukL∞ (R;H (s,a) ) ≤ Cku0 kH (s,a) . x
t
Proof
(s,a)
From the definition of Z s, a and Hx
x
, we can obtain Lemma 2.8.
Lemma 2.9 Let s, a ∈ R and u(t) = φ(t)
Z
t
W (t − s)f (s)ds. 0
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Then, we have that kukZ s, a + kukL∞(R;H (s,a) ) x t
≤ C F −1 hτ + (−1)n ξ 2n+1 i−1 F f Z s, a + C hξis−a |ξ|a hτ + (−1)n ξ 2n+1 i−1 F f L2 L1 . ξ
τ
For the proof of Lemma 2.9, we refer the reader to [6, 9, 11].
3
Dyadic Bilinear Estimates
1 , f and g are restricted to Aj1 and Aj2 Lemma 3.1 Suppose s = − n2 + 34 and a = − 2n with j1 , j2 ∈ N ∪ {0}, respectively. Then we have the estimates
hτ + (−1)n ξ 2n+1 i−1 ξ(ξ n f ∗ g) ˆs, a ≤ Ckf kZˆ s, a kgkZˆ s, a , (3.1) (Aj ) Z
for j ≥ 0 in the following cases. (i) At least two of j, j1 , j2 are less than 30 and C(j, j1 , j2 ) ∼ 1. 1 (ii) j1 , j2 ≥ 30, |j1 − j2 | ≤ 10, 0 < j < j1 − 9 and C(j, j1 , j2 ) ∼ 2− 2 j . (iii) j, j1 ≥ 30, |j − j1 | ≤ 10, 0 < j2 < j − 10 and C(j, j1 , j2 ) ∼ 2−δj2 + 2−δ(j−j2 ) for some δ > 0. (iv) j, j2 ≥ 30, |j − j2 | ≤ 10, 0 < j1 < j − 10 and C(j, j1 , j2 ) ∼ 2−δj1 + 2−δ(j−j1 ) for some δ > 0. (v) j, j1 , j2 ≥ 30, |j − j1 | ≤ 10, |j − j2 | ≤ 10 and C(j, j1 , j2 ) ∼ 1. (vi) j1 , j2 ≥ 30, j = 0 and C(j, j1 , j2 ) ∼ 1. (vii) j, j1 ≥ 30, j2 = 0 and C(j, j1 , j2 ) ∼ 1. (viii) j, j2 ≥ 30, j1 = 0, and C(j, j1 , j2 ) ∼ 1. Proof (i) In this case we may assume that j, j1 , j2 are all less than 40. By using the Young inequality and H¨ older inequality as well as Lemma 2.7, we have that the left hand side of (3.1) can be bounded kf ∗ gkL∞ L2 ≤ Ckf kL2 L4/3 kgkL2 L4/3 ≤ Ckf k ˆ 0,− ξ
τ
ξ
τ
ξ
X
τ
1 , n+1 2n 4n
kgk ˆ 0,− X
1 , n+1 2n 4n
≤ Ckf kZˆ s,a kgkZˆ s,a .
(ii) From Lemma 2.5, in this case, we have that 2kmax := 2max{k, k1 , k2 } ≥ C2j+2nj1 .
(3.2)
In this case, by using (3.2), the left hand side of (3.1) can be bounded by X n 7 C2(− 2 + 4 )j 2nj1 2−k/2 kf ∗ gkL2τ ξ (Aj ∩Bk ) .
(3.3)
k≥0
When 2k ≥ C2j+2nj1 , by using (2.3), since n ≥ 2, by using the definition of Zˆ s,a , we have that (3.3) can bounded by X n 7 3 C2(− 2 + 4 )j 2(2n− 2 )j1 2−k/2 k(hξis f ) ∗ (hξis g)kL2 (Bk ) τξ
k≥j+2nj1 +O(1)
n
5
n
3
3
≤ C2(− 2 + 4 )j 2(n− 2 )j1 k(hξis f ) ∗ (hξis g)kL2 ≤ C2(− 2 + 4 )j 2−j1 kf k ˆ s, 21 kgk ˆ s, 21 X(2,1)
≤ C2
−5j/4
X(2,1)
kf k ˆ s, 21 kgk ˆ s, 21 ≤ C2−5j/4 kf kZˆ s,a kgkZˆ s,a . X(2,1)
X(2,1)
(3.4)
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1
When 2k1 ≥ C2j+2nj1 , we have that 2 4n k1 2− 4n (j+2nj1 ) 2 4n (k1 −k) ≥ C, since n ≥ 2, by using the definition of Zˆ s,a and (2.16), (3.3) can be bounded by
X n 5 1
C2(− 2 + 4 + 4n )j 2(n−1)j1 2−(1+2n)k/4n hξis hτ + (−1)n ξ 2n+1 i1/2 f ∗ (hξis g) 2 L (Aj ∩Bk )
k≥0
≤ C2
5 1 (− n 2 + 4 + 4n )j
2(n−1)j1
X
2−k/4n kf k
k≥0 n
5
1
≤ C2(− 2 + 4 + 4n )j 2−j1 kf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
1
1
ˆ s, 2 X (2,1)
≤ C2− 2 j kf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
1
≤ C2− 2 j kf kZˆ s,a kgkZˆ s,a . When 2k2 ≥ C2j+2j1 , this case can be treated similarly to 2k1 ≥ C2j+2nj1 . (iii) In this case, from Lemma 2.5, we have that 2max ≥ C22nj+j1 . Thus, in this case, the left hand side of (3.1) can be bounded by X n 7 C2(− 2 + 4 )j 2nj1 2−k/2 kf ∗ gkL2 L2τ .
(3.5)
ξ
k≥0
When 2k ∼ 2kmax ≥ 22nj+j2 , by using (2.3), (3.5) can be bounded by X n 3 n 7 3n 3 C2( 2 − 4 )j2 2(− 2 + 4 )j 2( 2 − 4 )j1 2−k/2 k(hξis f ) ∗ (hξis g)kL2
τ ξ (Bk )
k≥2nj+j2 +O(1)
n
n
5
≤ C2( 2 − 4 )j2 2j1 k(hξis f ) ∗ (hξis g)kL2
τ ξ (Bk )
5
≤ C2( 2 − 4 )j2 2(1−n)j1 kf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
,
if n = 2, (3.5) can be bounded by 1
1
C2− 4 j2 2−j1 kf k ˆ s, 21 kgk ˆ s, 21 ≤ C2− 4 j2 2−j1 kf kZˆ s,a kgkZˆ s,a ; X(2,1)
X(2,1)
if n > 2, (3.5) can be bounded by n
n
5
n
1
n
5
1
2(− 2 + 4 )(j−j2 ) 2(− 2 − 4 )j kf k ˆ s, 21 kgk ˆ s, 21 ≤ C2(− 2 + 4 )(j−j2 ) 2(− 2 − 4 )j kf kZˆ s,a kgkZˆ s,a . X(2,1)
X(2,1)
When 2k1 ∼ 2kmax ≥ 22nj+j2 , with the aid of (2.16) and the fact that 2−
2n−1 4n (2nj+j2 )
1
2(2n−1)k1 /4n 2 4n (k1 −k) ≥ C,
(3.5) can be bounded by n
3
5
1
C2 2 j 2( 2 − 4 + 4n )j2
X
k≥0 3
n
5
1
2−(2n+1)k/4n hξis hτ + (−1)n ξ 2n+1 i1/2 f ∗ (hξis g)
L2 (Aj ∩Bk )
≤ C2(−n+ 2 )j 2( 2 − 4 + 4n )j2
X
2−k/4n kf k
k≥0
if n = 2, (3.6) can be bounded by X 1 1 C2− 2 j 2− 8 j2 2−k/4n kf k k≥0
1
ˆ s, 2 X (2,1)
1
ˆ s, 2 X (2,1)
kgk
kgk
1
ˆ s, 2 X (2,1)
,
1
1
ˆ s, 2 X (2,1)
(3.6)
1
≤ C2− 2 j 2− 8 j2 kf kZˆ s,a kgkZˆ s,a ,
if n > 2, (3.6) can be bounded by 1
3
C2(−n+ 4n +1)j1 2(−n+ 2 )j kf k
1
ˆ s, 2 X (2,1)
kgk
n
1
ˆ s, 2 X (2,1)
5
1
When 2k2 ∼ 2kmax ≥ 22nj+j2 , with the aid of (2.16) and the fact that 2−
2n−1 4n (2nj+j2 )
n
≤ C2−( 2 − 4 + 4n )(j−j2 ) 2−( 2 −
1
2(2n−1)k2 /4n 2 4n (k2 −k) ≥ C,
n+1 4n )j
kf kZˆ s,a kgkZˆ s,a .
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(3.5) can be bounded by n
3
5
1
C2 2 j 2( 2 − 4 + 4n )j2
X
k≥0 n
3
5
1
2−(2n+1)k/4n (hξis f ) ∗ (hξis hτ + (−1)n ξ 2n+1 i1/2 g)
L2 (Aj ∩Bk )
≤ C2(−n+ 2 )j 2( 2 − 4 + 4n )j2
X
2−k/4n kf k ˆ s, 12 kgk ˆ s, 21 , X(2,1)
k≥0
if n = 2, (3.7) can be bounded by X 1 1 C2− 2 j 2− 8 j2 2−k/4n kf k k≥0
1
ˆ s, 2 X (2,1)
kgk
(3.7)
X(2,1)
1
1
ˆ s, 2 X (2,1)
1
≤ C2− 2 j 2− 8 j2 kf kZˆ s,a kgkZˆ s,a ,
if n > 2, (3.7) can be bounded by 3
1
C2(−n+ 4n +1)j1 2(−n+ 2 )j kf k
1
ˆ s, 2 X (2,1)
kgk
n
1
ˆ s, 2 X (2,1)
5
n
1
≤ C2−( 2 − 4 + 4n )(j−j2 ) 2−( 2 −
n+1 4n )j
kf kZˆ s,a kgkZˆ s,a .
(iv) In this case, from Lemma 2.5, we have that 2max ≥ C22nj+j1 . Thus, in this case, the left hand side of (3.1) can be bounded by X n 7 C2(− 2 + 4 )j 2nj1 2−k/2 kf ∗ gkL2 L2τ . (3.8) ξ
k≥0
When 2k ∼ 2kmax ≥ 22nj+j1 , by using (2.3), (3.8) can be bounded by X n 3 n 7 3n 3 C2( 2 − 4 )j2 2(− 2 + 4 )j 2( 2 − 4 )j1 2−k/2 k(hξis f ) ∗ (hξis g)kL2
τ ξ (Bk )
k≥2nj+j1 +O(1)
≤ C2( ≤ C2
3n 5 2 − 4 )j1
5 ( 3n 2 − 4 )j2
≤ C2(−
2(1−n)j2 k(hξis f ) ∗ (hξis g)kL2
τ ξ (Bk )
2(1−2n)j1 kf k
3n 5 2 + 4 )(j−j1 )
n
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
2(−
3n 5 2 + 4 )(j−j1 )
n
1
2(− 2 − 4 )j kf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
1
2(− 2 − 4 )j kf kZˆ s,a kgkZˆ s,a .
When 2k1 ∼ 2kmax ≥ 22nj+j1 , with the aid of (2.16) and the fact that 2−
2n−1 4n (2nj+j1 )
1
2(2n−1)k1 /4n 2 4n (k1 −k) ≥ C,
(3.8) can be bounded by 3
n
5
1
C2(−n+ 2 )j 2( 2 − 4 + 4n )j1
X
k≥0 3
n
5
1
≤ C2(−n+ 2 )j 2( 2 − 4 + 4n )j1
X
k≥0
2−(2n+1)k/4n hξis hτ + (−1)n ξ 2n+1 i1/2 f ∗ (hξis g)
L2 (Aj ∩Bk )
2−k/4n kf k ˆ s, 21 kgk ˆ s, 12 , X(2,1)
X(2,1)
(3.9)
if n = 2, (3.9) can be bounded by X 1 1 1 1 C2− 2 j 2− 8 j1 2−k/4n kf k ˆ s, 21 kgk ˆ s, 12 ≤ C2− 2 j 2− 8 j1 kf kZˆ s,a kgkZˆ s,a , X(2,1)
k≥0
X(2,1)
if n > 2, (3.9) can be bounded by 1
3
C2(−n+ 4n +1)j1 2(−n+ 2 )j kf k
1
ˆ s, 2 X (2,1)
kgk
n
1
ˆ s, 2 X (2,1)
5
1
When 2k2 ∼ 2kmax ≥ 22nj+j1 , with the aid of (2.16) and the fact that 2−
2n−1 4n (2nj+j1 )
n
≤ C2−( 2 − 4 + 4n )(j−j1 ) 2−( 2 −
1
2(2n−1)k2 /4n 2 4n (k2 −k) ≥ C,
n+1 4n )j
kf kZˆ s,a kgkZˆ s,a .
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(3.8) can be bounded by n
3
5
1
C2(−n+ 2 )j 2( 2 − 4 + 4n )j1
X
k≥0 n
3
5
1
≤ C2(−n+ 2 )j 2( 2 − 4 + 4n )j1
X
2−(2n+1)k/4n (hξis f ) ∗ (hξis hτ + (−1)n ξ 2n+1 i1/2 g)
L2 (Aj ∩Bk )
2−k/4n kf k ˆ s, 21 kgk ˆ s, 12 , X(2,1)
X(2,1)
k≥0
(3.10)
if n = 2, (3.10) can be bounded by X 1 1 1 1 C2− 2 j 2− 8 j1 2−k/4n kf k ˆ s, 21 kgk ˆ s, 12 ≤ C2− 2 j 2− 8 j1 kf kZˆ s,a kgkZˆ s,a , X(2,1)
k≥0
X(2,1)
if n > 2, (3.10) can be bounded by n
3
1
5
n
1
C2(−n+ 4n +1)j1 2(−n+ 2 )j kf k ˆ s, 12 kgk ˆ s, 21 ≤ C2−( 2 − 4 + 4n )(j−j1 ) 2−( 2 − X(2,1)
n+1 4n )j
X(2,1)
kf kZˆ s,a kgkZˆ s,a .
(v) In this case, (3.1) can be bounded by X 7 n 2( 2 + 4 )j 2−k/2 kf ∗ gkL2τ ξ (Bk ) .
(3.11)
k≥0
(a) When 2k ∼ 2kmax ≥ 22nj1 +j , (3.8) can be bounded by X 6n+1 2−k/2 k(hξis f ) ∗ (hξis g)kL2 C2 4 j
τ ξ (Bk )
k≥(2n+1)j+O(1)
≤ C2
2n−1 j 4
k(hξis f ) ∗ (hξis g)kL2 ≤ Ckf k τξ
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.12)
with the aid of (2.2). (b) When 2k1 ∼ 2kmax ≥ 22nj1 +j , with the aid of (2.15), (3.11) can be bounded by 6n+1 X C2 4 j 2−k/2 k(hξis f ) ∗ (hξis g)kL2 (Bk ) τξ
k≥0
≤ C2
2n−1 j 4
X
k≥0
≤C
X
2−k/2 (hξis hτ + (−1)n ξ 2n+1 i1/2 f ) ∗ (hξis g)
L2τ ξ
2−k/4 kf k ˆ s, 12 kgk ˆ s, 21 ≤ Ckf k ˆ s, 12 kgk ˆ s, 12 ≤ Ckf kZˆ s,a kgkZˆ s,a . X(2,1)
k≥0
X(2,1)
X(2,1)
X(2,1)
(3.13)
(c) When 2k2 ≥ 22nj1 +j , this case can be proved similarly to 2k1 ∼ 2kmax ≥ 22nj1 +j . (vi) We prove
2nj1 hτ + (−1)n ξ 2n+1 i−1 ξf ∗ g Xˆ a ≤ Ckf k ˆ s, 21 kgk ˆ s, 12 . (3.14) X(2,1)
L
X(2,1)
When |ξ| ≤ 2−2nj1 , since
kf kXˆ a ≤ kf k L
ˆ a, X L
n+1 ,1 2n
,
(3.15)
by using the H¨older inequality and Young inequality as well as (2.31), we have that
n−1 3
2(2n− 2 )j1 |ξ|a+1 hτ i− 2n (hξis f ) ∗ (hξis g) 2 Lτ ξ
≤ C2
(2n− 32 )j1
≤ C2
−(n+ 12 )j1
a+1
|ξ|
2 k(hξis f ) ∗ (hξis g)kL∞ L2 L (|ξ|≤2−2nj1 ) ξ
ξ
s
khξi f kL2 L4/3 khξi gkL2 L4/3 ξ
τ
ξ
τ
≤ Ckf k ˆ s, 21 kgk ˆ s, 21 ≤ Ckf kZˆ s,a kgkZˆ s,a . X(2,1)
τ
s
X(2,1)
(3.16)
No.4
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When 2−2nj1 ≤ |ξ| ≤ 1, we consider 2k2 ∼ 2kmax , 2k1 ∼ 2kmax and 2k ∼ 2kmax , respectively. When 2k2 ∼ 2kmax , (3.14) can be bounded by X n−1
3 2(2n− 2 )j1 (3.17) 2− 2n k |ξ|a+1 (hξis f ) ∗ (hξis g) L2 (B ) , k
τξ
k≥0
since −s ≤ n(a + 1), by using 2k2 ≥ C|ξ|22nj1 , we have s
1
3
3
2−2sj1 |ξ|a+1 ≤ C(|ξ|22nj1 )− n ≤ C2( 2 − 4n )k2 ≤ C2k2 /2 2− 4n k , by using (2.16), we have that (3.17) can be bounded by
X 2n+1
C2nj1 2− 4n k (hξis f ) ∗ (hξis hτ + (−1)n ξ 2n+1 i1/2 g)
L2τ ξ (Bk )
k≥0
≤C
X
1
2− 4n k kf k
k≥0
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
≤ Ckf k
kgk
1
ˆ s, 2 X (2,1)
1
ˆ s, 2 X (2,1)
≤ Ckf kZˆ s,a kgkZˆ s,a .
When 2k1 ∼ 2kmax , this case can be proved similarly to case 2k2 ∼ 2kmax . When 2k = 2kmax ≤ 4max 2k1 , 2k2 , this case can be proved similarly to case 2k2 ∼ 2kmax . When 2k = 2kmax > 4max 2k1 , 2k2 , then 2k ∼ C|ξ|22nj1 , in this case, we consider (a) f ∗ g is restricted to D1 ; (b) f ∗ g is outside of D2 . 2n−1 2n+1 If f ∗ g is restricted to D1 , then 2− 2 j1 ≤ |ξ| ≤ 1 and 2 2 j1 ≤ |τ | ≤ 22nj1 , by using (2.3), we have that
1
n−1 1 1
2nj1 |ξ|1− 2n hτ i− 2n f ∗ g 2 ≤ C2(n− 2 )j1 |ξ| 2 (hξis f ) ∗ (hξis g) 2 Lτ ξ
Lτ ξ
≤ Ckf k ˆ s, 12 kgk ˆ s, 12 ≤ Ckf kZˆ s,a kgkZˆ s,a . X(2,1)
X(2,1)
− 2n−1 j1 2
2n+1
If f ∗ g is restricted to D2 , then 2−2nj1 ≤ |ξ| ≤ 2 and 1 ≤ |τ | ≤ 2 2 j1 , by using the H¨older inequality and the Young inequality, since |ξ| ∼ 2k−2nj1 , by using (2.31)–(2.32), we have that
X n−1 1
2nj1 2− 2n k |ξ|1− 2n f ∗ g 2 Lτ ξ (Bk )
k≤ 2n+1 j1 +O(1) 2 3
X
≤ C2(2n− 2 )j1
k≤ 2n+1 j1 +O(1) 2 3
X
≤ C2(2n− 2 )j1
n−1 1
2− 2n k |ξ|1− 2n (hξis f ) ∗ (hξis g)
L2τ ξ (Bk )
2−
n−1 2n k
1
k|ξ|1− 2n kL2ξ (|ξ|∼2k−2nj1 ) k(hξis f ) ∗ (hξis g)kL∞ L2τ ξ
k≤ 2n+1 j1 +O(1) 2 1
≤ C2(−n− 2 )j1
X
2k khξis f kL2ξ L1τ khξis gkL2τ ξ
k≤ 2n+1 j1 +O(1) 2
≤ Ckf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.18)
(vii) In this case, we consider |ξ2 | ≤ 2−2nj and 2−2nj ≤ |ξ2 | ≤ 1. When |ξ2 | ≤ 2−2nj , by using the Young inequality and H¨ older inequality and (2.32), we have that (3.1) can be bounded by X 2(n+1)j 2−k/2 k(hξis f ) ∗ gkL2 (Bk ) τξ
k≥0
X 1
≤ C2nj 2−k/2 (hξis f ) ∗ (|ξ2 |− 2n g) k≥0
L2τ ξ (Bk )
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1
− 2n
g
|ξ|
1
≤ C2nj khξis f kL2ξ L1τ k|ξ|− 2n gkL1ξ L2τ ≤ Ckf k ≤ Ckf k
kgk
1
ˆ s, 2 X (2,1)
1 ˆ s, 2 X (2,1)
L2ξ L2τ
≤ Ckf kZˆ s,a kgkZˆ s,a .
1
ˆ − 2n X L
(3.19)
Now we consider the case 2−2nj ≤ |ξ2 | ≤ 1. When 2k = 2kmax , since 1
1
2−j (|ξ2 |)− 2n 2ǫ(k−k2 ) 2 2n k ≥ C, by using (2.3) and (i) of Lemma 2.4, we have kξ(ξ n f ) ∗ gk ˆ s, − 12 X(2,1)
1
≤ C2(n+1)j hτ + (−1)n ξ 2n+1 i− 2 +ǫ (hξis f ) ∗ g 2 Lτ ξ
1 1
≤ C2nj hτ + (−1)n ξ 2n+1 i−( 2 − 2n )+2ǫ (hξis f ) ∗ (|ξ|−1/2n hτ + (−1)n ξ 2n+1 i−ǫ g)
L2τ ξ
≤ Ckf k
1
1
ˆ s, 2 X (2,1)
kχ|ξ|≥2−2nj |ξ|− 2n gk
1
ˆ 0, 2 −ǫ X (2,1)
≤ Ckf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ − 2n , X L
n+1 2n
≤ Ckf kZˆ s,a kgkZˆ s,a . k1
k
k2
1
When 2k1 = 2kmax , by using 2 n 2− 6n |ξ2 |− 2n 2−j 2− 3n ≥ C and (2.16), we have that in this case the left hand side of (3.1) can be bounded by X 2(n+1)j 2−k/2 k(hξis f ) ∗ gkL2 (Bk ) τξ
k≥0
≤ C2nj
X
2−
3n+1 6n k
L2τ ξ (Bk )
k≥0
≤C
X
1 1
(hξis hτ + (−1)n ξ 2n+1 i n f ) ∗ (|ξ|−1/2n hτ + (−1)n ξ 2n+1 i− 3n g)
1
2− 6n k kf k
k≥0
≤ Ckf k ˆ s, 21 kgk X(2,1)
1
ˆ s, n X (2,1) 1
ˆ − 2n , X L
kgk
n+1 2n
1
ˆ − 2n , X L
3n−2 6n
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.20)
When 2k2 = 2kmax , if 2k2 = 2kmax ≤ 4max 2k1 , 2k , this case can be proved similarly to 2k = 2kmax or 2k1 = 2kmax . If 2k2 = 2kmax > 4max 2k1 , 2k , then 2k2 = 2kmax ∼ |ξ2 |22nj , in this case, we consider that g is restricted to D2 and g is restricted to D1 . When g is restricted 2n−1 2n+1 to D2 , then 2−2nj1 ≤ |ξ| ≤ 2− 2 j1 and 1 ≤ |τ | ≤ 2 2 j1 , by using the H¨older inequality and the Young inequality, since |ξ| ∼ 2k−2nj1 , by using (2.22), we have that X k 2(n+1)j 2− 2 k(hξis f ) ∗ gkL2 (Bk ) τξ
k≥0
≤ C2(n+1)j khξis f kL2ξ L1τ kgkL1ξ L2τ
1
≤ C2(n+1)j kf k ˆ s, 12 |ξ2 | 2n 2
Lξ (|ξ2 |∼2k2 −2nj )
X(2,1)
≤ C2(n+1)j kf k ˆ s, 12
X(2,1)
≤ Ckf k ˆ s, 21 kgk X(2,1)
2
X 1
− 2n
g
|ξ2 |
L2τ ξ
k2
X n+1 1
2 2n (k2 −2nj) |ξ2 |− 2n g
L2τ ξ
k2
1 n+1 ˆ − 2n , 2n ,1 X L 2n−1
≤ Ckf kZˆ s,a kgkZˆ s,a .
2n+1
2
2
(3.21)
When g is restricted to D1 , then 2− 2 j ≤ |ξ2 | ≤ 1 and 2 2 j ≤ |τ2 | ≤ 22nj . When g is 2n−1 − 2n−1 j 2 restricted to B[ 2n+1 j, 2n+1 j+α] with 0 ≤ α ≤ 2n−1 ≤ |ξ2 | ≤ 2− 2 j+α , by 2 j. From 2 2
2
No.4
1075
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
using the H¨older inequality and the Young inequality as well as |τ2 | ≥ C22nj |ξ2 | which yields n+1 n+1 |τ2 | 2n ≥ C2(n+1)j |ξ2 | 2n , by using (2.22), we have that X ≤ C2(n+1)j 2−k/2 k(hξis f ) ∗ gkL2 (Bk ) kξ(ξ n f ) ∗ gk s, − 12 ˆ X (B≥2α ) (2,1)
τξ
k≥2α
≤ C2(n+1)j 2−α khξis f kL2ξ L1τ kgkL1ξ L2τ ≤ C2−α kf k ˆ s, 21 k1k X(2,1)
L2ξ (2−
√ ≤ C α2−α kf k ˆ s, 12 kgk X(2,1)
2n−1 j 2n−1 2 ≤|ξ|≤2− 2 j+α )
1
ˆ − 2n , X L
kgk
1
ˆ − 2n , X L
n+1 2n
√ ≤ C α2−α kf kZˆ s,a kgkZˆ s,a . When g is restricted to B[ 2n+1 j+γ,2nj] with 0 ≤ γ ≤ 2 which yields |τ2 |
n+1 2n
2
≥ C2(n+1)j |ξ2 |
kξ(ξ n f ) ∗ gk
1
ˆ s, − 2 (B≤2α ) X (2,1)
2n−1 j 2
n+1 2n
(3.22) , by using (2.16) and |τ2 | ≥ C22nj |ξ2 |
n+1 2n
, we have that X ≤ C2(n+1)j 2−k/2 k(hξis f ) ∗ gkL2
τ ξ (Bk )
k≤2α
3j
≤ C2 2
X
kf k
k≤2α
≤ Cα2−γ kf k
1
ˆ s, 2 X (2,1) 1
ˆ s, 2 X (2,1)
k|ξ|−1/2 gk
kgk
1
ˆ − 2n , X L
L2τ ξ (|ξ|≥2−
2n−1 j+γ 2 )
n+1 2n
≤ Cα2−γ kf kZˆ s,a kgkZˆ s,a .
(3.23)
If g is restricted to B[ 2n+1 j+γ, 2n+1 j+α] with γ ≤ α, from (3.22) and (3.23), we have that 2 2 √ n kξ(ξ f ) ∗ gk ˆ s, − 12 ≤ C( α2−α + α2−γ )kf kZˆ s,a kgkZˆ s,a . (3.24) X(2,1)
Let an (0 ≤ n ≤ N ) be the decreasing sequence satisfying a0 =
2n − 1 an 3 j, an+1 = , 0 < aN ≤ . 2 2 2
(3.25)
where N is the minimum integer satisfying N ≥ log2 nj . Firstly we take α = a0 and γ = a1 and next take α = a1 and γ = a2 . Repeating this procedure at the end we take α = aN and γ = 0. By using the fact that 2x ≥ x2 ≥ x with x ≥ 4, combining (3.24), we have that X 1 kf k ˆ s,a kgk ˆ s,a ≤ Ckf k ˆ s,a kgk ˆ s,a . kξ(ξ n f ) ∗ gk s, − 12 ≤ C 1 + Z Z Z Z ˆ (B≤2α ) an X (2,1) 0≤n≤N
(viii) This case can be proved similarly to case (vii). We have completed the proof of Lemma 3.1.
Lemma 3.2 Suppose f and g are restricted to Aj1 and Aj2 , respectively. Then we have the estimates
s−a
hξi hτ + (−1)n ξ 2n+1 i−1 |ξ|a+1 (ξ n f ) ∗ g) L2 L1 ≤ Ckf kZˆ s, a kgkZˆ s, a (3.26) ξ
τ
for j ≥ 0 in the following cases. (i) At least two of j, j1 , j2 are less than 30 and C(j, j1 , j2 ) ∼ 1. 3 (ii) j1 , j2 ≥ 30, |j1 − j2 | ≤ 10, 0 < j < j1 − 9 and C(j, j1 , j2 ) ∼ 2− 8 j . (iii) j, j1 ≥ 30, |j − j1 | ≤ 10, 0 < j2 < j − 10 and C(j, j1 , j2 ) ∼ 2−δj2 + 2−δ(j−j2 ) for some δ > 0.
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ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
(iv) j, j2 ≥ 30, |j − j2 | ≤ 10, 0 < j1 < j − 10 and C(j, j1 , j2 ) ∼ 2−δj1 + 2−δ(j−j1 ) for some δ > 0. (v) j, j1 , j2 ≥ 30, |j − j1 | ≤ 10, |j − j2 | ≤ 10 and C(j, j1 , j2 ) ∼ 1. (vi) j1 , j2 ≥ 30, j = 0 and C(j, j1 , j2 ) ∼ 1. (vii) j, j1 ≥ 30, j2 = 0 and C(j, j1 , j2 ) ∼ 1. (viii) j, j2 ≥ 30, j1 = 0, and C(j, j1 , j2 ) ∼ 1. Proof (i) In this case we may assume that j, j1 , j2 are all less than 40. By using the Young inequality and H¨ older inequality, we have that the left hand side of (3.26) can be bounded by kf ∗ gkL∞ L2 ≤ Ckf kL2 L4/3 kgkL2L4/3 ≤ Ckf k ˆ 0,− ξ
τ
ξ
τ
ξ
X
τ
1 , n+1 2n 4n
kgk ˆ 0,− X
1 , n+1 2n 4n
≤ Ckf kZˆ s,a kgkZˆ s,a .
Cases (ii)–(v), (vii), (viii) of Lemma 3.2 can be derived from Cases (ii)–(v), (vii), (viii) of Lemma 3.1. In fact, by using the triangle inequality and the Schwarz inequality, we have that X X (3.27) 2k/2 kf kL2τ (Bk ) . kf kL1τ (Bk ) ≤ C kf kL1τ ≤ C k≥0
k≥0
From (3.25), for j 6= 0, combining Lemma 3.1 with (3.27), we have that
s−a a+1
hξi |ξ| hτ + (−1)n ξ 2n+1 i−1 (ξ n f ) ∗ g 2 1 L L (Aj ) τ
ξ
≤ C kξ(ξ n f ) ∗ gk
1 ˆ s, − 2 X (2,1)
(Aj )
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.28)
Now consider the case (vi). We consider |ξ| ≤ 2−2nj1 and 2−2nj ≤ |ξ| ≤ 1, respectively. When |ξ| ≤ 2−2nj1 , by using the Young inequality and H¨older inequality, we have that
s−a a+1
hξi |ξ| hτ + (−1)n ξ 2n+1 i−1 (ξ n f ) ∗ g L2 L1 (Aj ) τ
ξ
nj1 a+1 n 2n+1 − n−1 ≤ C2 |ξ| hτ + (−1) ξ i 2n f ∗ g 2 2 Lξ Lτ (Aj )
n−1 3
≤ C2(2n− 2 )j1 |ξ|a+1 hτ i− 2n (hξis f ) ∗ (hξis g) 2 Lτ ξ
≤ C2
(2n− 32 )j1
≤ C2
−(n+ 12 )j1
a+1
|ξ|
2 k(hξis f ) ∗ (hξis g)kL∞ L2τ L (|ξ|≤2−2nj1 ) ξ
ξ
s
s
khξi f kL2 L4/3 khξi gkL2 L4/3 ξ
≤ Ckf k
1 ˆ s, 2 X (2,1)
kgk
1 ˆ s, 2 X (2,1)
τ
τ
ξ
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.29)
Case 2−2nj ≤ |ξ| ≤ 1, we consider cases 2k = 2kmax , 2k1 = 2kmax , 2k2 = 2kmax , respectively. (a) 2n−3 s Case 2k = 2kmax . Since |ξ|a+1 2−2sj1 ≤ C(|ξ|22nj1 )− n ≤ C2 4n k , by using the H¨older inequality and the Young inequality, we have that
s−a a+1
hξi |ξ| hτ + (−1)n ξ 2n+1 i−1 (ξ n f ) ∗ g L2 L1 (Aj ) ξ τ
3 ≤ C2(2n− 2 )j1 |ξ|a+1 hτ i−1 (hξis f ) ∗ (hξis g) L2 L1 ξ τ
− 2n+3 nj1 a+1 s s ≤ C2 |ξ| hτ i 4n (hξi f ) ∗ (hξi g) 2 Lξ L1τ
2n+3 3
≤ C2− 2 j1 |ξ|− 4n 2 k(hξis f ) ∗ (hξis g)kL∞ L1 Lξ (2−2nj ≤|ξ|)
s
s
τ
ξ
s
s
≤ C k(hξi f ) ∗ (hξi g)kL∞ L1τ ≤ C khξi f kL2 L1τ khξi gkL2 L1τ ξ
ξ
ξ
No.4
1077
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
≤ Ckf k ˆ s, 21 kgk ˆ s, 21 ≤ Ckf kZˆ s,a kgkZˆ s,a . X(2,1)
(3.30)
X(2,1)
s
2n−3
(b) Case 2k1 = 2kmax . Since |ξ|a+1 2−2sj1 ≤ C(|ξ|22nj1 )− n ≤ C2 4n k1 ≤ C2 using (2.16), we have that
s−a a+1
hξi |ξ| hτ + (−1)n ξ 2n+1 i−1 ((ξ n f ) ∗ g) L2 L1 (Aj ) ξ τ
X 2n+1 s s n 1/2 nj1 − 4n ≤ C2 2
(hξi f ) ∗ (hξi hτ + (−1) i g) 2 2
k1 2
Lξ Lτ
k≥0
2n+3
≤ C2nj1 |ξ|a+1 hτ i− 4n (hξis f ) ∗ (hξis g) 2 1 Lξ Lτ X 1 ≤C 2− 4n kf k ˆ s, 21 kgk ˆ s, 21 X(2,1)
k≥0
≤ Ckf k
1
ˆ s, 2 X (2,1)
3
2− 4n k , by
kgk
1
ˆ s, 2 X (2,1)
X(2,1)
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.31)
(c) Case 2k2 = 2kmax . This case can be proved similarly to case 2k1 = 2kmax . We have completed the proof of Lemma 3.2.
1 Lemma 3.3 Suppose s = − n2 + 43 and a = − 2n , f and g are restricted to Aj1 and Aj2 , respectively. Then we have the estimates
hτ + (−1)n ξ 2n+1 i−1 ξ n+1 (f ∗ g) ˆs, a ≤ Ckf kZˆ s, a kgkZˆ s, a , (3.32) Z (Aj )
for j ≥ 0 in cases (i)–(viii), where (i)–(viii) are defined as in Lemma 3.1. The proof of Lemma 3.3 can be proved similarly to Lemma 3.1.
Lemma 3.4 Suppose f and g are restricted to Aj1 and Aj2 , respectively. Then we have the estimates
s−a
hξi hτ + (−1)n ξ 2n+1 i−1 |ξ|a+1+n (f ∗ g) 2 1 ≤ Ckf k ˆ s, a kgk ˆ s, a (3.33) Z Z L L ξ
τ
for j ≥ 0, in cases (i)–(viii), where (i)–(viii) are defined as in Lemma 3.2. The proof of Lemma 3.4 can be proved similarly to Lemma 3.2.
1 Lemma 3.5 Let u, v ∈ Z s,a with s = − n2 + 43 and s = − 2n , then
i h
−1
−1 τ + (−1)n ξ 2n+1 F [∂x ((∂xn u)v)]
F Z s,a
i h
−1
n 2n+1 −1 n + F τ + (−1) ξ F [∂x ((∂x u)v)] ≤ CkukZ s,a kvkZ s,a .
(3.34)
Y
To prove (3.34), it suffices to prove that
h i −1
−1
τ + (−1)n ξ 2n+1 F [∂x ((∂xn u)v)] s,a ≤ CkukZ s,a kvkZ s,a .
F
h i Z
−1
n 2n+1 −1 n τ + (−1) ξ F [∂x ((∂x u)v)] ≤ CkukZ s,a kvkZ s,a .
F Y P 2 2 We first prove (3.35). By using kf kZˆ s,a = kIAj f kZˆ s,a , we have that Proof
(3.35) (3.36)
j≥0
h i 2 −1
−1
τ + (−1)n ξ 2n+1 F [∂x ((∂xn u)v)]
F s,a Z
=
2 X −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v)
ξ τ + (−1)n ξ 2n+1
j,j1 j2 ≥0
ˆ Z s,a
=
8 X j=1
Tj ,
1078
ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
where T1 =
X
j,j1 j2 ≥0,i
T2 =
X
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a ,
ξ τ + (−1)n ξ 2n+1 ˆ
j,j1 j2 ≥0,ii
T3 =
X
Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v)
ξ τ + (−1)n ξ 2n+1
ˆ s,a Z
j,j1 j2 ≥0,iii
T4 =
X
j,j1 j2 ≥0,iv
T5 =
X
j,j1 j2 ≥0,v
T6 =
X
X
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a ,
ξ τ + (−1)n ξ 2n+1 ˆ Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a ,
ξ τ + (−1)n ξ 2n+1 ˆ Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) ,
ξ τ + (−1)n ξ 2n+1 ˆ s,a
j,j1 j2 ≥0,vi
T7 =
Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a ,
ξ τ + (−1)n ξ 2n+1 ˆ
j,j1 j2 ≥0,vii
T8 =
X
,
Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) ,
ξ τ + (−1)n ξ 2n+1 ˆ s,a
j,j1 j2 ≥0,viii
Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a .
ξ τ + (−1)n ξ 2n+1 ˆ Z
here (i), (ii), (iii), (iv), (v), (vi), (vii), (viii) is case (i), (ii), (iii), (iv), (v), (vi), (vii), (viii) of Lemmas 3.1 and Lemma 3.2. Combining Tj (1 ≤ j ≤ 8, j ∈ N ), Lemmas 3.1 and Lemma 3.2 P with kf k2Zˆ s,a = j≥0 kIAj f k2Zˆ s,a , we easily obtain (3.35). By using a proof similarly to (3.35), we easily obtain (3.36). We have completed the proof of Lemma 3.5. 1 , then Lemma 3.6 Let u, v ∈ Z s,a with s = − n2 + 43 and s = − 2n
h
i −1 n+1
−1 τ + (−1)n ξ 2n+1 F ∂x (uv) s,a
F Z
h i n+1
−1
n 2n+1 −1 + F τ + (−1) ξ F ∂x (uv) ≤ CkukZ s,a kvkZ s,a . Y
(3.37)
Proof Combining Lemma 3.3 with Lemma 3.4, by using a similar proof to Lemma 3.5, we can obtain Lemma 3.6.
4
Proof of Theorem 1.1
Now we are in a position to prove Theorem 1.1. 1 In this section, we assume that s = − n2 + 34 and s = − 2n . −n If u(x, t) is the solution to (1.1)–(1.2), then uλ (x, t) = λ u solution to (1.1)–(1.2). By a direct calculation, we have that
x t λ , λ2n+1
with λ ≥ 1 is the
1
kuλ (x, 0)kH (s,a) ≤ Cλ−n+ 2 −a ku0 kH (s,a) . Thus, without loss of generality, we can assume that ku0 kH (s,a) is sufficiently small. (1.1)–(1.2) is equivalent to the following integral equation: Z t 2n+1 2n+1 u(t) = e−t∂x u0 + e−(t−s)∂x ∂x (us )ds. 0
(4.1)
(4.2)
No.4
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
1079
We define 2n+1
Φ(u) = e−t∂x
u0 +
Z
t
2n+1
e−(t−s)∂x
∂x (us )ds.
(4.3)
0
By using Lemmas 2.8–2.9 and Lemmas 3.5–3.6 and proposition 4.1 of [17] as well as (4.2)–(4.3), we can obtain the existence of the solution to (1.1)–(1.2). The uniqueness of the solution to (1.1)–(1.2) can be found in Proposition 5.1 of [13]. The rest of Theorem 1.1 can be proved similarly to Lemma 4.2 in [16].
5
Proof of Theorem 1.2 In this section, motivated by [1, 4, 13], we prove Theorem 1.2. We define ξ = ξ1 + ξ2 + ξ3 , Q1 = φ(ξ1 ) + φ(ξ2 ) − φ(ξ1 + ξ2 ), Q2 = φ(ξ2 ) + φ(ξ3 ) − φ(ξ2 + ξ3 ), Q3 = Q4 =
eit(φ(ξ1 )+φ(ξ2 )+φ(ξ3 )) − eitφ(ξ) , φ(ξ1 ) + φ(ξ2 ) + φ(ξ3 ) − φ(ξ)
eit(φ(ξ1 )+φ(ξ2 +ξ3 )) − eitφ(ξ) eit(φ(ξ3 )+φ(ξ1 +ξ2 )) − eitφ(ξ) , Q5 = . φ(ξ1 ) + φ(ξ2 + ξ3 ) − φ(ξ) φ(ξ3 ) + φ(ξ1 + ξ2 ) − φ(ξ)
In this section, we always assume ξ(ξ2 + ξ3 )n+1 [2(ξ2 + ξ3 )n + ξ2n + ξ3n ] , Q2 ξξ n (ξ1 + ξ2 ) [2(ξ1 + ξ2 )n + ξ1n + ξ2n ] H2 (ξ, ξ1 , ξ2 , ξ3 ) = 3 , Q1 ξ n+1 (ξ1 + ξ2 ) [2(ξ1 + ξ2 )n + ξ1n + ξ2n ] H3 (ξ, ξ1 , ξ2 , ξ3 ) = , Q1 ξ n+1 (ξ2 + ξ3 ) [2(ξ2 + ξ3 )n + ξ2n + ξ3n ] . H4 (ξ, ξ1 , ξ2 , ξ3 ) = Q2
H1 (ξ, ξ1 , ξ2 , ξ3 ) =
By contradiction, we assume that the solution map of (1.1)–(1.2) St : u0 ∈ H (s, a) (R) → u ∈ C([0, T ]; H (s, a) (R)) is C 3 at zero with s < − n2 + 34 and a ∈ R. Following the general well-posedness principle of [1], we have sup kB3 (u0 )kH (s,a) ≤ Cku0 k3H (s,a) .
(5.1)
t∈[0,T ]
where B1 (u0 )(x, t) = W (t)u0 , Z t B2 (u0 )(x, t) = 2 W (t − τ )∂xn+1 (B1 (u0 )B1 (u0 ))dτ 0 Z t +2 W (t − τ )∂x (B1 (u0 )∂xn B1 (u0 ))dτ, 0 Z t B3 (u0 )(x, t) = 3 W (t − τ )∂xn+1 (B1 (u0 )B2 (u0 ))dτ 0 Z t +3 W (t − τ )∂xn+1 (B2 (u0 )B1 (u0 ))dτ 0 Z t +3 W (t − τ )∂x (B1 (u0 )∂xn B2 (u0 ))dτ 0
(5.2)
(5.3)
1080
ACTA MATHEMATICA SCIENTIA
+3
t
Z
0
We consider the initial data u0 (x) := N
−s+ 2n−1 4
From (5.2), we have that
e−iN x
Vol.37 Ser.B
W (t − τ )∂x ((∂xn B1 (u0 ))B2 (u0 ))dτ.
Z
N−
2n−1 2
eixξ dξ + eiN x
Z
2N −
N−
0
F u0 (ξ) = N −s+
2n−1 4
2n−1 2
2n−1 2
(5.4)
eixξ dξ .
(χI1 (ξ) + χI2 (ξ)) ,
(5.5)
(5.6)
where o n 2n−1 , I1 =: −N, −N + N − 2
n o 2n−1 2n−1 I2 =: N + N − 2 , N + 2N − 2 .
(5.7)
Let Ω = I1 ∪ I2 , by a direct computation, we have that ku0 kH (s, a) ∼ 1.
(5.8)
kB3 (u0 )kH (s,a) = C k[J1 − J2 + J(3) − J(4) + J(5) − J(6) + J(7) − J(8)]kH (s,a) ,
(5.9)
From (5.2)–(5.4), we have that
where J1 = N
−3s+ 3(2n−1) 4
Z
ξ1 ∈Ω
J2 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J3 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J4 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J5 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J6 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J7 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J8 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
eixξ H1 Q3 dξ1 dξ2 dξ3 ,
(5.10)
eixξ H1 Q4 dξ1 dξ2 dξ3 ,
(5.11)
eixξ H2 Q3 dξ1 dξ2 dξ3 ,
(5.12)
eixξ H2 Q5 dξ1 dξ2 dξ3 ,
(5.13)
eixξ H3 Q3 dξ1 dξ2 dξ3 ,
(5.14)
eixξ H3 Q5 dξ1 dξ2 dξ3 ,
(5.15)
eixξ H4 Q3 dξ1 dξ2 dξ3 ,
(5.16)
eixξ H4 Q4 dξ1 dξ2 dξ3 .
(5.17)
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
To estimate kB3 (u0 )kH (s,a) , we consider the following three cases. (1) : ξj (j = 1, 2, 3) ∈ I1 , (2) : ξj (j = 1, 2, 3) ∈ I2 , (3) : ξj (j = 1, 2) ∈ I1 , ξ3 ∈ I2 , (4) : ξ1 ∈ I1 , ξj (j = 2, 3) ∈ I2 , (5) : ξj (j = 1, 2) ∈ I2 , ξ3 ∈ I1 , (6) : ξ1 ∈ I2 , ξj (j = 2, 3) ∈ I1 , (7) : ξ1 , ξ3 ∈ I1 , ξ2 ∈ I2 , (8) : ξ1 , ξ3 ∈ I2 , ξ2 ∈ I1 . When (1), (2) occurs, we have that kB3 (u0 )kH (s,a) ≤ CN −3s+
3(2n−1) 4
N s+
5(2n−1) −2n 4
= CN −2s+2n−2 .
(5.18)
When (3) occurs, by using Lemmas 2.5, 2.6 and n is even, we have that |Q1 | ∼ N 2n+1 , |Q2 | ∼ N
2n+1 2
, |Q3 | ∼ 1,
(5.19)
No.4
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
|Q4 | ≤ CN − |H1 | ∼ N
2n+1 2
, |Q5 | ≤ CN −2n−1 ,
− 12 n−n2 +1
1081 (5.20)
, |H2 | ∼ N, |H3 | ∼ N, |H4 | ∼ N.
(5.21)
By using (5.19)–(5.21), we have that 2
kJ1 kH (s,a) ∼ N −2s−n kJ2 kH (s,a) ≤ CN
− 3n−3 2
,
2 −2s− n−2 2 −n
kJ3 kH (s,a) ∼ CN
−2s−(n− 23 )
kJ4 kH (s,a) ∼ CN
−2s−3n+ 12
(5.22) ,
(5.23)
,
(5.24)
,
(5.25)
kJ5 kH (s,a) ∼ CN
−2s−(n− 23 )
,
(5.26)
kJ6 kH (s,a) ∼ CN
−2s−3n+ 12
,
(5.27)
kJ7 kH (s,a) ∼ CN
−2s−(n− 23 )
,
(5.28)
kJ8 kH (s,a) ∼ CN −2s−(2n−1) .
(5.29)
Consequently, when (3) occurs, we have that 1 ∼ ku0 k3H (s,a) ≥ kB3 (u0 )kH (s,a) ≥ kJ3 kH (s,a) + kJ5 kH (s,a) + kJ7 kH (s,a) −kJ1 kH (s,a) − kJ2 kH (s,a) − kJ4 kH (s,a) − kJ6 kH (s,a) − kJ8 kH (s,a) 3
≥ CN −2s−(n− 2 ) ,
(5.30)
when s < − n2 + 34 , letting N −→ ∞ yields a contraction in (5.30). By using a proof similar to the case (3), we can obtain that in cases (4)-(8), 3
1 ∼ ku0 k3H (s,a) ≥ kB3 (u0 )kH (s,a) ≥ CN −2s−(n− 2 ) ,
(5.31)
when s < − n2 + 34 , letting N −→ ∞ yields a contraction in (5.31). We have completed the proof of Theorem 1.2. References [1] Bejenaru I, Tao T. Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr¨ odinger equation. J Funct Anal, 2006, 233: 228–259 [2] Bourgain J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation. Geom Funct Anal, 1993, 3: 107–156 [3] Bourgain J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation. Geom Funct Anal, 1993, 3: 209–262 [4] Bourgain J. Periodic Korteweg-de Vries equation with measures as initial data. Selecta Math, 1997, 3: 115–159 [5] Constantin P, Saut J C. Local smoothing properties of dispersive equations. J Amer Math Soc, 1988, 1: 413–439 [6] Ginibre J, Tsutsumi Y, Velo G. On the Cauchy problem for the Zakharov System. J Funct Anal, 1997, 151: 384–436 [7] Guo Z H. Global well-posedness of the Korteweg-de Vries equation in H −3/4 . J Math Pures Appl, 2009, 91: 583–597 [8] Guo Z H, Kwak C, Kwon S. Rough solutions of the fifth-order KdV equations. J Funct Anal, 2013, 265: 2791–2829 [9] Guo Z H, Wang B X. Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation. J Diff Eqns, 2009, 246: 3864–3901
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[10] Gr¨ unrock A. On the hierarchies of higher order mKdV and KdV equations. Central European J Math, 2010, 8: 500–536 [11] Ionescu A D, Kenig C E. Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J Amer Math Soc, 2007, 20: 753–798 [12] Ionescu A D, Kenig C E, Tataru D. Global well-posedness of the KP-I initial-value problem in the energy space. Invent Math, 2008, 173: 265–304 [13] Kato T. Well-posedness for the fifith order KdV equation. Funkcial Ekvac, 2012, 55: 17–53 [14] Kenig C E, Ponce G, Vega L. Higher-order nonlinear dispersive equations. Proc Amer Math Soc, 1994, 122: 157–166 [15] Kenig C E, Pilod D. Well-posedness for the fifth-order KdV equation in the energy space. Trans Amer Math Soc, 2015, 367: 2551–2612 [16] Kishimoto N. Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Diff Int Eqns, 2009, 22: 447–464 [17] Kishimoto N, Tsugawa K. Local well-posedness for quadratic nonlinear Schr¨ odinger equations and the “good” Boussinesq equation. Diff Int Eqns, 2010, 23: 463–493 [18] Li Y S, Huang J H, Yan W. The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity. J Diff Eqns, 2015, 259: 1379–1408 [19] Pilod D. On the Cauchy problem for higher-order nonlinear dispersive equations. J Diff Eqns, 2008, 245: 2055–2077 [20] Wang H, Cui S. Well-posedness of the Cauchy problem of a water wave equation with low regularity initial data. Math. Computer Mode, 2007, 45: 1118–1132 [21] Yan W, Li Y S, Li S M. Sharp well-posedness and ill-posedness of a higher-ordermodified Camassa-Holm equation. Diff Int Eqns, 2012, 25: 1053–1074 [22] Yan W, Li Y S. Ill-posedness of madified Kawahara equation and Kaup-Kupershmidt equation. Acta Math Sci, 2012, 32B: 710–716
Appendix When n is odd, motivated by the Appendix of [13], we have the following result. Example 1 (high × high7→ low interaction) Let 1 2 − 2n−1 2n+1 2 P1 := (τ, ξ) ∈ R ; |ξ − N | ≤ N , τ −ξ ≤ , 2 P2 := (τ, ξ) ∈ R2 ; (−τ, −ξ) ∈ P1 , f (τ, ξ) := χP1 (τ, ξ), f ∗ g(τ, ξ) ≥ CN
g(τ, ξ) := χP2 (τ, ξ),
− 2n−1 2
χR1 (τ, ξ),
where N is a sufficiently large positive number and 1 − 2n−1 3 − 2n−1 2 2 2 R1 := (τ, ξ) ∈ R ; ξ ∈ N , N , 2 4 By a direct computation, we have that s− 1 , b = kgk − n + 3 , 1 , b ∼ N kf k ˆ − n2 + 34 , 2n ˆ 2 4 2n X
X
F (f g) ≥ CN k∂x ((∂xn f )
− 2n−1 2
τ − 1 N 2n ξ ≤ 1 2 16 2n−1 4
,
IR1 , 5
a
n−( 4 − 2 )(2n−1)+ 1 , b−1 ∼ N ∗ g)k ˆ − n2 + 34 , 2n X
2n+1 (b−1) 2
.
1 , b−1 ≤ Ckf k − n + 3 , 1 , b kgk − n + 3 , 1 , b is invalid Then, we have that k∂x ((∂xn f )g)k ˆ − n2 + 34 , 2n ˆ 2 4 2n ˆ 2 4 2n X X X n+1 for b < 2n+1 .
SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION∗
ö¯#)
Minjie JIANG (
A%)
Wei YAN (
†
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China E-mail :[email protected]; [email protected]
ܬ)
Yimin ZHANG (
School of Science, Wuhan University of Technology, Wuhan 430070, China E-mail : [email protected] This current paper is devoted to the Cauchy problem for higher order dispersive
Abstract equation
ut + ∂x2n+1 u = ∂x (u∂xn u) + ∂xn−1 (u2x ),
n ≥ 2, n ∈ N+ .
By using Besov-type spaces, we prove that the associated problem is locally well-posed in n 3 1 H (− 2 + 4 ,− 2n ) (R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H (s,a) (R) with s < − n2 + 34 and all a ∈ R. Key words
Cauchy problem; sharp well-posedness; modified Bourgain spaces
2010 MR Subject Classification
1
35K30; 35G25
Introduction
In this paper, we investigate the Cauchy problem for the following higher-order dispersive equation ut + ∂x2n+1 u = ∂x (u∂xn u) + ∂xn−1 (u2x ),
n ≥ 2, n ∈ N, x, t ∈ R,
u(x, 0) = u0 (x).
(1.1) (1.2)
In 1988, Constantin and Saut [5] studied the Cauchy problem for the following equations ut + iP (D)u = F, ∗ Received February 24, 2016; revised December 24, 2016. This work is supported by Natural Science Foundation of China NSFC (11401180 and 11471330). The second author is also supported by the Young Core Teachers Program of Henan Normal University (15A110033). The third author is also supported by the Fundamental Research Funds for the Central Universities (WUT: 2017 IVA 075). † Corresponding author: Wei Yan
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∂ , ···, ∂x∂ n ) and P (D)u = where D = 1i ( ∂x 1
R
Rn
Vol.37 Ser.B
e2πihx,ξi p(ξ)Fx u(ξ)dξ. Kenig et al. [14] considered
ut + ∂x2n+1 u + P (u, ∂x u, · · ·, ∂x2n u) = 0,
j ∈ N, x, t ∈ R,
u(x, 0) = u0 (x),
(1.3) (1.4)
where P : R2n+1 −→ R (or P : C2n+1 −→ C) is a polynomial having no constants or linear terms. They proved that the problem (1.3)–(1.4) is well-posed in some weighted Sobolev spaces for small initial data and for arbitrary initial data with the aid of a gauge transformation. By using the Fourier restriction norm method introduced by Bourgain [2, 3], Wang and Cui [20] 1 proved that the case of n = 2 in (1.1)–(1.2) is well-posed in modified Sobolev space H (s,− 4 ) with s > − 41 . Recently, the following problem has been investigated: P ut + ∂x2n+1 u = an1 ,n2 ∂xn1 u∂xn2 u, n ∈ N, x, t ∈ R, (1.5) 0≤n1 +n2 ≤2n
u(x, 0) = u0 (x).
(1.6)
Pilod [19] studied the well-posedness of problem (1.5)–(1.6) in Besov paces and proved that the problem (1.5)–(1.6) is well-posed in the space H k (R) ∩ H k−2n (R; x2 dx) for small initial data with k > 2n + 14 . Pilod also proved some ill-posedness results when a0, k 6= 0 for some k > n in the sense that (1.5)–(1.6) cannot have flow map C 2 at the origin in H s (R) for any s ∈ R. In [22], Yan and Li studied the ill-posedness of modified Kawahara equation and Kaup-Kupershmidt equation. Recently, Kato [13] considered the Cauchy problem for ut − ∂x5 u + c1 ∂x (u3 ) + c2 ∂x (u2x ) + c3 ∂x (uuxx ) = 0,
(1.7)
where cj ∈ R, j = 1, 2, 3 with c3 6= 0, in modified Sobolev space. Very recently, Guo et al. [8] and Kenig and Pilod [15] studied the Cauchy problem for (1.7) in Sobolev space with the aid of the short time Xs, b space which is defined below. In this paper, as in [1, 7, 8, 11–13, 16], we use the modified Bourgain space to establish some new dyadic bilinear estimates, then we prove (1.1)–(1.2) is locally well-posed in n 3 1 H (− 2 + 4 ,− 2n ) (R). When n is even, we also prove that the associated equation is ill-posed in H (s,a) (R) with s < − n2 + 34 and all a ∈ R. We give some notations before presenting the main results. In this paper, IΩ denotes the 1 characteristic function of a set Ω. 0 < ǫ < 10000 . j, k ∈ N+ . C is a positive constant which may vary from line to line. A ∼ B means that |B| ≤ |A| ≤ 4|B|. A ≫ B means that |A| ≥ 4|B|. Let ψ(t) be the smooth function supported in [−1, 2] and equals to 1 in [0, 1]. F u denotes the Fourier transformation of u with respect to its all variables. F −1 u denotes the Fourier inverse transformation of u with respect to its all variables. Fx u denotes the Fourier transformation of u with respect to its space variable. Fx−1 u denotes the Fourier inverse transformation of u with respect to its space variable. Let φ(t) is a smooth cut-off function φ(t) satisfying suppφ ⊂ [−2, 2] and φ(t) = 1 for |t| < 1, Z n+1 2n+1 −t∂x2n+1 W (t)u0 = Ce u0 = C ei(−1) tξ Fx u0 (ξ)dξ, R
and
n o 2n+1 D1 := (τ, ξ) ∈ R2 : |ξ| ≤ 1, |τ | ≥ |ξ|− 2n−1 , n o 2n+1 D2 := (τ, ξ) ∈ R2 : |ξ| ≤ 1, |τ | ≤ |ξ|− 2n−1 ,
No.4
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M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
Aj = (τ, ξ) : 2j ≤ hξi < 2j+1 ,
Bk = (τ, ξ) : 2k ≤ τ + (−1)n ξ 2n+1 < 2k+1
for nonnegative integers j, k. The modified Sobolev space is defined by n o
H (s,a) := u ∈ S ′ (R) : kukH (s, a) = hξis−a |ξ|a Fx u(ξ) L2 < ∞ . ξ
Bourgain spaces is defined by n
Xs, b := u ∈ S ′ (R2 ) : kukXs, b = hξis hτ + (−1)n ξ 2n+1 ib F u L2
τξ
(R2 )
ˆ s, a, b is defined by The modified Bourgain space X n
ˆ s, a, b := u ∈ S ′ (R2 ) : kuk ˆ s, a, b = hξis−a |ξ|a hτ + (−1)n ξ 2n+1 ib u 2 X X L
τξ
1 − 2n , n+1 2n
ˆ We denote X L
o <∞ . (R2 )
1 − 2n , n+1 2n ,1
ˆ ,X L
by
n+1 1
kf k − 1 , n+1 := |ξ|− 2n hτ + (−1)n ξ 2n+1 i 2n f 2 , ˆ 2n 2n Lτ ξ (A0 ) X L
X n+1 n+1 1
2 2n k |ξ|− 2n hτ + (−1)n ξ 2n+1 i 2n f kf k − 1 , n+1 ,1 := 2n 2n ˆ X L
o <∞ .
L2τ ξ (A0 ∩Bk )
k≥0
,
ˆ s, b is then defined by the norm respectively. The space X (2,1)
b
s
kukXˆ s, b = hξi τ + (−1)n ξ 2n+1 u 2
(2,1)
Lτ ξ (Aj ∩Bk ) j, k≥0 2
ιj ι1k
X ∼ 2js j
X
2bk kukL2τ ξ(Aj ∩Bk )
k
!2 1/2
1
We define the function space Z (s,− 2n ) as follows:
1 = χ|ξ|≥1 u kuk ˆ s,− 2n
1
ˆ s, 2 X (2,1)
Z
+ kuk
1
ˆ − 2n X L
.
,
where
kuk
1
ˆ − 2n X L
= kχD1 uk
1
ˆ − 2n , X L
n+1 2n
+ kχD2 uk
1
ˆ − 2n , X L
n+1 ,1 2n
.
We define
n 3 1 1
kf kY = hξi(− 2 + 4 )+ 2n |ξ|− 2n F u
L2ξ L1τ
, kF f kXˆ = kf kX , kf k
1
Z s,− 2n
1 , = kF f k ˆ s,− 2n
Z
kF f kXˆ s, b = kf kXs, b , kF f kXˆ s, a, b = kf kX s, a , b . (s,−
1
)
We denote the function space ZT 2n by 1 := 1 : kuk s,− 2n kvk s,− 2n ZT
ZT
u(t) = v(t)
on t ∈ [0, T ] .
The main results of this paper are as follows: n 3 1 Theorem 1.1 Let u0 (x) ∈ H (− 2 + 4 ,− 2n ) (R). Then the Cauchy problem for (1.1) is locally well-posed in the sense of Hadamard. Theorem 1.2 Let n be even and s < − n2 + 34 and a ∈ R. Then there is no T such that the flow map for (1.1), u0 → u(t) is C 3 as a map from Br (H (s,a) ) to H (s,a) for any t ∈ [0, T ].
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Remark Theorem 1 is sharp in the sense of Theorem 2. The rest of paper is arranged as follows. In Section 2, we present some preliminaries. In Section 3, we give some crucial bilinear estimates. In Section 4, we prove Theorem 1.1. In Section 5, we prove Theorem 1.2. In Section 6, we give Appendix which contains the example where the bilinear estimates is invalid in modified Bourgain spaces.
2
Preliminaries In this section, we make some preparations for the proof of bilinear estimates of the Section
3. Lemma 2.1 Assume that n ∈ N + . Then, for ξ = ξ1 + ξ2 , τ = τ1 + τ2 , we have that 2n+1 nξ n 2n+1 n 2n+1 τ + (−1) − τ + (−1) ξ − τ + (−1) ξ 1 2 1 2 4n 2 ξ ξ 2n+1 2n+1 2n+1 n+1 n+1 = (−1) = (−1) ξ ξ1 − F1 (ξ, ξ1 ), (2.1) ξ1 + ξ2 − 4n 2
where
"
2 2n−2 2n−2 1 1 ξ 4 ξ 2n−2 + C2n+1 ξ 2n−4 ξ1 − 2 2 2 2n−2 # ξ 2n + · · · + C2n+1 . ξ1 − 2
2 F1 (ξ, ξ1 ) = C2n+1
We omit the process of proof since the proof of Lemma 2.1 is easy. Lemma 2.2 Assume that each f and g is supported on Aj1 , Aj2 , respectively. Then k|ξ|
2n−1 4
f ∗ gkL2 (R2 ) ≤ Ckf k
1
ˆ 0, 2 X (2,1)
kgk
1
ˆ 0, 2 X (2,1)
.
(2.2)
Moreover, if K := inf {|ξ1 − ξ2 | : τ1 , τ2 , s.t.(τ1 , ξ1 ) ∈ suppf, (τ2 , ξ2 ) ∈ suppg} > 0, then, we have that k|ξ|1/2 f ∗ gkL2 (R2 ) ≤ CK − Proof
2n−1 2
kf k
1
ˆ 0, 2 X (2,1)
kgk
1
ˆ 0, 2 X (2,1)
.
To obtain (2.2) and (2.3), we firstly prove that Z Z (χBk1 f )(τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2
(2.3)
τ =τ1 +τ2
≤ C2
k1 +k2 2
kf kL2τ ξ (Bk1 ) kgkL2τ ξ (Bk2 ) k|ξ|−
2n−1 4
hkL2τ ξ
(2.4)
and Z Z (χBk1 f )(τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
≤ CK
− 2n−1 2
2
k1 +k2 2
1
kf kL2τ ξ (Bk1 ) kgkL2τ ξ (Bk2 ) k|ξ|− 2 hkL2τ ξ ,
(2.5)
where f, g are restricted to Bk1 , BK2 for k1 , k2 ≥ 0. By using the Cauchy-Schwartz inequality twice and the Fubini’s Theorem, we have that Z Z (χBk1 f )(τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
No.4
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M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
≤C
sup (τ,
ξ)∈R2
m(τ, ξ)1/2 kf kL2τ ξ (Bk1 ) kgkL2τ ξ (Bk2 ) k|ξ|−1/2 hkL2τ ξ ,
(2.6)
where Z m(τ, ξ) = χΛ1 (τ1 , ξ1 , τ, ξ)dτ1 dξ1 , τ = τ1 + τ2 , ξ = ξ1 + ξ2 , Λ1 := (τ1 , ξ1 , τ, ξ) ∈ R4 , (τ1 , ξ1 ) ∈ supp(χBk1 ∩Aj1 f ), (τ2 , ξ2 ) ∈ supp(χBk2 ∩Aj2 g) .
Hence, (2.4) and (2.5) are reduced to o n 2n−1 m(τ, ξ) ≤ Cmin |ξ|− 2 2k1 +k2 , K −(2n−1) |ξ|−1 2k1 +k2 .
(2.7)
Now we prove (2.7). We fix τ, ξ 6= 0 and assume that G1 and G2 be the projections of Λ1 onto the ξ1 -axis and τ1 -axis, respectively. From (2.1), we have that 4|M − C(2k1 + 2k2 )| 2 |M + C(2k1 + 2k2 )| max ,K , (2.8) ≤ |2ξ1 − ξ|2 ≤ 4 |ξ|F1 (ξ, ξ1 ) |ξ|F1 (ξ, ξ1 ) 2n+1 where M = τ + (−1)n ξ 4n and C is some positive constant. 1/2 k1 +2k2 )| When K ≥ 4|M−C(2 , then the variation of |2ξ1 − ξ| is bounded by |ξ|F1 (ξ,ξ1 )
4|M + C(2k1 + 2k2 )| |ξ|F1 (ξ, ξ1 )
≤ |ξ|F1 (ξ, ξ1 ) ≤
1/2
−K =
C(2k1 + 2k2 )
4|M+C(2k1 +2k2 )| |ξ|F1 (ξ,ξ1 )
1/2
4|M+C(2k1 +2k2 )| |ξ|F1 (ξ,ξ1 )
4|M+C(2k1 +2k2 )| |ξ|F1 (ξ,ξ1 )
+K
C(2k1 + 2k2 ) C(2k1 + 2k2 ) ≤ . |ξ|F1 (ξ, ξ1 )K |ξ|K 2n−1
− K2
1/2
+K (2.9)
(2.10)
Moreover, from (2.9), the length of the interval |2ξ1 − ξ| is bounded by C(2k1 + 2k2 ) 1/2
|ξ|1/2 F1 (ξ, ξ1 )1/2 (|M + C(2k1 + 2k2 ))
≤
C(2k1 /2 + 2k2 /2 ) |ξ|
2n−1 2
.
From (2.10) and (2.11), we have that n o 2n−1 mes G1 ≤ Cmin |ξ|− 2 (2k1 /2 + 2k2 /2 ), K −(2n−1) |ξ|−1 (2k1 + 2k2 ) . When K ≤ by
4|M−C(2k1 +2k2 )| |ξ|F1 (ξ,ξ1 )
1/2
(2.11)
(2.12)
, we have that the length of the interval of |2ξ1 − ξ| is bounded
1/2 1/2 4|M + C(2k1 + 2k2 )| 4|M − C(2k1 + 2k2 )| − |ξ|F1 (ξ, ξ1 ) |ξ|F1 (ξ, ξ1 ) n o 2n−1 ≤ Cmin |ξ|− 2 (2k1 /2 + 2k2 /2 ), K −(2n−1) |ξ|−1 (2k1 + 2k2 ) .
From the above inequality, we have that in this case o n 2n−1 mes G1 ≤ Cmin |ξ|− 2 (2k1 /2 + 2k2 /2 ), K −(2n−1) |ξ|−1 (2k1 + 2k2 ) .
From
|τ1 + (−1)n ξ12n+1 | ≤ C2k1 ,
|τ − τ1 + (−1)n (ξ − ξ1 )2n+1 +
1 | ≤ C2k2 , ξ − ξ1
(2.13)
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we have that mes G2 ≤ Cmin 2k1 , 2k2 .
(2.14)
Combining (2.11), (2.13) with (2.14), we have (2.7).
We have completed the proof of Lemma 2.2. Lemma 2.3 Assume that f ∈ S (R), g ∈ S ′ (R) with suppg ⊂ Aj with j ≥ 0. Then, we have kf ∗ gkL2 (Bk ) ≤ C2k/4 kgk ˆ 0, 12 k|ξ|− X(2,1)
2n−1 4
f kL2 (R2 ) .
(2.15)
Moreover, if a non-empty set Ω ∈ R2 satisfies K1 := inf {|ξ2 + ξ| : τ, τ2 , s.t.(τ, ξ) ∈ Ω, (τ2 , ξ2 ) ∈ suppg} > 0, then, we have that − 2n−1 2
kf ∗ gkL2 (Ω∩Bk ) ≤ C2k/2 K1 Proof
Ck|ξ|−1/2 f kL2 (R2 ) kgkXˆ
0, 1 ,1 2
To obtain (2.15) and (2.16), we firstly prove that Z Z f (τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2
.
(2.16)
τ =τ1 +τ2
≤ C2
k2 /2+k/2
k|ξ|−
2n−1 4
f kL2τ ξ kgkL2τ ξ (Bk2 ) khkL2τ ξ
(2.17)
for h ∈ L2τ ξ (Bk ) and Z Z f (τ1 , ξ1 )(χBk2 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
≤
− 2n−1 CK1 2 2k/2+k3 /2 kf kL2τ ξ kgkL2τ ξ (Bk2 ) k|ξ|−1/2 hkL2τ ξ
(2.18)
for h ∈ L2τ ξ (Bk ∩ Ω). By using the Cauchy-Schwartz inequality twice and the Fubini’s Theorem, we have that Z Z f (τ1 , ξ1 )(g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
≤C
sup
(τ, ξ)∈R2
where m1 (τ, ξ) =
m1 (τ, ξ)1/2 kf kL2τ ξ kgkL2τ ξ khkL2τ ξ ,
(2.19)
R
χΛ2 (τ1 , ξ1 , τ, ξ)dτ1 dξ1 , and Λ2 := (τ1 , ξ1 , τ, ξ) ∈ R4 , (τ2 , ξ2 ) ∈ suppf, (τ, ξ) ∈ supph .
Hence, (2.15) and (2.16) are reduced to o n 2n−1 m1 (τ, ξ) ≤ Cmin |ξ|− 2 2k+k2 , K −(2n−1) |ξ1 |−1 2k+k2 .
(2.20)
Now we prove (2.20). We fix τ, ξ 6= 0 and assume that H1 and H2 be the projections of Λ2 onto the ξ1 -axis and τ1 -axis, respectively. It is easily checked that 2n+1 n ξ1 τ1 + (−1) − τ + (−1)n ξ 2n+1 + τ2 + (−1)n ξ22n+1 4n 2 ξ 2n+1 ξ = (−1)n+1 ξ 2n+1 − ξ22n+1 − 1 n = (−1)n+1 ξ1 ξ1 − F2 (ξ, ξ1 ), (2.21) 4 2
No.4
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
1067
where F2 (ξ, ξ1 ) 2n−2 2n−2 2 2n−2 ξ 1 1 ξ 2n−2 2n−4 2 4 2n = C2n+1 ξ1 + C2n+1 ξ1 ξ1 − + · · · + C2n+1 ξ1 − . 2 2 2 2 From (2.21), we have that |M + C(2k + 2k2 )| 4|M − C(2k + 2k2 )| 2 max ≤ |ξ − 2ξ1 |2 ≤ 4 ,K , (2.22) |ξ|F2 (ξ, ξ1 ) |ξ|F2 (ξ, ξ1 ) ξ 2n+1 where M1 = τ1 + (−1)n 14n and C is some positive constant. 1/2 k +2k2 )| , we have that the variation of |2ξ − ξ1 | is bounded by When K ≥ 4|M−C(2 |ξ1 |F2 (ξ,ξ1 )
4|M + C(2k + 2k2 )| |ξ1 |F2 (ξ, ξ1 )
≤ |ξ1 |F2 (ξ, ξ1 ) ≤
1/2
−K =
C(2k + 2k2 )
4|M+C(2k +2k2 )| |ξ1 |F2 (ξ,ξ1 )
4|M+C(2k1 +2k2 )| |ξ1 |F2 (ξ,ξ1 )
4|M+C(2k +2k2 )| |ξ1 |F2 (ξ,ξ1 )
1/2
+K
C(2k + 2k2 ) C(2k + 2k2 ) ≤ . |ξ1 |F2 (ξ, ξ1 )K |ξ1 |K 2n−1
− K2
1/2
+K (2.23)
(2.24)
Moreover, from (2.23), the variation of |ξ − 2ξ1 | is also bounded by C(2k + 2k2 ) |ξ1 |1/2 F2 (ξ, ξ1 )1/2 (|M + C(2k + 2k2 ))
1/2
≤
C(2k/2 + 2k2 /2 ) |ξ1 |
2n−1 2
.
From (2.24) and (2.25), we have that o n 2n−1 mes H1 ≤ Cmin |ξ1 |− 2 2k/2 + 2k2 /2 ), K −(2n−1) |ξ1 |−1 (2k + 2k2 ) . When K ≤ by
4|M−C(2k +2k2 )| |ξ1 |F2 (ξ,ξ1 )
1/2
(2.25)
(2.26)
, we have that the length of the interval of |2ξ1 − ξ| is bounded
1/2 1/2 4|M − C(2k1 + 2k2 )| 4|M + C(2k1 + 2k2 )| − |ξ|F2 (ξ, ξ1 ) |ξ|F2 (ξ, ξ1 ) n o 2n−1 ≤ Cmin |ξ|− 2 2k1 /2 + 2k2 /2 ), K −(2n−1) |ξ|−1 (2k1 + 2k2 ) .
In this case,
From
o n 2n−1 mes H1 ≤ Cmin |ξ1 |− 2 2k/2 + 2k2 /2 ), K −(2n−1) |ξ1 |−1 (2k + 2k2 ) . n 2n+1
|τ + (−1) ξ
k
| ≤ C2 ,
we have that
τ − τ1 + (−1)n (ξ − ξ1 )2n+1 + 1 ≤ C2k2 , ξ − ξ1
mes H2 ≤ Cmin 2k , 2k2 .
By using the Cauchy-Schwartz inequality and the triangle inequality, we have that Z Z f (τ1 , ξ1 )g(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ R2 ξ=ξ1 +ξ2 τ =τ1 +τ2
(2.27)
(2.28)
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X Z Z ≤C f (τ1 , ξ1 )(χBk3 g)(τ2 , ξ2 )h(τ, ξ)dτ1 dξ1 dτ dξ . R2 ξ=ξ1 +ξ2 k3
(2.29)
τ =τ1 +τ2
Combining (2.17)–(2.18), (2.26)–(2.27) with (2.28)–(2.29), we have (2.15)–(2.16). We have completed Lemma 2.3. 0,
1 2
ˆ Lemma 2.4 The space X (2,1) has the following properties. (i) For any b > 1/2, there exists C > 0 such that k|f k
≤ Ckf kXˆ 0, b .
1
ˆ 0, 2 X (2,1)
(2.30)
(ii) For 1 < p ≤ 2, there exists C > 0 such that kf kL2ξ Lpτ ≤ Ckf k ˆ 0, 21 ,
(2.31)
kf kL2ξ L1τ ≤ Ckf k
(2.32)
X(2,1) 1
ˆ 0, 2 X (2,1)
.
Lemma 2.4 can be proved similarly to Lemma 2.9 of [18]. 2 2 P P Lemma 2.5 Let τ = τj , ξ = ξj . Then, we have that j=1
j=1
τ + (−1)n ξ 2n+1 − τ1 + (−1)n ξ 2n+1 − τ2 + (−1)n ξ 2n+1 1 2 = ξ 2n+1 − ξ12n+1 − ξ22n+1 ∼ |ξmin ||ξmax |2n ,
where
|ξmin | := min {|ξ|, |ξ1 |, |ξ2 |} ,
|ξmax | := max {|ξ|, |ξ1 |, |ξ2 |} .
For the proof of Lemma 2.6, we refer the reader to Lemma 2.5 of [21]. 3 P Lemma 2.6 Let ξ = ξj . Then, we have that j=1
ξ
2n+1
−
ξ12n+1
− ξ22n+1 − ξ32n+1 = C(ξ1 + ξ2 )(ξ1 + ξ3 )(ξ2 + ξ3 )f (ξ1 , ξ2 , ξ3 ),
where f (ξ1 , ξ2 , ξ3 ) ≥ C (ξ1 + ξ2 )2n−2 + (ξ1 + ξ3 )2n−2 + (ξ2 + ξ3 )2n−2 .
For the proof of Lemma 2.6, we refer the reader to Lemma 5 of [10]. Lemma 2.7 Let ǫ > 0. Then, we have that ˆ s, a, X Proof
n+1 2n +ǫ
ˆ s, a, n+1 4n . ֒→ Zˆ s, a ֒→ X
From the definition, it is easily checked that Lemma 2.7 is true.
Lemma 2.8 Let s, a ∈ R and u(t) = φ(t)W (t)u0 . Then, we have that kukZ s, a + kukL∞ (R;H (s,a) ) ≤ Cku0 kH (s,a) . x
t
Proof
(s,a)
From the definition of Z s, a and Hx
x
, we can obtain Lemma 2.8.
Lemma 2.9 Let s, a ∈ R and u(t) = φ(t)
Z
t
W (t − s)f (s)ds. 0
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Then, we have that kukZ s, a + kukL∞(R;H (s,a) ) x t
≤ C F −1 hτ + (−1)n ξ 2n+1 i−1 F f Z s, a + C hξis−a |ξ|a hτ + (−1)n ξ 2n+1 i−1 F f L2 L1 . ξ
τ
For the proof of Lemma 2.9, we refer the reader to [6, 9, 11].
3
Dyadic Bilinear Estimates
1 , f and g are restricted to Aj1 and Aj2 Lemma 3.1 Suppose s = − n2 + 34 and a = − 2n with j1 , j2 ∈ N ∪ {0}, respectively. Then we have the estimates
hτ + (−1)n ξ 2n+1 i−1 ξ(ξ n f ∗ g) ˆs, a ≤ Ckf kZˆ s, a kgkZˆ s, a , (3.1) (Aj ) Z
for j ≥ 0 in the following cases. (i) At least two of j, j1 , j2 are less than 30 and C(j, j1 , j2 ) ∼ 1. 1 (ii) j1 , j2 ≥ 30, |j1 − j2 | ≤ 10, 0 < j < j1 − 9 and C(j, j1 , j2 ) ∼ 2− 2 j . (iii) j, j1 ≥ 30, |j − j1 | ≤ 10, 0 < j2 < j − 10 and C(j, j1 , j2 ) ∼ 2−δj2 + 2−δ(j−j2 ) for some δ > 0. (iv) j, j2 ≥ 30, |j − j2 | ≤ 10, 0 < j1 < j − 10 and C(j, j1 , j2 ) ∼ 2−δj1 + 2−δ(j−j1 ) for some δ > 0. (v) j, j1 , j2 ≥ 30, |j − j1 | ≤ 10, |j − j2 | ≤ 10 and C(j, j1 , j2 ) ∼ 1. (vi) j1 , j2 ≥ 30, j = 0 and C(j, j1 , j2 ) ∼ 1. (vii) j, j1 ≥ 30, j2 = 0 and C(j, j1 , j2 ) ∼ 1. (viii) j, j2 ≥ 30, j1 = 0, and C(j, j1 , j2 ) ∼ 1. Proof (i) In this case we may assume that j, j1 , j2 are all less than 40. By using the Young inequality and H¨ older inequality as well as Lemma 2.7, we have that the left hand side of (3.1) can be bounded kf ∗ gkL∞ L2 ≤ Ckf kL2 L4/3 kgkL2 L4/3 ≤ Ckf k ˆ 0,− ξ
τ
ξ
τ
ξ
X
τ
1 , n+1 2n 4n
kgk ˆ 0,− X
1 , n+1 2n 4n
≤ Ckf kZˆ s,a kgkZˆ s,a .
(ii) From Lemma 2.5, in this case, we have that 2kmax := 2max{k, k1 , k2 } ≥ C2j+2nj1 .
(3.2)
In this case, by using (3.2), the left hand side of (3.1) can be bounded by X n 7 C2(− 2 + 4 )j 2nj1 2−k/2 kf ∗ gkL2τ ξ (Aj ∩Bk ) .
(3.3)
k≥0
When 2k ≥ C2j+2nj1 , by using (2.3), since n ≥ 2, by using the definition of Zˆ s,a , we have that (3.3) can bounded by X n 7 3 C2(− 2 + 4 )j 2(2n− 2 )j1 2−k/2 k(hξis f ) ∗ (hξis g)kL2 (Bk ) τξ
k≥j+2nj1 +O(1)
n
5
n
3
3
≤ C2(− 2 + 4 )j 2(n− 2 )j1 k(hξis f ) ∗ (hξis g)kL2 ≤ C2(− 2 + 4 )j 2−j1 kf k ˆ s, 21 kgk ˆ s, 21 X(2,1)
≤ C2
−5j/4
X(2,1)
kf k ˆ s, 21 kgk ˆ s, 21 ≤ C2−5j/4 kf kZˆ s,a kgkZˆ s,a . X(2,1)
X(2,1)
(3.4)
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1
When 2k1 ≥ C2j+2nj1 , we have that 2 4n k1 2− 4n (j+2nj1 ) 2 4n (k1 −k) ≥ C, since n ≥ 2, by using the definition of Zˆ s,a and (2.16), (3.3) can be bounded by
X n 5 1
C2(− 2 + 4 + 4n )j 2(n−1)j1 2−(1+2n)k/4n hξis hτ + (−1)n ξ 2n+1 i1/2 f ∗ (hξis g) 2 L (Aj ∩Bk )
k≥0
≤ C2
5 1 (− n 2 + 4 + 4n )j
2(n−1)j1
X
2−k/4n kf k
k≥0 n
5
1
≤ C2(− 2 + 4 + 4n )j 2−j1 kf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
1
1
ˆ s, 2 X (2,1)
≤ C2− 2 j kf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
1
≤ C2− 2 j kf kZˆ s,a kgkZˆ s,a . When 2k2 ≥ C2j+2j1 , this case can be treated similarly to 2k1 ≥ C2j+2nj1 . (iii) In this case, from Lemma 2.5, we have that 2max ≥ C22nj+j1 . Thus, in this case, the left hand side of (3.1) can be bounded by X n 7 C2(− 2 + 4 )j 2nj1 2−k/2 kf ∗ gkL2 L2τ .
(3.5)
ξ
k≥0
When 2k ∼ 2kmax ≥ 22nj+j2 , by using (2.3), (3.5) can be bounded by X n 3 n 7 3n 3 C2( 2 − 4 )j2 2(− 2 + 4 )j 2( 2 − 4 )j1 2−k/2 k(hξis f ) ∗ (hξis g)kL2
τ ξ (Bk )
k≥2nj+j2 +O(1)
n
n
5
≤ C2( 2 − 4 )j2 2j1 k(hξis f ) ∗ (hξis g)kL2
τ ξ (Bk )
5
≤ C2( 2 − 4 )j2 2(1−n)j1 kf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
,
if n = 2, (3.5) can be bounded by 1
1
C2− 4 j2 2−j1 kf k ˆ s, 21 kgk ˆ s, 21 ≤ C2− 4 j2 2−j1 kf kZˆ s,a kgkZˆ s,a ; X(2,1)
X(2,1)
if n > 2, (3.5) can be bounded by n
n
5
n
1
n
5
1
2(− 2 + 4 )(j−j2 ) 2(− 2 − 4 )j kf k ˆ s, 21 kgk ˆ s, 21 ≤ C2(− 2 + 4 )(j−j2 ) 2(− 2 − 4 )j kf kZˆ s,a kgkZˆ s,a . X(2,1)
X(2,1)
When 2k1 ∼ 2kmax ≥ 22nj+j2 , with the aid of (2.16) and the fact that 2−
2n−1 4n (2nj+j2 )
1
2(2n−1)k1 /4n 2 4n (k1 −k) ≥ C,
(3.5) can be bounded by n
3
5
1
C2 2 j 2( 2 − 4 + 4n )j2
X
k≥0 3
n
5
1
2−(2n+1)k/4n hξis hτ + (−1)n ξ 2n+1 i1/2 f ∗ (hξis g)
L2 (Aj ∩Bk )
≤ C2(−n+ 2 )j 2( 2 − 4 + 4n )j2
X
2−k/4n kf k
k≥0
if n = 2, (3.6) can be bounded by X 1 1 C2− 2 j 2− 8 j2 2−k/4n kf k k≥0
1
ˆ s, 2 X (2,1)
1
ˆ s, 2 X (2,1)
kgk
kgk
1
ˆ s, 2 X (2,1)
,
1
1
ˆ s, 2 X (2,1)
(3.6)
1
≤ C2− 2 j 2− 8 j2 kf kZˆ s,a kgkZˆ s,a ,
if n > 2, (3.6) can be bounded by 1
3
C2(−n+ 4n +1)j1 2(−n+ 2 )j kf k
1
ˆ s, 2 X (2,1)
kgk
n
1
ˆ s, 2 X (2,1)
5
1
When 2k2 ∼ 2kmax ≥ 22nj+j2 , with the aid of (2.16) and the fact that 2−
2n−1 4n (2nj+j2 )
n
≤ C2−( 2 − 4 + 4n )(j−j2 ) 2−( 2 −
1
2(2n−1)k2 /4n 2 4n (k2 −k) ≥ C,
n+1 4n )j
kf kZˆ s,a kgkZˆ s,a .
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(3.5) can be bounded by n
3
5
1
C2 2 j 2( 2 − 4 + 4n )j2
X
k≥0 n
3
5
1
2−(2n+1)k/4n (hξis f ) ∗ (hξis hτ + (−1)n ξ 2n+1 i1/2 g)
L2 (Aj ∩Bk )
≤ C2(−n+ 2 )j 2( 2 − 4 + 4n )j2
X
2−k/4n kf k ˆ s, 12 kgk ˆ s, 21 , X(2,1)
k≥0
if n = 2, (3.7) can be bounded by X 1 1 C2− 2 j 2− 8 j2 2−k/4n kf k k≥0
1
ˆ s, 2 X (2,1)
kgk
(3.7)
X(2,1)
1
1
ˆ s, 2 X (2,1)
1
≤ C2− 2 j 2− 8 j2 kf kZˆ s,a kgkZˆ s,a ,
if n > 2, (3.7) can be bounded by 3
1
C2(−n+ 4n +1)j1 2(−n+ 2 )j kf k
1
ˆ s, 2 X (2,1)
kgk
n
1
ˆ s, 2 X (2,1)
5
n
1
≤ C2−( 2 − 4 + 4n )(j−j2 ) 2−( 2 −
n+1 4n )j
kf kZˆ s,a kgkZˆ s,a .
(iv) In this case, from Lemma 2.5, we have that 2max ≥ C22nj+j1 . Thus, in this case, the left hand side of (3.1) can be bounded by X n 7 C2(− 2 + 4 )j 2nj1 2−k/2 kf ∗ gkL2 L2τ . (3.8) ξ
k≥0
When 2k ∼ 2kmax ≥ 22nj+j1 , by using (2.3), (3.8) can be bounded by X n 3 n 7 3n 3 C2( 2 − 4 )j2 2(− 2 + 4 )j 2( 2 − 4 )j1 2−k/2 k(hξis f ) ∗ (hξis g)kL2
τ ξ (Bk )
k≥2nj+j1 +O(1)
≤ C2( ≤ C2
3n 5 2 − 4 )j1
5 ( 3n 2 − 4 )j2
≤ C2(−
2(1−n)j2 k(hξis f ) ∗ (hξis g)kL2
τ ξ (Bk )
2(1−2n)j1 kf k
3n 5 2 + 4 )(j−j1 )
n
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
2(−
3n 5 2 + 4 )(j−j1 )
n
1
2(− 2 − 4 )j kf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
1
2(− 2 − 4 )j kf kZˆ s,a kgkZˆ s,a .
When 2k1 ∼ 2kmax ≥ 22nj+j1 , with the aid of (2.16) and the fact that 2−
2n−1 4n (2nj+j1 )
1
2(2n−1)k1 /4n 2 4n (k1 −k) ≥ C,
(3.8) can be bounded by 3
n
5
1
C2(−n+ 2 )j 2( 2 − 4 + 4n )j1
X
k≥0 3
n
5
1
≤ C2(−n+ 2 )j 2( 2 − 4 + 4n )j1
X
k≥0
2−(2n+1)k/4n hξis hτ + (−1)n ξ 2n+1 i1/2 f ∗ (hξis g)
L2 (Aj ∩Bk )
2−k/4n kf k ˆ s, 21 kgk ˆ s, 12 , X(2,1)
X(2,1)
(3.9)
if n = 2, (3.9) can be bounded by X 1 1 1 1 C2− 2 j 2− 8 j1 2−k/4n kf k ˆ s, 21 kgk ˆ s, 12 ≤ C2− 2 j 2− 8 j1 kf kZˆ s,a kgkZˆ s,a , X(2,1)
k≥0
X(2,1)
if n > 2, (3.9) can be bounded by 1
3
C2(−n+ 4n +1)j1 2(−n+ 2 )j kf k
1
ˆ s, 2 X (2,1)
kgk
n
1
ˆ s, 2 X (2,1)
5
1
When 2k2 ∼ 2kmax ≥ 22nj+j1 , with the aid of (2.16) and the fact that 2−
2n−1 4n (2nj+j1 )
n
≤ C2−( 2 − 4 + 4n )(j−j1 ) 2−( 2 −
1
2(2n−1)k2 /4n 2 4n (k2 −k) ≥ C,
n+1 4n )j
kf kZˆ s,a kgkZˆ s,a .
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(3.8) can be bounded by n
3
5
1
C2(−n+ 2 )j 2( 2 − 4 + 4n )j1
X
k≥0 n
3
5
1
≤ C2(−n+ 2 )j 2( 2 − 4 + 4n )j1
X
2−(2n+1)k/4n (hξis f ) ∗ (hξis hτ + (−1)n ξ 2n+1 i1/2 g)
L2 (Aj ∩Bk )
2−k/4n kf k ˆ s, 21 kgk ˆ s, 12 , X(2,1)
X(2,1)
k≥0
(3.10)
if n = 2, (3.10) can be bounded by X 1 1 1 1 C2− 2 j 2− 8 j1 2−k/4n kf k ˆ s, 21 kgk ˆ s, 12 ≤ C2− 2 j 2− 8 j1 kf kZˆ s,a kgkZˆ s,a , X(2,1)
k≥0
X(2,1)
if n > 2, (3.10) can be bounded by n
3
1
5
n
1
C2(−n+ 4n +1)j1 2(−n+ 2 )j kf k ˆ s, 12 kgk ˆ s, 21 ≤ C2−( 2 − 4 + 4n )(j−j1 ) 2−( 2 − X(2,1)
n+1 4n )j
X(2,1)
kf kZˆ s,a kgkZˆ s,a .
(v) In this case, (3.1) can be bounded by X 7 n 2( 2 + 4 )j 2−k/2 kf ∗ gkL2τ ξ (Bk ) .
(3.11)
k≥0
(a) When 2k ∼ 2kmax ≥ 22nj1 +j , (3.8) can be bounded by X 6n+1 2−k/2 k(hξis f ) ∗ (hξis g)kL2 C2 4 j
τ ξ (Bk )
k≥(2n+1)j+O(1)
≤ C2
2n−1 j 4
k(hξis f ) ∗ (hξis g)kL2 ≤ Ckf k τξ
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.12)
with the aid of (2.2). (b) When 2k1 ∼ 2kmax ≥ 22nj1 +j , with the aid of (2.15), (3.11) can be bounded by 6n+1 X C2 4 j 2−k/2 k(hξis f ) ∗ (hξis g)kL2 (Bk ) τξ
k≥0
≤ C2
2n−1 j 4
X
k≥0
≤C
X
2−k/2 (hξis hτ + (−1)n ξ 2n+1 i1/2 f ) ∗ (hξis g)
L2τ ξ
2−k/4 kf k ˆ s, 12 kgk ˆ s, 21 ≤ Ckf k ˆ s, 12 kgk ˆ s, 12 ≤ Ckf kZˆ s,a kgkZˆ s,a . X(2,1)
k≥0
X(2,1)
X(2,1)
X(2,1)
(3.13)
(c) When 2k2 ≥ 22nj1 +j , this case can be proved similarly to 2k1 ∼ 2kmax ≥ 22nj1 +j . (vi) We prove
2nj1 hτ + (−1)n ξ 2n+1 i−1 ξf ∗ g Xˆ a ≤ Ckf k ˆ s, 21 kgk ˆ s, 12 . (3.14) X(2,1)
L
X(2,1)
When |ξ| ≤ 2−2nj1 , since
kf kXˆ a ≤ kf k L
ˆ a, X L
n+1 ,1 2n
,
(3.15)
by using the H¨older inequality and Young inequality as well as (2.31), we have that
n−1 3
2(2n− 2 )j1 |ξ|a+1 hτ i− 2n (hξis f ) ∗ (hξis g) 2 Lτ ξ
≤ C2
(2n− 32 )j1
≤ C2
−(n+ 12 )j1
a+1
|ξ|
2 k(hξis f ) ∗ (hξis g)kL∞ L2 L (|ξ|≤2−2nj1 ) ξ
ξ
s
khξi f kL2 L4/3 khξi gkL2 L4/3 ξ
τ
ξ
τ
≤ Ckf k ˆ s, 21 kgk ˆ s, 21 ≤ Ckf kZˆ s,a kgkZˆ s,a . X(2,1)
τ
s
X(2,1)
(3.16)
No.4
1073
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
When 2−2nj1 ≤ |ξ| ≤ 1, we consider 2k2 ∼ 2kmax , 2k1 ∼ 2kmax and 2k ∼ 2kmax , respectively. When 2k2 ∼ 2kmax , (3.14) can be bounded by X n−1
3 2(2n− 2 )j1 (3.17) 2− 2n k |ξ|a+1 (hξis f ) ∗ (hξis g) L2 (B ) , k
τξ
k≥0
since −s ≤ n(a + 1), by using 2k2 ≥ C|ξ|22nj1 , we have s
1
3
3
2−2sj1 |ξ|a+1 ≤ C(|ξ|22nj1 )− n ≤ C2( 2 − 4n )k2 ≤ C2k2 /2 2− 4n k , by using (2.16), we have that (3.17) can be bounded by
X 2n+1
C2nj1 2− 4n k (hξis f ) ∗ (hξis hτ + (−1)n ξ 2n+1 i1/2 g)
L2τ ξ (Bk )
k≥0
≤C
X
1
2− 4n k kf k
k≥0
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
≤ Ckf k
kgk
1
ˆ s, 2 X (2,1)
1
ˆ s, 2 X (2,1)
≤ Ckf kZˆ s,a kgkZˆ s,a .
When 2k1 ∼ 2kmax , this case can be proved similarly to case 2k2 ∼ 2kmax . When 2k = 2kmax ≤ 4max 2k1 , 2k2 , this case can be proved similarly to case 2k2 ∼ 2kmax . When 2k = 2kmax > 4max 2k1 , 2k2 , then 2k ∼ C|ξ|22nj1 , in this case, we consider (a) f ∗ g is restricted to D1 ; (b) f ∗ g is outside of D2 . 2n−1 2n+1 If f ∗ g is restricted to D1 , then 2− 2 j1 ≤ |ξ| ≤ 1 and 2 2 j1 ≤ |τ | ≤ 22nj1 , by using (2.3), we have that
1
n−1 1 1
2nj1 |ξ|1− 2n hτ i− 2n f ∗ g 2 ≤ C2(n− 2 )j1 |ξ| 2 (hξis f ) ∗ (hξis g) 2 Lτ ξ
Lτ ξ
≤ Ckf k ˆ s, 12 kgk ˆ s, 12 ≤ Ckf kZˆ s,a kgkZˆ s,a . X(2,1)
X(2,1)
− 2n−1 j1 2
2n+1
If f ∗ g is restricted to D2 , then 2−2nj1 ≤ |ξ| ≤ 2 and 1 ≤ |τ | ≤ 2 2 j1 , by using the H¨older inequality and the Young inequality, since |ξ| ∼ 2k−2nj1 , by using (2.31)–(2.32), we have that
X n−1 1
2nj1 2− 2n k |ξ|1− 2n f ∗ g 2 Lτ ξ (Bk )
k≤ 2n+1 j1 +O(1) 2 3
X
≤ C2(2n− 2 )j1
k≤ 2n+1 j1 +O(1) 2 3
X
≤ C2(2n− 2 )j1
n−1 1
2− 2n k |ξ|1− 2n (hξis f ) ∗ (hξis g)
L2τ ξ (Bk )
2−
n−1 2n k
1
k|ξ|1− 2n kL2ξ (|ξ|∼2k−2nj1 ) k(hξis f ) ∗ (hξis g)kL∞ L2τ ξ
k≤ 2n+1 j1 +O(1) 2 1
≤ C2(−n− 2 )j1
X
2k khξis f kL2ξ L1τ khξis gkL2τ ξ
k≤ 2n+1 j1 +O(1) 2
≤ Ckf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ s, 2 X (2,1)
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.18)
(vii) In this case, we consider |ξ2 | ≤ 2−2nj and 2−2nj ≤ |ξ2 | ≤ 1. When |ξ2 | ≤ 2−2nj , by using the Young inequality and H¨ older inequality and (2.32), we have that (3.1) can be bounded by X 2(n+1)j 2−k/2 k(hξis f ) ∗ gkL2 (Bk ) τξ
k≥0
X 1
≤ C2nj 2−k/2 (hξis f ) ∗ (|ξ2 |− 2n g) k≥0
L2τ ξ (Bk )
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ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
1
− 2n
g
|ξ|
1
≤ C2nj khξis f kL2ξ L1τ k|ξ|− 2n gkL1ξ L2τ ≤ Ckf k ≤ Ckf k
kgk
1
ˆ s, 2 X (2,1)
1 ˆ s, 2 X (2,1)
L2ξ L2τ
≤ Ckf kZˆ s,a kgkZˆ s,a .
1
ˆ − 2n X L
(3.19)
Now we consider the case 2−2nj ≤ |ξ2 | ≤ 1. When 2k = 2kmax , since 1
1
2−j (|ξ2 |)− 2n 2ǫ(k−k2 ) 2 2n k ≥ C, by using (2.3) and (i) of Lemma 2.4, we have kξ(ξ n f ) ∗ gk ˆ s, − 12 X(2,1)
1
≤ C2(n+1)j hτ + (−1)n ξ 2n+1 i− 2 +ǫ (hξis f ) ∗ g 2 Lτ ξ
1 1
≤ C2nj hτ + (−1)n ξ 2n+1 i−( 2 − 2n )+2ǫ (hξis f ) ∗ (|ξ|−1/2n hτ + (−1)n ξ 2n+1 i−ǫ g)
L2τ ξ
≤ Ckf k
1
1
ˆ s, 2 X (2,1)
kχ|ξ|≥2−2nj |ξ|− 2n gk
1
ˆ 0, 2 −ǫ X (2,1)
≤ Ckf k
1
ˆ s, 2 X (2,1)
kgk
1
ˆ − 2n , X L
n+1 2n
≤ Ckf kZˆ s,a kgkZˆ s,a . k1
k
k2
1
When 2k1 = 2kmax , by using 2 n 2− 6n |ξ2 |− 2n 2−j 2− 3n ≥ C and (2.16), we have that in this case the left hand side of (3.1) can be bounded by X 2(n+1)j 2−k/2 k(hξis f ) ∗ gkL2 (Bk ) τξ
k≥0
≤ C2nj
X
2−
3n+1 6n k
L2τ ξ (Bk )
k≥0
≤C
X
1 1
(hξis hτ + (−1)n ξ 2n+1 i n f ) ∗ (|ξ|−1/2n hτ + (−1)n ξ 2n+1 i− 3n g)
1
2− 6n k kf k
k≥0
≤ Ckf k ˆ s, 21 kgk X(2,1)
1
ˆ s, n X (2,1) 1
ˆ − 2n , X L
kgk
n+1 2n
1
ˆ − 2n , X L
3n−2 6n
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.20)
When 2k2 = 2kmax , if 2k2 = 2kmax ≤ 4max 2k1 , 2k , this case can be proved similarly to 2k = 2kmax or 2k1 = 2kmax . If 2k2 = 2kmax > 4max 2k1 , 2k , then 2k2 = 2kmax ∼ |ξ2 |22nj , in this case, we consider that g is restricted to D2 and g is restricted to D1 . When g is restricted 2n−1 2n+1 to D2 , then 2−2nj1 ≤ |ξ| ≤ 2− 2 j1 and 1 ≤ |τ | ≤ 2 2 j1 , by using the H¨older inequality and the Young inequality, since |ξ| ∼ 2k−2nj1 , by using (2.22), we have that X k 2(n+1)j 2− 2 k(hξis f ) ∗ gkL2 (Bk ) τξ
k≥0
≤ C2(n+1)j khξis f kL2ξ L1τ kgkL1ξ L2τ
1
≤ C2(n+1)j kf k ˆ s, 12 |ξ2 | 2n 2
Lξ (|ξ2 |∼2k2 −2nj )
X(2,1)
≤ C2(n+1)j kf k ˆ s, 12
X(2,1)
≤ Ckf k ˆ s, 21 kgk X(2,1)
2
X 1
− 2n
g
|ξ2 |
L2τ ξ
k2
X n+1 1
2 2n (k2 −2nj) |ξ2 |− 2n g
L2τ ξ
k2
1 n+1 ˆ − 2n , 2n ,1 X L 2n−1
≤ Ckf kZˆ s,a kgkZˆ s,a .
2n+1
2
2
(3.21)
When g is restricted to D1 , then 2− 2 j ≤ |ξ2 | ≤ 1 and 2 2 j ≤ |τ2 | ≤ 22nj . When g is 2n−1 − 2n−1 j 2 restricted to B[ 2n+1 j, 2n+1 j+α] with 0 ≤ α ≤ 2n−1 ≤ |ξ2 | ≤ 2− 2 j+α , by 2 j. From 2 2
2
No.4
1075
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
using the H¨older inequality and the Young inequality as well as |τ2 | ≥ C22nj |ξ2 | which yields n+1 n+1 |τ2 | 2n ≥ C2(n+1)j |ξ2 | 2n , by using (2.22), we have that X ≤ C2(n+1)j 2−k/2 k(hξis f ) ∗ gkL2 (Bk ) kξ(ξ n f ) ∗ gk s, − 12 ˆ X (B≥2α ) (2,1)
τξ
k≥2α
≤ C2(n+1)j 2−α khξis f kL2ξ L1τ kgkL1ξ L2τ ≤ C2−α kf k ˆ s, 21 k1k X(2,1)
L2ξ (2−
√ ≤ C α2−α kf k ˆ s, 12 kgk X(2,1)
2n−1 j 2n−1 2 ≤|ξ|≤2− 2 j+α )
1
ˆ − 2n , X L
kgk
1
ˆ − 2n , X L
n+1 2n
√ ≤ C α2−α kf kZˆ s,a kgkZˆ s,a . When g is restricted to B[ 2n+1 j+γ,2nj] with 0 ≤ γ ≤ 2 which yields |τ2 |
n+1 2n
2
≥ C2(n+1)j |ξ2 |
kξ(ξ n f ) ∗ gk
1
ˆ s, − 2 (B≤2α ) X (2,1)
2n−1 j 2
n+1 2n
(3.22) , by using (2.16) and |τ2 | ≥ C22nj |ξ2 |
n+1 2n
, we have that X ≤ C2(n+1)j 2−k/2 k(hξis f ) ∗ gkL2
τ ξ (Bk )
k≤2α
3j
≤ C2 2
X
kf k
k≤2α
≤ Cα2−γ kf k
1
ˆ s, 2 X (2,1) 1
ˆ s, 2 X (2,1)
k|ξ|−1/2 gk
kgk
1
ˆ − 2n , X L
L2τ ξ (|ξ|≥2−
2n−1 j+γ 2 )
n+1 2n
≤ Cα2−γ kf kZˆ s,a kgkZˆ s,a .
(3.23)
If g is restricted to B[ 2n+1 j+γ, 2n+1 j+α] with γ ≤ α, from (3.22) and (3.23), we have that 2 2 √ n kξ(ξ f ) ∗ gk ˆ s, − 12 ≤ C( α2−α + α2−γ )kf kZˆ s,a kgkZˆ s,a . (3.24) X(2,1)
Let an (0 ≤ n ≤ N ) be the decreasing sequence satisfying a0 =
2n − 1 an 3 j, an+1 = , 0 < aN ≤ . 2 2 2
(3.25)
where N is the minimum integer satisfying N ≥ log2 nj . Firstly we take α = a0 and γ = a1 and next take α = a1 and γ = a2 . Repeating this procedure at the end we take α = aN and γ = 0. By using the fact that 2x ≥ x2 ≥ x with x ≥ 4, combining (3.24), we have that X 1 kf k ˆ s,a kgk ˆ s,a ≤ Ckf k ˆ s,a kgk ˆ s,a . kξ(ξ n f ) ∗ gk s, − 12 ≤ C 1 + Z Z Z Z ˆ (B≤2α ) an X (2,1) 0≤n≤N
(viii) This case can be proved similarly to case (vii). We have completed the proof of Lemma 3.1.
Lemma 3.2 Suppose f and g are restricted to Aj1 and Aj2 , respectively. Then we have the estimates
s−a
hξi hτ + (−1)n ξ 2n+1 i−1 |ξ|a+1 (ξ n f ) ∗ g) L2 L1 ≤ Ckf kZˆ s, a kgkZˆ s, a (3.26) ξ
τ
for j ≥ 0 in the following cases. (i) At least two of j, j1 , j2 are less than 30 and C(j, j1 , j2 ) ∼ 1. 3 (ii) j1 , j2 ≥ 30, |j1 − j2 | ≤ 10, 0 < j < j1 − 9 and C(j, j1 , j2 ) ∼ 2− 8 j . (iii) j, j1 ≥ 30, |j − j1 | ≤ 10, 0 < j2 < j − 10 and C(j, j1 , j2 ) ∼ 2−δj2 + 2−δ(j−j2 ) for some δ > 0.
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ACTA MATHEMATICA SCIENTIA
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(iv) j, j2 ≥ 30, |j − j2 | ≤ 10, 0 < j1 < j − 10 and C(j, j1 , j2 ) ∼ 2−δj1 + 2−δ(j−j1 ) for some δ > 0. (v) j, j1 , j2 ≥ 30, |j − j1 | ≤ 10, |j − j2 | ≤ 10 and C(j, j1 , j2 ) ∼ 1. (vi) j1 , j2 ≥ 30, j = 0 and C(j, j1 , j2 ) ∼ 1. (vii) j, j1 ≥ 30, j2 = 0 and C(j, j1 , j2 ) ∼ 1. (viii) j, j2 ≥ 30, j1 = 0, and C(j, j1 , j2 ) ∼ 1. Proof (i) In this case we may assume that j, j1 , j2 are all less than 40. By using the Young inequality and H¨ older inequality, we have that the left hand side of (3.26) can be bounded by kf ∗ gkL∞ L2 ≤ Ckf kL2 L4/3 kgkL2L4/3 ≤ Ckf k ˆ 0,− ξ
τ
ξ
τ
ξ
X
τ
1 , n+1 2n 4n
kgk ˆ 0,− X
1 , n+1 2n 4n
≤ Ckf kZˆ s,a kgkZˆ s,a .
Cases (ii)–(v), (vii), (viii) of Lemma 3.2 can be derived from Cases (ii)–(v), (vii), (viii) of Lemma 3.1. In fact, by using the triangle inequality and the Schwarz inequality, we have that X X (3.27) 2k/2 kf kL2τ (Bk ) . kf kL1τ (Bk ) ≤ C kf kL1τ ≤ C k≥0
k≥0
From (3.25), for j 6= 0, combining Lemma 3.1 with (3.27), we have that
s−a a+1
hξi |ξ| hτ + (−1)n ξ 2n+1 i−1 (ξ n f ) ∗ g 2 1 L L (Aj ) τ
ξ
≤ C kξ(ξ n f ) ∗ gk
1 ˆ s, − 2 X (2,1)
(Aj )
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.28)
Now consider the case (vi). We consider |ξ| ≤ 2−2nj1 and 2−2nj ≤ |ξ| ≤ 1, respectively. When |ξ| ≤ 2−2nj1 , by using the Young inequality and H¨older inequality, we have that
s−a a+1
hξi |ξ| hτ + (−1)n ξ 2n+1 i−1 (ξ n f ) ∗ g L2 L1 (Aj ) τ
ξ
nj1 a+1 n 2n+1 − n−1 ≤ C2 |ξ| hτ + (−1) ξ i 2n f ∗ g 2 2 Lξ Lτ (Aj )
n−1 3
≤ C2(2n− 2 )j1 |ξ|a+1 hτ i− 2n (hξis f ) ∗ (hξis g) 2 Lτ ξ
≤ C2
(2n− 32 )j1
≤ C2
−(n+ 12 )j1
a+1
|ξ|
2 k(hξis f ) ∗ (hξis g)kL∞ L2τ L (|ξ|≤2−2nj1 ) ξ
ξ
s
s
khξi f kL2 L4/3 khξi gkL2 L4/3 ξ
≤ Ckf k
1 ˆ s, 2 X (2,1)
kgk
1 ˆ s, 2 X (2,1)
τ
τ
ξ
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.29)
Case 2−2nj ≤ |ξ| ≤ 1, we consider cases 2k = 2kmax , 2k1 = 2kmax , 2k2 = 2kmax , respectively. (a) 2n−3 s Case 2k = 2kmax . Since |ξ|a+1 2−2sj1 ≤ C(|ξ|22nj1 )− n ≤ C2 4n k , by using the H¨older inequality and the Young inequality, we have that
s−a a+1
hξi |ξ| hτ + (−1)n ξ 2n+1 i−1 (ξ n f ) ∗ g L2 L1 (Aj ) ξ τ
3 ≤ C2(2n− 2 )j1 |ξ|a+1 hτ i−1 (hξis f ) ∗ (hξis g) L2 L1 ξ τ
− 2n+3 nj1 a+1 s s ≤ C2 |ξ| hτ i 4n (hξi f ) ∗ (hξi g) 2 Lξ L1τ
2n+3 3
≤ C2− 2 j1 |ξ|− 4n 2 k(hξis f ) ∗ (hξis g)kL∞ L1 Lξ (2−2nj ≤|ξ|)
s
s
τ
ξ
s
s
≤ C k(hξi f ) ∗ (hξi g)kL∞ L1τ ≤ C khξi f kL2 L1τ khξi gkL2 L1τ ξ
ξ
ξ
No.4
1077
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
≤ Ckf k ˆ s, 21 kgk ˆ s, 21 ≤ Ckf kZˆ s,a kgkZˆ s,a . X(2,1)
(3.30)
X(2,1)
s
2n−3
(b) Case 2k1 = 2kmax . Since |ξ|a+1 2−2sj1 ≤ C(|ξ|22nj1 )− n ≤ C2 4n k1 ≤ C2 using (2.16), we have that
s−a a+1
hξi |ξ| hτ + (−1)n ξ 2n+1 i−1 ((ξ n f ) ∗ g) L2 L1 (Aj ) ξ τ
X 2n+1 s s n 1/2 nj1 − 4n ≤ C2 2
(hξi f ) ∗ (hξi hτ + (−1) i g) 2 2
k1 2
Lξ Lτ
k≥0
2n+3
≤ C2nj1 |ξ|a+1 hτ i− 4n (hξis f ) ∗ (hξis g) 2 1 Lξ Lτ X 1 ≤C 2− 4n kf k ˆ s, 21 kgk ˆ s, 21 X(2,1)
k≥0
≤ Ckf k
1
ˆ s, 2 X (2,1)
3
2− 4n k , by
kgk
1
ˆ s, 2 X (2,1)
X(2,1)
≤ Ckf kZˆ s,a kgkZˆ s,a .
(3.31)
(c) Case 2k2 = 2kmax . This case can be proved similarly to case 2k1 = 2kmax . We have completed the proof of Lemma 3.2.
1 Lemma 3.3 Suppose s = − n2 + 43 and a = − 2n , f and g are restricted to Aj1 and Aj2 , respectively. Then we have the estimates
hτ + (−1)n ξ 2n+1 i−1 ξ n+1 (f ∗ g) ˆs, a ≤ Ckf kZˆ s, a kgkZˆ s, a , (3.32) Z (Aj )
for j ≥ 0 in cases (i)–(viii), where (i)–(viii) are defined as in Lemma 3.1. The proof of Lemma 3.3 can be proved similarly to Lemma 3.1.
Lemma 3.4 Suppose f and g are restricted to Aj1 and Aj2 , respectively. Then we have the estimates
s−a
hξi hτ + (−1)n ξ 2n+1 i−1 |ξ|a+1+n (f ∗ g) 2 1 ≤ Ckf k ˆ s, a kgk ˆ s, a (3.33) Z Z L L ξ
τ
for j ≥ 0, in cases (i)–(viii), where (i)–(viii) are defined as in Lemma 3.2. The proof of Lemma 3.4 can be proved similarly to Lemma 3.2.
1 Lemma 3.5 Let u, v ∈ Z s,a with s = − n2 + 43 and s = − 2n , then
i h
−1
−1 τ + (−1)n ξ 2n+1 F [∂x ((∂xn u)v)]
F Z s,a
i h
−1
n 2n+1 −1 n + F τ + (−1) ξ F [∂x ((∂x u)v)] ≤ CkukZ s,a kvkZ s,a .
(3.34)
Y
To prove (3.34), it suffices to prove that
h i −1
−1
τ + (−1)n ξ 2n+1 F [∂x ((∂xn u)v)] s,a ≤ CkukZ s,a kvkZ s,a .
F
h i Z
−1
n 2n+1 −1 n τ + (−1) ξ F [∂x ((∂x u)v)] ≤ CkukZ s,a kvkZ s,a .
F Y P 2 2 We first prove (3.35). By using kf kZˆ s,a = kIAj f kZˆ s,a , we have that Proof
(3.35) (3.36)
j≥0
h i 2 −1
−1
τ + (−1)n ξ 2n+1 F [∂x ((∂xn u)v)]
F s,a Z
=
2 X −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v)
ξ τ + (−1)n ξ 2n+1
j,j1 j2 ≥0
ˆ Z s,a
=
8 X j=1
Tj ,
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ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
where T1 =
X
j,j1 j2 ≥0,i
T2 =
X
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a ,
ξ τ + (−1)n ξ 2n+1 ˆ
j,j1 j2 ≥0,ii
T3 =
X
Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v)
ξ τ + (−1)n ξ 2n+1
ˆ s,a Z
j,j1 j2 ≥0,iii
T4 =
X
j,j1 j2 ≥0,iv
T5 =
X
j,j1 j2 ≥0,v
T6 =
X
X
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a ,
ξ τ + (−1)n ξ 2n+1 ˆ Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a ,
ξ τ + (−1)n ξ 2n+1 ˆ Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) ,
ξ τ + (−1)n ξ 2n+1 ˆ s,a
j,j1 j2 ≥0,vi
T7 =
Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a ,
ξ τ + (−1)n ξ 2n+1 ˆ
j,j1 j2 ≥0,vii
T8 =
X
,
Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) ,
ξ τ + (−1)n ξ 2n+1 ˆ s,a
j,j1 j2 ≥0,viii
Z
2 −1
IAj (IAj1 F (∂xn u)) ∗ (IAj2 F v) s,a .
ξ τ + (−1)n ξ 2n+1 ˆ Z
here (i), (ii), (iii), (iv), (v), (vi), (vii), (viii) is case (i), (ii), (iii), (iv), (v), (vi), (vii), (viii) of Lemmas 3.1 and Lemma 3.2. Combining Tj (1 ≤ j ≤ 8, j ∈ N ), Lemmas 3.1 and Lemma 3.2 P with kf k2Zˆ s,a = j≥0 kIAj f k2Zˆ s,a , we easily obtain (3.35). By using a proof similarly to (3.35), we easily obtain (3.36). We have completed the proof of Lemma 3.5. 1 , then Lemma 3.6 Let u, v ∈ Z s,a with s = − n2 + 43 and s = − 2n
h
i −1 n+1
−1 τ + (−1)n ξ 2n+1 F ∂x (uv) s,a
F Z
h i n+1
−1
n 2n+1 −1 + F τ + (−1) ξ F ∂x (uv) ≤ CkukZ s,a kvkZ s,a . Y
(3.37)
Proof Combining Lemma 3.3 with Lemma 3.4, by using a similar proof to Lemma 3.5, we can obtain Lemma 3.6.
4
Proof of Theorem 1.1
Now we are in a position to prove Theorem 1.1. 1 In this section, we assume that s = − n2 + 34 and s = − 2n . −n If u(x, t) is the solution to (1.1)–(1.2), then uλ (x, t) = λ u solution to (1.1)–(1.2). By a direct calculation, we have that
x t λ , λ2n+1
with λ ≥ 1 is the
1
kuλ (x, 0)kH (s,a) ≤ Cλ−n+ 2 −a ku0 kH (s,a) . Thus, without loss of generality, we can assume that ku0 kH (s,a) is sufficiently small. (1.1)–(1.2) is equivalent to the following integral equation: Z t 2n+1 2n+1 u(t) = e−t∂x u0 + e−(t−s)∂x ∂x (us )ds. 0
(4.1)
(4.2)
No.4
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
1079
We define 2n+1
Φ(u) = e−t∂x
u0 +
Z
t
2n+1
e−(t−s)∂x
∂x (us )ds.
(4.3)
0
By using Lemmas 2.8–2.9 and Lemmas 3.5–3.6 and proposition 4.1 of [17] as well as (4.2)–(4.3), we can obtain the existence of the solution to (1.1)–(1.2). The uniqueness of the solution to (1.1)–(1.2) can be found in Proposition 5.1 of [13]. The rest of Theorem 1.1 can be proved similarly to Lemma 4.2 in [16].
5
Proof of Theorem 1.2 In this section, motivated by [1, 4, 13], we prove Theorem 1.2. We define ξ = ξ1 + ξ2 + ξ3 , Q1 = φ(ξ1 ) + φ(ξ2 ) − φ(ξ1 + ξ2 ), Q2 = φ(ξ2 ) + φ(ξ3 ) − φ(ξ2 + ξ3 ), Q3 = Q4 =
eit(φ(ξ1 )+φ(ξ2 )+φ(ξ3 )) − eitφ(ξ) , φ(ξ1 ) + φ(ξ2 ) + φ(ξ3 ) − φ(ξ)
eit(φ(ξ1 )+φ(ξ2 +ξ3 )) − eitφ(ξ) eit(φ(ξ3 )+φ(ξ1 +ξ2 )) − eitφ(ξ) , Q5 = . φ(ξ1 ) + φ(ξ2 + ξ3 ) − φ(ξ) φ(ξ3 ) + φ(ξ1 + ξ2 ) − φ(ξ)
In this section, we always assume ξ(ξ2 + ξ3 )n+1 [2(ξ2 + ξ3 )n + ξ2n + ξ3n ] , Q2 ξξ n (ξ1 + ξ2 ) [2(ξ1 + ξ2 )n + ξ1n + ξ2n ] H2 (ξ, ξ1 , ξ2 , ξ3 ) = 3 , Q1 ξ n+1 (ξ1 + ξ2 ) [2(ξ1 + ξ2 )n + ξ1n + ξ2n ] H3 (ξ, ξ1 , ξ2 , ξ3 ) = , Q1 ξ n+1 (ξ2 + ξ3 ) [2(ξ2 + ξ3 )n + ξ2n + ξ3n ] . H4 (ξ, ξ1 , ξ2 , ξ3 ) = Q2
H1 (ξ, ξ1 , ξ2 , ξ3 ) =
By contradiction, we assume that the solution map of (1.1)–(1.2) St : u0 ∈ H (s, a) (R) → u ∈ C([0, T ]; H (s, a) (R)) is C 3 at zero with s < − n2 + 34 and a ∈ R. Following the general well-posedness principle of [1], we have sup kB3 (u0 )kH (s,a) ≤ Cku0 k3H (s,a) .
(5.1)
t∈[0,T ]
where B1 (u0 )(x, t) = W (t)u0 , Z t B2 (u0 )(x, t) = 2 W (t − τ )∂xn+1 (B1 (u0 )B1 (u0 ))dτ 0 Z t +2 W (t − τ )∂x (B1 (u0 )∂xn B1 (u0 ))dτ, 0 Z t B3 (u0 )(x, t) = 3 W (t − τ )∂xn+1 (B1 (u0 )B2 (u0 ))dτ 0 Z t +3 W (t − τ )∂xn+1 (B2 (u0 )B1 (u0 ))dτ 0 Z t +3 W (t − τ )∂x (B1 (u0 )∂xn B2 (u0 ))dτ 0
(5.2)
(5.3)
1080
ACTA MATHEMATICA SCIENTIA
+3
t
Z
0
We consider the initial data u0 (x) := N
−s+ 2n−1 4
From (5.2), we have that
e−iN x
Vol.37 Ser.B
W (t − τ )∂x ((∂xn B1 (u0 ))B2 (u0 ))dτ.
Z
N−
2n−1 2
eixξ dξ + eiN x
Z
2N −
N−
0
F u0 (ξ) = N −s+
2n−1 4
2n−1 2
2n−1 2
(5.4)
eixξ dξ .
(χI1 (ξ) + χI2 (ξ)) ,
(5.5)
(5.6)
where o n 2n−1 , I1 =: −N, −N + N − 2
n o 2n−1 2n−1 I2 =: N + N − 2 , N + 2N − 2 .
(5.7)
Let Ω = I1 ∪ I2 , by a direct computation, we have that ku0 kH (s, a) ∼ 1.
(5.8)
kB3 (u0 )kH (s,a) = C k[J1 − J2 + J(3) − J(4) + J(5) − J(6) + J(7) − J(8)]kH (s,a) ,
(5.9)
From (5.2)–(5.4), we have that
where J1 = N
−3s+ 3(2n−1) 4
Z
ξ1 ∈Ω
J2 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J3 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J4 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J5 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J6 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J7 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
J8 = N −3s+
3(2n−1) 4
Z
ξ1 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
ξ2 ∈Ω
Z
eixξ H1 Q3 dξ1 dξ2 dξ3 ,
(5.10)
eixξ H1 Q4 dξ1 dξ2 dξ3 ,
(5.11)
eixξ H2 Q3 dξ1 dξ2 dξ3 ,
(5.12)
eixξ H2 Q5 dξ1 dξ2 dξ3 ,
(5.13)
eixξ H3 Q3 dξ1 dξ2 dξ3 ,
(5.14)
eixξ H3 Q5 dξ1 dξ2 dξ3 ,
(5.15)
eixξ H4 Q3 dξ1 dξ2 dξ3 ,
(5.16)
eixξ H4 Q4 dξ1 dξ2 dξ3 .
(5.17)
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
Z
ξ3 ∈Ω
To estimate kB3 (u0 )kH (s,a) , we consider the following three cases. (1) : ξj (j = 1, 2, 3) ∈ I1 , (2) : ξj (j = 1, 2, 3) ∈ I2 , (3) : ξj (j = 1, 2) ∈ I1 , ξ3 ∈ I2 , (4) : ξ1 ∈ I1 , ξj (j = 2, 3) ∈ I2 , (5) : ξj (j = 1, 2) ∈ I2 , ξ3 ∈ I1 , (6) : ξ1 ∈ I2 , ξj (j = 2, 3) ∈ I1 , (7) : ξ1 , ξ3 ∈ I1 , ξ2 ∈ I2 , (8) : ξ1 , ξ3 ∈ I2 , ξ2 ∈ I1 . When (1), (2) occurs, we have that kB3 (u0 )kH (s,a) ≤ CN −3s+
3(2n−1) 4
N s+
5(2n−1) −2n 4
= CN −2s+2n−2 .
(5.18)
When (3) occurs, by using Lemmas 2.5, 2.6 and n is even, we have that |Q1 | ∼ N 2n+1 , |Q2 | ∼ N
2n+1 2
, |Q3 | ∼ 1,
(5.19)
No.4
M.J. Jiang et al: WELL-POSEDNESS OF DISPERSIVE EQUATION
|Q4 | ≤ CN − |H1 | ∼ N
2n+1 2
, |Q5 | ≤ CN −2n−1 ,
− 12 n−n2 +1
1081 (5.20)
, |H2 | ∼ N, |H3 | ∼ N, |H4 | ∼ N.
(5.21)
By using (5.19)–(5.21), we have that 2
kJ1 kH (s,a) ∼ N −2s−n kJ2 kH (s,a) ≤ CN
− 3n−3 2
,
2 −2s− n−2 2 −n
kJ3 kH (s,a) ∼ CN
−2s−(n− 23 )
kJ4 kH (s,a) ∼ CN
−2s−3n+ 12
(5.22) ,
(5.23)
,
(5.24)
,
(5.25)
kJ5 kH (s,a) ∼ CN
−2s−(n− 23 )
,
(5.26)
kJ6 kH (s,a) ∼ CN
−2s−3n+ 12
,
(5.27)
kJ7 kH (s,a) ∼ CN
−2s−(n− 23 )
,
(5.28)
kJ8 kH (s,a) ∼ CN −2s−(2n−1) .
(5.29)
Consequently, when (3) occurs, we have that 1 ∼ ku0 k3H (s,a) ≥ kB3 (u0 )kH (s,a) ≥ kJ3 kH (s,a) + kJ5 kH (s,a) + kJ7 kH (s,a) −kJ1 kH (s,a) − kJ2 kH (s,a) − kJ4 kH (s,a) − kJ6 kH (s,a) − kJ8 kH (s,a) 3
≥ CN −2s−(n− 2 ) ,
(5.30)
when s < − n2 + 34 , letting N −→ ∞ yields a contraction in (5.30). By using a proof similar to the case (3), we can obtain that in cases (4)-(8), 3
1 ∼ ku0 k3H (s,a) ≥ kB3 (u0 )kH (s,a) ≥ CN −2s−(n− 2 ) ,
(5.31)
when s < − n2 + 34 , letting N −→ ∞ yields a contraction in (5.31). We have completed the proof of Theorem 1.2. References [1] Bejenaru I, Tao T. Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr¨ odinger equation. J Funct Anal, 2006, 233: 228–259 [2] Bourgain J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation. Geom Funct Anal, 1993, 3: 107–156 [3] Bourgain J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation. Geom Funct Anal, 1993, 3: 209–262 [4] Bourgain J. Periodic Korteweg-de Vries equation with measures as initial data. Selecta Math, 1997, 3: 115–159 [5] Constantin P, Saut J C. Local smoothing properties of dispersive equations. J Amer Math Soc, 1988, 1: 413–439 [6] Ginibre J, Tsutsumi Y, Velo G. On the Cauchy problem for the Zakharov System. J Funct Anal, 1997, 151: 384–436 [7] Guo Z H. Global well-posedness of the Korteweg-de Vries equation in H −3/4 . J Math Pures Appl, 2009, 91: 583–597 [8] Guo Z H, Kwak C, Kwon S. Rough solutions of the fifth-order KdV equations. J Funct Anal, 2013, 265: 2791–2829 [9] Guo Z H, Wang B X. Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation. J Diff Eqns, 2009, 246: 3864–3901
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[10] Gr¨ unrock A. On the hierarchies of higher order mKdV and KdV equations. Central European J Math, 2010, 8: 500–536 [11] Ionescu A D, Kenig C E. Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J Amer Math Soc, 2007, 20: 753–798 [12] Ionescu A D, Kenig C E, Tataru D. Global well-posedness of the KP-I initial-value problem in the energy space. Invent Math, 2008, 173: 265–304 [13] Kato T. Well-posedness for the fifith order KdV equation. Funkcial Ekvac, 2012, 55: 17–53 [14] Kenig C E, Ponce G, Vega L. Higher-order nonlinear dispersive equations. Proc Amer Math Soc, 1994, 122: 157–166 [15] Kenig C E, Pilod D. Well-posedness for the fifth-order KdV equation in the energy space. Trans Amer Math Soc, 2015, 367: 2551–2612 [16] Kishimoto N. Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Diff Int Eqns, 2009, 22: 447–464 [17] Kishimoto N, Tsugawa K. Local well-posedness for quadratic nonlinear Schr¨ odinger equations and the “good” Boussinesq equation. Diff Int Eqns, 2010, 23: 463–493 [18] Li Y S, Huang J H, Yan W. The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity. J Diff Eqns, 2015, 259: 1379–1408 [19] Pilod D. On the Cauchy problem for higher-order nonlinear dispersive equations. J Diff Eqns, 2008, 245: 2055–2077 [20] Wang H, Cui S. Well-posedness of the Cauchy problem of a water wave equation with low regularity initial data. Math. Computer Mode, 2007, 45: 1118–1132 [21] Yan W, Li Y S, Li S M. Sharp well-posedness and ill-posedness of a higher-ordermodified Camassa-Holm equation. Diff Int Eqns, 2012, 25: 1053–1074 [22] Yan W, Li Y S. Ill-posedness of madified Kawahara equation and Kaup-Kupershmidt equation. Acta Math Sci, 2012, 32B: 710–716
Appendix When n is odd, motivated by the Appendix of [13], we have the following result. Example 1 (high × high7→ low interaction) Let 1 2 − 2n−1 2n+1 2 P1 := (τ, ξ) ∈ R ; |ξ − N | ≤ N , τ −ξ ≤ , 2 P2 := (τ, ξ) ∈ R2 ; (−τ, −ξ) ∈ P1 , f (τ, ξ) := χP1 (τ, ξ), f ∗ g(τ, ξ) ≥ CN
g(τ, ξ) := χP2 (τ, ξ),
− 2n−1 2
χR1 (τ, ξ),
where N is a sufficiently large positive number and 1 − 2n−1 3 − 2n−1 2 2 2 R1 := (τ, ξ) ∈ R ; ξ ∈ N , N , 2 4 By a direct computation, we have that s− 1 , b = kgk − n + 3 , 1 , b ∼ N kf k ˆ − n2 + 34 , 2n ˆ 2 4 2n X
X
F (f g) ≥ CN k∂x ((∂xn f )
− 2n−1 2
τ − 1 N 2n ξ ≤ 1 2 16 2n−1 4
,
IR1 , 5
a
n−( 4 − 2 )(2n−1)+ 1 , b−1 ∼ N ∗ g)k ˆ − n2 + 34 , 2n X
2n+1 (b−1) 2
.
1 , b−1 ≤ Ckf k − n + 3 , 1 , b kgk − n + 3 , 1 , b is invalid Then, we have that k∂x ((∂xn f )g)k ˆ − n2 + 34 , 2n ˆ 2 4 2n ˆ 2 4 2n X X X n+1 for b < 2n+1 .