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Analytic well-posedness of periodic gKdV
J. Differential Equations 253 (2012) 3101–3112
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Journal of Differential Equations www.elsevier.com/locate/jde
Analytic well-posedness of periodic gKdV A. Alexandrou Himonas a,∗ , Gerson Petronilho b,1 a b
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 December 2011 Revised 23 August 2012 Available online 3 September 2012
In the periodic case, it is proved that the Cauchy problem for the generalized Korteweg–de Vries equation (gKdV) is locally well-posed in a class of analytic functions that can be extended holomorphically in a symmetric strip of the complex plane around the x-axis. Thus, the uniform analyticity radius of the solution does not change as time progresses. Also, information about the regularity of the solution in the time variable is provided. © 2012 Elsevier Inc. All rights reserved.
MSC: primary 35Q53 secondary 35B65 Keywords: Analytic spaces Generalized Korteweg–de Vries equation Initial value problem Multilinear estimates Uniform analyticity radius
1. Introduction For k = 1, 2, 3, . . . we consider the initial value problem for the generalized Korteweg–de Vries equation (gKdV)
∂t u + ∂x3 u + uk ∂x u = 0, u (x, 0) = ϕ (x),
x ∈ T, t ∈ R,
(1.1)
with initial data ϕ (x) belonging in a class of 2π -periodic analytic functions that can be extended . holomorphically in a symmetric strip S δ = {x + iy: | y | < δ}, δ > 0, of the complex plane around the x-axis. Our main result is that the gKdV equation is locally well-posed in these spaces. A consequence
* 1
Corresponding author. Fax: +1 574 631 6579. E-mail addresses: [email protected] (A.A. Himonas), [email protected] (G. Petronilho). Partially supported by CNPq and Fapesp.
0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2012.08.024
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of this result is that the uniform analyticity radius of the solution u (x, t ) during its local lifespan is the same as that of the initial data u (x, 0). The motivation for this work comes from the results in [13], where well-posedness of gKdV was studied in Gevrey and analytic spaces, but no precise information about the uniform analyticity radius of the solution was provided. Concerning well-posedness for the nonperiodic gKdV equation, Kato and Masuda [19] showed that if the initial data of the gKdV equation has an analytic continuation which is bounded in a strip containing the real axis, then the solution has the same property for all time, although the width of the strip might decrease with time. Results of this type have been also obtained by Hayashi [14]. Using the analytic spaces G δ,s introduced by Foias and Temam [8] which are defined by the norm
ϕ 2G δ,s =
2s
1 + |ξ |
2
e 2δ(1+|ξ |) ϕˆ (ξ ) dξ < ∞,
(1.2)
R
Grujic´ and Kalisch [11] showed that for given initial data that are analytic in a symmetric strip { z = x + iy: | y | < δ} in the complex plane of width 2δ , there exists a time T > 0 such that the corresponding gKdV solution is analytic in the same strip during the time period [0, T ]. In other words, in the nonperiodic case, the uniform radius of spatial analyticity does not shrink as time progresses. Further results on the uniform radius of spatial analyticity have been established by Bona, Grujic´ and Kalisch [1]. As we have mentioned earlier, in the periodic case, the Cauchy problem for gKdV was studied in [13] using Bourgain type analytic and Gevrey spaces. However, the uniform radius of spatial analyticity obtained could decrease immediately after the initial time. In this paper, using a periodic version of the analytic spaces used in [11] and a modified version of the Bourgain spaces used by Colliander, Keel, Staffilani, Takaoka, and Tao [4], we show that the uniform radius of analyticity does not shrink as time progresses. This is a consequence of the following well-posedness result, which for simplicity we state for sufficiently small initial data so that the corresponding lifespan is equal to one. Theorem 1.1. Let s 1/2 and δ > 0. For small initial data in the space
G δ,s (T) =
f ∈ L 2 (T): f 2G δ,s (T) =
2 |k|2s e 2δ|k| ˆf (k) < ∞ ,
(1.3)
k∈Z
the Cauchy problem for gKdV (1.1) is locally well-posed in the space C ([−1, 1], G δ,s (T)). Spacial analyticity of the periodic KdV equation was proved first by Trubowitz [26]. Construction of non-analytic solutions in time with analytic initial data for KdV has been obtained in [3] and for gKdV in [10] and [17]. When the initial data are analytic then the time-regularity of the solutions is of class Gevrey three. Furthermore, this result is sharp in the sense that there exist initial data that are analytic and 2π -periodic but the corresponding solution to (1.1) does belong to G r (R) for 1 r < 3 [13]. New such examples are presented here using the spaces G δ,s (T) defined by (1.3). For additional results concerning well-posedness and regularity properties of gKdV we refer the reader to Bourgain [2], De Bouard, Hayashi and Kato [7], Kato [18], Kato and Masuda [19], Kato and Ogawa [20], Kenig, Ponce and Vega [21,22] and the references therein. Finally, we would like to point out that there are many works treating the analytic smoothing effect, that is to say, the phenomenon that solutions become analytic with respect to the spacial variable even if the initial data is not analytic but decays rapidly enough. For instance, Tarama [25] demonstrated this phenomenon for the KdV on the line. More precisely, he showed that if the initial data ϕ ∈ L 2 (R) satisfy the decay conditions
∞ −∞
1 + |x| ϕ (x) dx < ∞ and
∞ 0
e δ|x|
1/2
ϕ (x)2 dx < ∞,
(1.4)
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where δ is some positive number, then the corresponding KdV solution u (x, t ) becomes analytic with respect to the variable x for any t > 0. In the periodic case the decay conditions (1.4) do not give any additional information for the initial data ϕ beyond that it belongs to L 2 [0, 2π ]. Taking the initial data such that ϕˆ (k) = 1/(1 + k2 ) we have ϕ ∈ L 2 [0, 2π ]. The solution of the linear part of KdV 3
that corresponds to this initial data has Fourier transform with respect to x equal to e ik t /(1 + k2 ). Therefore, it is not analytic for any t > 0. Adding the nonlinearity u ∂x u to this equation, which gives KdV, should not change this conclusion. For more information about the analytic smoothing effect of dispersive equations we refer the reader to [7,6,15,16,20]. For other smoothing effects and the general theory of KdV and gKdV we refer the reader to [23] and [24]. The paper is organized as follows. Section 2 is dedicated to the proof of Theorem 1.1. In Section 2.1 we introduce the spaces needed and prove the existence of the solution. Then, in Section 2.2 we prove the uniqueness, and in Section 2.3 we prove continuous dependence on initial data. In Section 3 we demonstrate that the uniform radius of analyticity does not change as time progresses. Finally, in the last section we discuss regularity in the time variable. 2. Proof of Theorem 1.1 2.1. Existence Taking the Fourier transform with respect to x in (1.1), solving the resulting differential equation in t and using the inverse Fourier transform reduces the Cauchy problem (1.1) to the following integral equation
t u (x, t ) = W (t )ϕ (x) −
W (t − τ ) w (x, τ ) dτ ,
(2.5)
0
where W (t ) = e −t ∂x and w = uk ∂x u. Next we localize in the time variable by using a cut-off function ψ(t ) ∈ C 0∞ (−2, 2) with 0 ψ 1 and such that ψ(t ) ≡ 1 for |t | < 1. Multiplying (2.5) by ψ , we have 3
t ψ(t )u (x, t ) = ψ(t ) W (t )ϕ (x) − ψ(t )
W (t − τ ) w (x, τ ) dτ ,
(2.6)
0
which can be written as
ψ(t )u (x, t ) = ψ(t )
3 e i (nx+n t ) ϕˆ (n)
n∈Z
+ i ψ(t )
e
i (nx+n3 t )
n∈Z
∞
−∞
3 e i (λ−n )t − 1
λ − n3
ˆ (n, λ) dλ. w
(2.7)
Here, for simplicity, we shall restrict our attention to mean zero data
ϕˆ (0) =
1 2π
ϕ (x) dx = 0. T
(2.8)
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Next, we define the right-hand side of the integral equation (2.6) by T u and decompose it as follows
T u (x, t ) = ψ(t )
3 e i (nx+n t ) ϕˆ (n)
(2.9)
n∈Z∗
+i
∞ j j i t j =1
+ i ψ(t )
j!
ψ(t )
− i ψ(t )
3 e i (nx+n t )
n∈Z∗
e inx
n∈Z∗
R
j −1 ˆ (n, λ) dλ ψ λ − n3 λ − n3 w
(2.10)
R
(1 − ψ)(λ − n3 ) i λt ˆ (n, λ) dλ e w λ − n3
3 e i (nx+n t )
n∈Z∗
R
(2.11)
(1 − ψ)(λ − n3 ) ˆ (n, λ) dλ, w λ − n3
(2.12)
where k∂ u ( u
ˆ (n, λ) = u w x ˆ ∗ uˆ ∗ · · · ∗ u ˆ ∗∂x u )(n, λ) k
and
Z∗ = Z\{0}. Now, our goal is to solve the equation T u = u. We begin by defining the spaces needed. They can be thought as the periodic version of the spaces used in [11]. The spaces. Let δ > 0, s 0. We set
X δ,s = X δ,s (T × R) = v ∈ L 2 (T × R): v 2X δ,s < ∞ , where
v 2X δ,s =
2
1 + τ − k3 |k|2s e 2δ|k| vˆ (k, τ ) dτ .
k∈Z R
The space X δ,s (T × R) is the natural periodic extension of the space X δ,s,1/2 (see [11]) for the nonperiodic case. It is well known that in the periodic case we really need b = 1/2 in order to prove multi-linear estimates in the spaces X s , which correspond to the case δ = 0 (see [22]). But when b = 1/2 we no longer have the continuous embedding X δ,s (T × R) → C ([0, T ], G δ,s (T)) used in [11]. In order to fix it, we shall introduce the following family of spaces. Definition 2.1. Let δ > 0, s 0. We set
Y δ,s = Y δ,s (T × R) = v ∈ L 2 (T × R): v 2Y δ,s < ∞ , where
v Y δ,s = v X δ,s +
k∈Z
|k|2s e 2δ|k|
R
vˆ (k, τ ) dτ
2 12 .
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Choosing δ = 0 gives the Y s spaces introduced by Colliander, Keel, Staffilani, Takaoka, and Tao for studying well-posedness of KdV, mKdV in [4] and gKdV in [5]. The spaces Y δ,s possess the following important property. Lemma 2.2. Y δ,s (T × R) → C ([− T , T ], G δ,s (T)) for any T > 0. Proof. Let T > 0 be given. Recalling that the norm of u ∈ C ([− T , T ], G δ,s (T)) is given by
|u |C T ,δ,s = sup u (·, t )G δ,s (T) , |t | T
we have
u (·, t )
G δ,s (T)
=
k∈Z
2 1 1 2 |k|2s e 2δ|k| e i τ t uˆ (k, τ ) dτ 2π
1 2π
|k|2s e 2δ|k|
R
k∈Z
uˆ (k, τ ) dτ
2 12
R
1 2π
u Y δ,s .
Thus, u (·, t )G δ,s (T) 21π u Y δ,s , ∀t ∈ R, which implies that
1 |u |C T ,δ,s = sup u (·, t )G δ,s (T) u Y δ,s . 2π
|t | T
This completes the proof of Lemma 2.2.
2
Next, computing the Y δ,s norm of T u, as in [9], we obtain the following result. Lemma 2.3. If s 1/2 then there is a constant c ψ > 0 such that
T u Y δ,s c ψ w Z δ,s + ϕ G δ,s (T) ,
for all u ∈ Y δ,s ,
(2.13)
where w = uk ∂x u and
. w Z δ,s =
|n|2s e 2δ|n|
n∈Z∗
+
R
|n|2s e 2δ|n|
n∈Z∗
12
ˆ (n, λ)|2 |w dλ 1 + |λ − n3 | R
2 12
ˆ (n, λ)| |w dλ 1 + |λ − n3 |
.
(2.14)
Proposition 2.4. For s 1/2 and v 1 , v 2 , . . . , v k+1 ∈ Y δ,s , we have
v 1 · v 2 · . . . · v k · ∂x ( v k+1 )
Z δ,s
v 1 Y δ,s v 2 Y δ,s . . . v k+1 Y δ,s .
(2.15)
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Proof. First, we observe that the operator A defined by
Au (n, λ) = e δ|n| uˆ (n, λ) satisfies the relation
u Y δ,s = Au Y s ,
for all u ∈ Y δ,s .
(2.16)
Also, for any v 1 · . . . · v k+1 ∈ Y δ,s it satisfies the relation
v 1 · . . . · v k · ∂x v k+1 Z δ,s = A ( v 1 · . . . · v k · ∂x v k+1 ) Z .
(2.17)
s
Therefore, to prove (2.15) it suffices to show the following inequality
A ( v 1 · . . . · v k · ∂x v k+1 )
Zs
( A v 1 ) · . . . · ( A v k ) · ∂x ( A v k+1 ) Z .
(2.18)
s
In fact, applying Proposition 1 from [5] we obtain
( A v 1 ) · . . . · ( A v k ) · ∂x ( A v k+1 ) C A v 1 Y · . . . · A v k+1 Y , s s Z
(2.19)
s
which combined with identity (2.16) gives the desired multi-linear estimate (2.15). Let us prove (2.18). Letting w = v 1 · . . . · v k · ∂x v k+1 we have
A ( v 1 · . . . · v k · ∂x v k+1 )
Zs
= A wZs 12 2 12 | A w (n, λ)|2 | A w (n, λ)| 2s 2s = |n| dλ + |n| dλ 1 + |λ − n3 | 1 + |λ − n3 | ∗ ∗ n∈Z
=
n∈Z
R
|n|
n∈Z∗
2s
R
ˆ (n, λ)|2 |e δ|n| w
1 2
dλ
1 + |λ − n3 |
+
n∈Z∗
R
|n|
2s
R
2 12
ˆ (n, λ)| |e δ|n| w dλ 1 + |λ − n3 |
.
Next, we analyze the first part of the last sum. The analysis of the second part is similar. By the ˆ (n, λ) ( v1 ∗ v2 ∗ · · · ∗ vk ∗ ∂ definition of w x v k+1 )(n, λ) as a convolution we have
|n|2s
n∈Z∗
R
=
|n|2s
n∈Z∗
12
ˆ (n, λ)|2 |e δ|n| w dλ 1 + |λ − n3 |
δ|n| e ... R
n1 ∈Z R 1
nk−1 ∈Z R
k −1
v1 (n − n1 , λ − λ1 ) . . .
nk ∈Z R k
2 12 3 −1 v k+1 (nk , λk ) dλ1 . . . dλk 1 + λ − n dλ × vk (nk−1 − nk , λk−1 − λk )nk |n|2s ... e δ|n−n1 | v1 (n − n1 , λ − λ1 ) . . . n∈Z∗
R n1 ∈Z R1
nk−1 ∈Z R
k −1
nk ∈Z R k
A.A. Himonas, G. Petronilho / J. Differential Equations 253 (2012) 3101–3112
2 × e δ|nk−1 −nk | vk (nk−1 − nk , λk−1 − λk )nk e δ|nk | v k+1 (nk , λk ) dλ1 . . . dλk
n∈Z∗
2s |n| ... R
n1 ∈Z R 1
nk−1 ∈Z R
3107
12
dλ 1 + |λ − n3 |
A v 1 (n − n1 , λ − λ1 ) . . .
nk ∈Z R k
k −1
2 12 3 −1
× A v k (nk−1 − nk , λk−1 − λk )nk A v k+1 (nk , λk ) dλ1 . . . dλk 1 + λ − n dλ
|n|2s
n∈Z∗
12
−1 2 A
v1 ∗ · · · ∗ A v k ∗ ∂x ( A v k+1 ) (n, λ) 1 + λ − n3 dλ
.
R
Letting w A = ( A v 1 ) · . . . · ( A v k ) · (∂x ( A v k+1 )), it follows from the above that
12
ˆ (n, λ)|2 |e δ|n| w dλ 1 + |λ − n3 |
|n|2s
n∈Z∗
R
12
| w A (n, λ)|2 dλ 1 + |λ − n3 |
|n|2s
n∈Z∗
R
.
Similarly, we obtain
|n|
n∈Z∗
2 12
ˆ (n, λ)| |e δ|n| w dλ 1 + |λ − n3 |
2s
R
|n|
2s
n∈Z∗
R
2 12
| w A (n, λ)| dλ 1 + |λ − n3 |
.
Adding the above two inequalities gives (2.18). This completes the proof of Proposition 2.4.
2
Combining Lemma 2.3 and Proposition 2.4 gives the following result. Proposition 2.5. If s 1/2 then there is a constant c ψ > 0 such that
T u Y δ,s c ψ u kY+δ,1s + c ψ ϕ G δ,s (T) ,
u ∈ Y δ,s ,
(2.20)
and
T u − T v Y δ,s c ψ
k
u kY− v Y δ,s δ,s
u − v Y δ,s ,
u , v ∈ Y δ,s .
(2.21)
=0
Proof. Estimate (2.20) follows from (2.13) and (2.15) applied with v 1 = · · · = v k+1 = u. To prove estimate (2.21) we observe that
T u − T v = i ψ(t )
3 e i (nx+n t )
n∈Z
3 e i (λ−n )t − 1
R
λ − n3
ˆ (n, λ) dλ, w
(2.22)
where w now is given by
w=
1 k+1
∂x u
k +1
−v
k +1
=
1 k+1
∂ x (u − v )
k
=0
u
k−
v
.
(2.23)
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Thus, applying estimate (2.13) with ϕ = 0 and multi-linear estimate (2.15) for each one of the (k + 1) terms of the sum (2.23) we obtain (2.21) completing the proof of Proposition 2.5. 2 The next proposition shows that our map T is in fact a contraction. Proposition 2.6. Let s 1/2. For initial data ϕ satisfying the smallness condition
3k − 1
ϕ G δ,s (T)
k +1 k
3k+1 c ψ (k + 1)
(2.24)
k +1 k
.
1/k
if we choose the ball B(0, r ) = {u ∈ Y δ,s : u Y δ,s r } with radius r = 1/[3c ψ (k + 1)1/k ] then T : B(0, r ) → B(0, r ) is a contraction. Proof. In fact, applying Proposition 2.5 we get
T u Y δ,s c ψ u kY+δ,1s + c ψ ϕ G δ,s (T) cψ
=
k+1
1 1
1
3c ψk (k + 1) k c ψ · 3k k +1 k
3k+1 c ψ (k + 1)
k +1 k
+ cψ
3k − 1 k +1
3k+1 c ψk (k + 1) 1
1 k
1
3c ψ (k + 1) k
k +1 k
= r.
Thus T maps B(0, r ) into B(0, r ). Also, it is a contraction, since
T u − T v Y δ,s c ψ cψ
k
u kY− v Y δ,s δ,s
=0 k
u − v Y δ,s
r
k−
r
u − v Y δ,s
=0
= c ψ r k (k + 1)u − v Y δ,s k 1 = cψ (k + 1)u − v Y δ,s 1 1 3c ψk (k + 1) k k =
1 3
u − v Y δ,s .
2
End of the proof of the existence. By Proposition 2.6 we see that for ϕ G δ,s (T) sufficiently small, the operator T is a contraction on a small ball centered at the origin in · Y δ,s . Hence the transformation T has a unique fixed point u in a · Y δ,s -neighborhood of 0. Since ψ(t ) = 1, |t | 1 it follows that u (x, t ) solves the gKdV initial value problem (1.1). Finally, thanks to Lemma 2.2, with T = 1, we have proved the existence of a solution to our Cauchy problem which belongs to the space C ([−1, 1], G δ,s (T)). 2
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2.2. Uniqueness Uniqueness of the solution in C ([−1, 1], G δ,s (T)) can be proved by the following standard argument. Lemma 2.7. Suppose that u , v ∈ C ([−1, 1], G δ,s (T)) are solutions to the gKdV Cauchy problem (1.1) with u (·, 0) = v (·, 0) in G δ,s (T) and s 1/2. Then u = v. Proof. Setting w = u − v, we see that w solves the Cauchy problem
∂t w + ∂x3 w +
1 k+1
∂ x ( f w ) = 0,
w (0) = 0,
(2.25)
where
f = uk + uk−1 v + · · · + uv k−1 v + v k .
(2.26)
Then using Eq. (2.25) we form the following identity for the L 2 -energy of w
1 d 2 dt
w (t )22
w ∂x3 w dx −
=−
L (T)
T
1 k+1
w ∂x ( f w ) dx.
(2.27)
T
Integrating by parts we obtain that T w ∂x3 w dx = 0. Using this and again integrating by parts, from Eq. (2.27) we get
d dt
w (t )22 L
=− (T)
1
(k + 1)
∂x f · w 2 dx, T
from which we deduce the inequality
d 2 1 w (t )22 ∂x f L ∞ w (t ) L 2 (T) . dt L (T) (k + 1)
(2.28)
Since u , v ∈ C ([−1, 1], G δ,s (T)) we have that u and v are continuous in t on the compact set [−1, 1] and are C ∞ in x on the compact torus. Thus the L ∞ norm of both u and v is finite. This implies that the L ∞ norm of ∂x f is finite, that is
∂x f (t ) ∞ c 0 < ∞. L
(2.29)
Therefore, from (2.28) and (2.29) we obtain the differential inequality
d w (t )22 c w (t )22 , dt L (T) L (T)
|t | 1,
(2.30)
c
where c = k+01 . Solving it gives
w (t )22
L (T)
2 ec w (0) L 2 (T) ,
|t | 1.
Since w (0) L 2 (T) = 0, from (2.31) we obtain that w (t ) = 0 or u = v.
(2.31)
2
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2.3. Continuous dependence of the initial data The next result implies the continuity of the data-to-solution map. Lemma 2.8. Suppose that u and v are solutions to (1.1) corresponding to initial data ϕ and θ respectively with the norms ϕ G δ,s (T) , θG δ,s (T) small and s 1/2. Then
|u − v |C 1 ,δ,s
3c ψ 2
ϕ − θG δ,s (T) .
Proof. We have
|u − v |C 1 ,δ,s = sup (u − v )(·, t )G δ,s (T) t ∈[−1,1]
u − v Y δ,s = T u − T v Y δ,s ,
(2.32)
taking u and v in a small · Y δ,s -neighborhood of 0 we have
1
u − v Y δ,s = T u − T v Y δ,s u − v Y δ,s + c ψ ϕ − θG δ,s (T) . 3
Thus,
3
u − v Y δ,s c ψ ϕ − θG δ,s (T) . 2
(2.33)
It follows from (2.32) and (2.33) that
3
|u − v |C T ,δ,s c ψ ϕ − θG δ,s (T) 2
and therefore the proof of continuous dependence is complete.
2
This completes the proof of Theorem 1.1. 3. Uniform radius of analyticity Next we show that the radius of analyticity of the solution u (·, t ) does not change as time progresses. If ϕ ∈ G δ,s (T), then it follows from the definition of the space G δ,s (T) that there exists a positive constant L such that the following inequality holds true
ϕˆ (n) Le −δ|n| ,
∀n ∈ Z.
(3.34)
. ∞
ˆ n) = ϕˆ (n), ∀n ∈ Z. Then, from (3.34) it follows that ψ(x) = n=0 ϕˆ (n)e inx ∈ C ω (T). Furthermore, ψ( Since ϕ ∈ L 2 (T), we conclude that ϕ ∈ C ω (T). We also have the following analytic continuation result. Lemma 3.1. If ϕ ∈ G δ,s (T), then ϕ has an analytic extension in a symmetric strip around the real axis and its width is equal to δ .
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Proof. Since by (3.34) we can write
ϕ (x) =
e inx ϕˆ (n),
n∈Z
we define
ϕ˜ (x + iy ) =
e in(x+iy ) ϕˆ (n) =
n∈Z
e inx e − yn ϕˆ (n),
n∈Z
which gives ϕ˜ (x + i0) = ϕ (x). Next we show that ϕ˜ is holomorphic in the strip | y | < δ . In fact, given y such that | y | < δ there exists L > 0 such that
− yn | y ||n| −δ|n| ϕ˜ (x + iy ) ϕˆ (n) e e Le n∈Z
=L
n∈Z
e
−(δ−| y |)|n|
< ∞.
n∈Z
Differentiating the series defining ϕ˜ (x + iy ), we can show similarly that the resulting series converges absolutely. Therefore, we can apply the Cauchy–Riemann operator ∂¯ term by term to see that ∂¯ ϕ˜ = 0, which shows that ϕ˜ is analytic in | y | < δ and 2π -periodic in x. This completes the proof of Lemma 3.1. 2 Following the lines of the proof of Lemma 3.1 one can easily show that the radius of analyticity of the solution u (·, t ) does not change as time progresses. 4. Regularity in time variable It follows from Theorem 1.1 that for the initial data ϕ ∈ G δ,s (T), the solution u (x, t ) is analytic in x ∈ T and is only continuous in the time t variable. But this regularity can be improved by using the results proved in [13, Section 4]. In fact, for initial data ϕ ∈ G δ,s (T) the results in [13] guarantee that the solution u (x, t ) is Gevrey of order 3 in the time variable t near zero, i.e., u (x, ·) ∈ G 3 . In [17] non-analytic in time solutions to the Cauchy problem (1.1) with real-valued analytic initial data are constructed when k is not a multiple of four. Next, we provide a few new examples where the initial data are in G δ,s (T). If for k ∈ {1, 3, 5, . . .} we define
ϕ (x) = −Re
∞
e
−2δn inx
e
,
n =1
and for k = 4r + 2, r = 0, 1, 2, . . . we define
ψ(x) = Re i
∞
e
−2δn inx
e
,
n =1
then it is easily seen that ϕ and ψ belong to the space G δ,s (T). Furthermore, following [17] one can prove that the solutions to the Cauchy problem (1.1) with initial data ϕ and ψ , respectively, are not G σ in t, for t near zero and for 1 σ < 3.
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For k = 4 , = 1, 2, . . . the following modification of an example given in [12]
θ(x) = i 1/2
∞
e −2δn e inx
n =1
provides initial data θ ∈ G δ,s (T). Now, following [13] one can prove that the solution to the Cauchy problem (1.1) with initial data θ is not G σ in t, for t near zero and for 1 σ < 3. Acknowledgments The authors would like to thank the referee(s) for constructive comments that led to improvements in this paper. References ´ H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV [1] J. Bona, Z. Grujic, equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (6) (2005) 783–797. [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (3) (1993) 209–262. [3] P. Byers, A. Himonas, Nonanalytic solutions of the KdV equation, Abstr. Appl. Anal. 2004 (6) (2004) 453–460. [4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003) 705–749. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal. 211 (2004) 173–218. [6] W. Craig, T. Kappeler, W.A. Strauss, Gain of regularity for equations of KdV type, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1994) 147–186. [7] A. De Bouard, N. Hayashi, K. Kato, Gevrey regularizing effect for the (generalized) Korteweg–de Vries equation and nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (6) (1995) 673–725. [8] C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal. 87 (2) (1989) 359–369. [9] J. Gorsky, A. Himonas, On analyticity in space variable of solutions to the KdV equation, in: Geometric Analysis of PDE and Several Complex Variables, in: Contemp. Math., vol. 368, American Mathematical Society, 2005, pp. 233–247. [10] J. Gorsky, A. Himonas, Construction of non-analytic solutions for the generalized KdV equation, J. Math. Anal. Appl. 303 (2) (2005) 522–529. ´ H. Kalisch, Local well-posedness of the generalized Korteweg–de Vries equation in spaces of analytic functions, [11] Z. Grujic, Differential Integral Equations 15 (11) (2002) 1325–1334. [12] H. Hannah, A. Himonas, G. Petronilho, Gevrey regularity in time for generalized KdV type equations, Contemp. Math. 400 (2006) 117–127. [13] H. Hannah, A. Himonas, G. Petronilho, Gevrey regularity of the periodic gKdV equation, J. Differential Equations 250 (2011) 2581–2600. [14] N. Hayashi, Analyticity of solutions of the Korteweg–de Vries equation, SIAM J. Math. Anal. 22 (6) (1991) 1738–1743. [15] N. Hayashi, K. Kato, Analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations, Comm. Math. Phys. 184 (1997) 273–300. [16] N. Hayashi, K. Kato, Global existence of small analytic solutions to Schröodinger equations with quadratic nonlinearity, Comm. Partial Differential Equations 22 (1997) 773–798. [17] A. Himonas, G. Petronilho, Real-valued non-analytic solutions for the generalized Korteweg–de Vries equation, Proc. Amer. Math. Soc. 139 (8) (2011) 2759–2766. [18] T. Kato, On the Cauchy problem for the (generalized) Korteweg–de Vries equation, Adv. Math. Suppl. Stud. 8 (1983) 93–128. [19] T. Kato, K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (6) (1986) 455–467. [20] K. Kato, T. Ogawa, Analyticity and smoothing effect for the Korteweg–de Vries equation with a single point singularity, Math. Ann. 316 (3) (2000) 577–608. [21] C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (4) (1993) 527–620. [22] C.E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (2) (1996) 573–603. [23] F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, 2009. [24] T. Tao, Nonlinear Dispersive Equations: Local and Global Solutions, American Mathematical Society, Providence, RI, 2006. [25] S. Tarama, Analyticity of solutions of the Korteweg–de Vries equation, J. Math. Kyoto Univ. 44 (1) (2004) 1–32. [26] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (3) (1977) 321–337.
Contents lists available at SciVerse ScienceDirect
Journal of Differential Equations www.elsevier.com/locate/jde
Analytic well-posedness of periodic gKdV A. Alexandrou Himonas a,∗ , Gerson Petronilho b,1 a b
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 December 2011 Revised 23 August 2012 Available online 3 September 2012
In the periodic case, it is proved that the Cauchy problem for the generalized Korteweg–de Vries equation (gKdV) is locally well-posed in a class of analytic functions that can be extended holomorphically in a symmetric strip of the complex plane around the x-axis. Thus, the uniform analyticity radius of the solution does not change as time progresses. Also, information about the regularity of the solution in the time variable is provided. © 2012 Elsevier Inc. All rights reserved.
MSC: primary 35Q53 secondary 35B65 Keywords: Analytic spaces Generalized Korteweg–de Vries equation Initial value problem Multilinear estimates Uniform analyticity radius
1. Introduction For k = 1, 2, 3, . . . we consider the initial value problem for the generalized Korteweg–de Vries equation (gKdV)
∂t u + ∂x3 u + uk ∂x u = 0, u (x, 0) = ϕ (x),
x ∈ T, t ∈ R,
(1.1)
with initial data ϕ (x) belonging in a class of 2π -periodic analytic functions that can be extended . holomorphically in a symmetric strip S δ = {x + iy: | y | < δ}, δ > 0, of the complex plane around the x-axis. Our main result is that the gKdV equation is locally well-posed in these spaces. A consequence
* 1
Corresponding author. Fax: +1 574 631 6579. E-mail addresses: [email protected] (A.A. Himonas), [email protected] (G. Petronilho). Partially supported by CNPq and Fapesp.
0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2012.08.024
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of this result is that the uniform analyticity radius of the solution u (x, t ) during its local lifespan is the same as that of the initial data u (x, 0). The motivation for this work comes from the results in [13], where well-posedness of gKdV was studied in Gevrey and analytic spaces, but no precise information about the uniform analyticity radius of the solution was provided. Concerning well-posedness for the nonperiodic gKdV equation, Kato and Masuda [19] showed that if the initial data of the gKdV equation has an analytic continuation which is bounded in a strip containing the real axis, then the solution has the same property for all time, although the width of the strip might decrease with time. Results of this type have been also obtained by Hayashi [14]. Using the analytic spaces G δ,s introduced by Foias and Temam [8] which are defined by the norm
ϕ 2G δ,s =
2s
1 + |ξ |
2
e 2δ(1+|ξ |) ϕˆ (ξ ) dξ < ∞,
(1.2)
R
Grujic´ and Kalisch [11] showed that for given initial data that are analytic in a symmetric strip { z = x + iy: | y | < δ} in the complex plane of width 2δ , there exists a time T > 0 such that the corresponding gKdV solution is analytic in the same strip during the time period [0, T ]. In other words, in the nonperiodic case, the uniform radius of spatial analyticity does not shrink as time progresses. Further results on the uniform radius of spatial analyticity have been established by Bona, Grujic´ and Kalisch [1]. As we have mentioned earlier, in the periodic case, the Cauchy problem for gKdV was studied in [13] using Bourgain type analytic and Gevrey spaces. However, the uniform radius of spatial analyticity obtained could decrease immediately after the initial time. In this paper, using a periodic version of the analytic spaces used in [11] and a modified version of the Bourgain spaces used by Colliander, Keel, Staffilani, Takaoka, and Tao [4], we show that the uniform radius of analyticity does not shrink as time progresses. This is a consequence of the following well-posedness result, which for simplicity we state for sufficiently small initial data so that the corresponding lifespan is equal to one. Theorem 1.1. Let s 1/2 and δ > 0. For small initial data in the space
G δ,s (T) =
f ∈ L 2 (T): f 2G δ,s (T) =
2 |k|2s e 2δ|k| ˆf (k) < ∞ ,
(1.3)
k∈Z
the Cauchy problem for gKdV (1.1) is locally well-posed in the space C ([−1, 1], G δ,s (T)). Spacial analyticity of the periodic KdV equation was proved first by Trubowitz [26]. Construction of non-analytic solutions in time with analytic initial data for KdV has been obtained in [3] and for gKdV in [10] and [17]. When the initial data are analytic then the time-regularity of the solutions is of class Gevrey three. Furthermore, this result is sharp in the sense that there exist initial data that are analytic and 2π -periodic but the corresponding solution to (1.1) does belong to G r (R) for 1 r < 3 [13]. New such examples are presented here using the spaces G δ,s (T) defined by (1.3). For additional results concerning well-posedness and regularity properties of gKdV we refer the reader to Bourgain [2], De Bouard, Hayashi and Kato [7], Kato [18], Kato and Masuda [19], Kato and Ogawa [20], Kenig, Ponce and Vega [21,22] and the references therein. Finally, we would like to point out that there are many works treating the analytic smoothing effect, that is to say, the phenomenon that solutions become analytic with respect to the spacial variable even if the initial data is not analytic but decays rapidly enough. For instance, Tarama [25] demonstrated this phenomenon for the KdV on the line. More precisely, he showed that if the initial data ϕ ∈ L 2 (R) satisfy the decay conditions
∞ −∞
1 + |x| ϕ (x) dx < ∞ and
∞ 0
e δ|x|
1/2
ϕ (x)2 dx < ∞,
(1.4)
A.A. Himonas, G. Petronilho / J. Differential Equations 253 (2012) 3101–3112
3103
where δ is some positive number, then the corresponding KdV solution u (x, t ) becomes analytic with respect to the variable x for any t > 0. In the periodic case the decay conditions (1.4) do not give any additional information for the initial data ϕ beyond that it belongs to L 2 [0, 2π ]. Taking the initial data such that ϕˆ (k) = 1/(1 + k2 ) we have ϕ ∈ L 2 [0, 2π ]. The solution of the linear part of KdV 3
that corresponds to this initial data has Fourier transform with respect to x equal to e ik t /(1 + k2 ). Therefore, it is not analytic for any t > 0. Adding the nonlinearity u ∂x u to this equation, which gives KdV, should not change this conclusion. For more information about the analytic smoothing effect of dispersive equations we refer the reader to [7,6,15,16,20]. For other smoothing effects and the general theory of KdV and gKdV we refer the reader to [23] and [24]. The paper is organized as follows. Section 2 is dedicated to the proof of Theorem 1.1. In Section 2.1 we introduce the spaces needed and prove the existence of the solution. Then, in Section 2.2 we prove the uniqueness, and in Section 2.3 we prove continuous dependence on initial data. In Section 3 we demonstrate that the uniform radius of analyticity does not change as time progresses. Finally, in the last section we discuss regularity in the time variable. 2. Proof of Theorem 1.1 2.1. Existence Taking the Fourier transform with respect to x in (1.1), solving the resulting differential equation in t and using the inverse Fourier transform reduces the Cauchy problem (1.1) to the following integral equation
t u (x, t ) = W (t )ϕ (x) −
W (t − τ ) w (x, τ ) dτ ,
(2.5)
0
where W (t ) = e −t ∂x and w = uk ∂x u. Next we localize in the time variable by using a cut-off function ψ(t ) ∈ C 0∞ (−2, 2) with 0 ψ 1 and such that ψ(t ) ≡ 1 for |t | < 1. Multiplying (2.5) by ψ , we have 3
t ψ(t )u (x, t ) = ψ(t ) W (t )ϕ (x) − ψ(t )
W (t − τ ) w (x, τ ) dτ ,
(2.6)
0
which can be written as
ψ(t )u (x, t ) = ψ(t )
3 e i (nx+n t ) ϕˆ (n)
n∈Z
+ i ψ(t )
e
i (nx+n3 t )
n∈Z
∞
−∞
3 e i (λ−n )t − 1
λ − n3
ˆ (n, λ) dλ. w
(2.7)
Here, for simplicity, we shall restrict our attention to mean zero data
ϕˆ (0) =
1 2π
ϕ (x) dx = 0. T
(2.8)
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Next, we define the right-hand side of the integral equation (2.6) by T u and decompose it as follows
T u (x, t ) = ψ(t )
3 e i (nx+n t ) ϕˆ (n)
(2.9)
n∈Z∗
+i
∞ j j i t j =1
+ i ψ(t )
j!
ψ(t )
− i ψ(t )
3 e i (nx+n t )
n∈Z∗
e inx
n∈Z∗
R
j −1 ˆ (n, λ) dλ ψ λ − n3 λ − n3 w
(2.10)
R
(1 − ψ)(λ − n3 ) i λt ˆ (n, λ) dλ e w λ − n3
3 e i (nx+n t )
n∈Z∗
R
(2.11)
(1 − ψ)(λ − n3 ) ˆ (n, λ) dλ, w λ − n3
(2.12)
where k∂ u ( u
ˆ (n, λ) = u w x ˆ ∗ uˆ ∗ · · · ∗ u ˆ ∗∂x u )(n, λ) k
and
Z∗ = Z\{0}. Now, our goal is to solve the equation T u = u. We begin by defining the spaces needed. They can be thought as the periodic version of the spaces used in [11]. The spaces. Let δ > 0, s 0. We set
X δ,s = X δ,s (T × R) = v ∈ L 2 (T × R): v 2X δ,s < ∞ , where
v 2X δ,s =
2
1 + τ − k3 |k|2s e 2δ|k| vˆ (k, τ ) dτ .
k∈Z R
The space X δ,s (T × R) is the natural periodic extension of the space X δ,s,1/2 (see [11]) for the nonperiodic case. It is well known that in the periodic case we really need b = 1/2 in order to prove multi-linear estimates in the spaces X s , which correspond to the case δ = 0 (see [22]). But when b = 1/2 we no longer have the continuous embedding X δ,s (T × R) → C ([0, T ], G δ,s (T)) used in [11]. In order to fix it, we shall introduce the following family of spaces. Definition 2.1. Let δ > 0, s 0. We set
Y δ,s = Y δ,s (T × R) = v ∈ L 2 (T × R): v 2Y δ,s < ∞ , where
v Y δ,s = v X δ,s +
k∈Z
|k|2s e 2δ|k|
R
vˆ (k, τ ) dτ
2 12 .
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3105
Choosing δ = 0 gives the Y s spaces introduced by Colliander, Keel, Staffilani, Takaoka, and Tao for studying well-posedness of KdV, mKdV in [4] and gKdV in [5]. The spaces Y δ,s possess the following important property. Lemma 2.2. Y δ,s (T × R) → C ([− T , T ], G δ,s (T)) for any T > 0. Proof. Let T > 0 be given. Recalling that the norm of u ∈ C ([− T , T ], G δ,s (T)) is given by
|u |C T ,δ,s = sup u (·, t )G δ,s (T) , |t | T
we have
u (·, t )
G δ,s (T)
=
k∈Z
2 1 1 2 |k|2s e 2δ|k| e i τ t uˆ (k, τ ) dτ 2π
1 2π
|k|2s e 2δ|k|
R
k∈Z
uˆ (k, τ ) dτ
2 12
R
1 2π
u Y δ,s .
Thus, u (·, t )G δ,s (T) 21π u Y δ,s , ∀t ∈ R, which implies that
1 |u |C T ,δ,s = sup u (·, t )G δ,s (T) u Y δ,s . 2π
|t | T
This completes the proof of Lemma 2.2.
2
Next, computing the Y δ,s norm of T u, as in [9], we obtain the following result. Lemma 2.3. If s 1/2 then there is a constant c ψ > 0 such that
T u Y δ,s c ψ w Z δ,s + ϕ G δ,s (T) ,
for all u ∈ Y δ,s ,
(2.13)
where w = uk ∂x u and
. w Z δ,s =
|n|2s e 2δ|n|
n∈Z∗
+
R
|n|2s e 2δ|n|
n∈Z∗
12
ˆ (n, λ)|2 |w dλ 1 + |λ − n3 | R
2 12
ˆ (n, λ)| |w dλ 1 + |λ − n3 |
.
(2.14)
Proposition 2.4. For s 1/2 and v 1 , v 2 , . . . , v k+1 ∈ Y δ,s , we have
v 1 · v 2 · . . . · v k · ∂x ( v k+1 )
Z δ,s
v 1 Y δ,s v 2 Y δ,s . . . v k+1 Y δ,s .
(2.15)
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Proof. First, we observe that the operator A defined by
Au (n, λ) = e δ|n| uˆ (n, λ) satisfies the relation
u Y δ,s = Au Y s ,
for all u ∈ Y δ,s .
(2.16)
Also, for any v 1 · . . . · v k+1 ∈ Y δ,s it satisfies the relation
v 1 · . . . · v k · ∂x v k+1 Z δ,s = A ( v 1 · . . . · v k · ∂x v k+1 ) Z .
(2.17)
s
Therefore, to prove (2.15) it suffices to show the following inequality
A ( v 1 · . . . · v k · ∂x v k+1 )
Zs
( A v 1 ) · . . . · ( A v k ) · ∂x ( A v k+1 ) Z .
(2.18)
s
In fact, applying Proposition 1 from [5] we obtain
( A v 1 ) · . . . · ( A v k ) · ∂x ( A v k+1 ) C A v 1 Y · . . . · A v k+1 Y , s s Z
(2.19)
s
which combined with identity (2.16) gives the desired multi-linear estimate (2.15). Let us prove (2.18). Letting w = v 1 · . . . · v k · ∂x v k+1 we have
A ( v 1 · . . . · v k · ∂x v k+1 )
Zs
= A wZs 12 2 12 | A w (n, λ)|2 | A w (n, λ)| 2s 2s = |n| dλ + |n| dλ 1 + |λ − n3 | 1 + |λ − n3 | ∗ ∗ n∈Z
=
n∈Z
R
|n|
n∈Z∗
2s
R
ˆ (n, λ)|2 |e δ|n| w
1 2
dλ
1 + |λ − n3 |
+
n∈Z∗
R
|n|
2s
R
2 12
ˆ (n, λ)| |e δ|n| w dλ 1 + |λ − n3 |
.
Next, we analyze the first part of the last sum. The analysis of the second part is similar. By the ˆ (n, λ) ( v1 ∗ v2 ∗ · · · ∗ vk ∗ ∂ definition of w x v k+1 )(n, λ) as a convolution we have
|n|2s
n∈Z∗
R
=
|n|2s
n∈Z∗
12
ˆ (n, λ)|2 |e δ|n| w dλ 1 + |λ − n3 |
δ|n| e ... R
n1 ∈Z R 1
nk−1 ∈Z R
k −1
v1 (n − n1 , λ − λ1 ) . . .
nk ∈Z R k
2 12 3 −1 v k+1 (nk , λk ) dλ1 . . . dλk 1 + λ − n dλ × vk (nk−1 − nk , λk−1 − λk )nk |n|2s ... e δ|n−n1 | v1 (n − n1 , λ − λ1 ) . . . n∈Z∗
R n1 ∈Z R1
nk−1 ∈Z R
k −1
nk ∈Z R k
A.A. Himonas, G. Petronilho / J. Differential Equations 253 (2012) 3101–3112
2 × e δ|nk−1 −nk | vk (nk−1 − nk , λk−1 − λk )nk e δ|nk | v k+1 (nk , λk ) dλ1 . . . dλk
n∈Z∗
2s |n| ... R
n1 ∈Z R 1
nk−1 ∈Z R
3107
12
dλ 1 + |λ − n3 |
A v 1 (n − n1 , λ − λ1 ) . . .
nk ∈Z R k
k −1
2 12 3 −1
× A v k (nk−1 − nk , λk−1 − λk )nk A v k+1 (nk , λk ) dλ1 . . . dλk 1 + λ − n dλ
|n|2s
n∈Z∗
12
−1 2 A
v1 ∗ · · · ∗ A v k ∗ ∂x ( A v k+1 ) (n, λ) 1 + λ − n3 dλ
.
R
Letting w A = ( A v 1 ) · . . . · ( A v k ) · (∂x ( A v k+1 )), it follows from the above that
12
ˆ (n, λ)|2 |e δ|n| w dλ 1 + |λ − n3 |
|n|2s
n∈Z∗
R
12
| w A (n, λ)|2 dλ 1 + |λ − n3 |
|n|2s
n∈Z∗
R
.
Similarly, we obtain
|n|
n∈Z∗
2 12
ˆ (n, λ)| |e δ|n| w dλ 1 + |λ − n3 |
2s
R
|n|
2s
n∈Z∗
R
2 12
| w A (n, λ)| dλ 1 + |λ − n3 |
.
Adding the above two inequalities gives (2.18). This completes the proof of Proposition 2.4.
2
Combining Lemma 2.3 and Proposition 2.4 gives the following result. Proposition 2.5. If s 1/2 then there is a constant c ψ > 0 such that
T u Y δ,s c ψ u kY+δ,1s + c ψ ϕ G δ,s (T) ,
u ∈ Y δ,s ,
(2.20)
and
T u − T v Y δ,s c ψ
k
u kY− v Y δ,s δ,s
u − v Y δ,s ,
u , v ∈ Y δ,s .
(2.21)
=0
Proof. Estimate (2.20) follows from (2.13) and (2.15) applied with v 1 = · · · = v k+1 = u. To prove estimate (2.21) we observe that
T u − T v = i ψ(t )
3 e i (nx+n t )
n∈Z
3 e i (λ−n )t − 1
R
λ − n3
ˆ (n, λ) dλ, w
(2.22)
where w now is given by
w=
1 k+1
∂x u
k +1
−v
k +1
=
1 k+1
∂ x (u − v )
k
=0
u
k−
v
.
(2.23)
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Thus, applying estimate (2.13) with ϕ = 0 and multi-linear estimate (2.15) for each one of the (k + 1) terms of the sum (2.23) we obtain (2.21) completing the proof of Proposition 2.5. 2 The next proposition shows that our map T is in fact a contraction. Proposition 2.6. Let s 1/2. For initial data ϕ satisfying the smallness condition
3k − 1
ϕ G δ,s (T)
k +1 k
3k+1 c ψ (k + 1)
(2.24)
k +1 k
.
1/k
if we choose the ball B(0, r ) = {u ∈ Y δ,s : u Y δ,s r } with radius r = 1/[3c ψ (k + 1)1/k ] then T : B(0, r ) → B(0, r ) is a contraction. Proof. In fact, applying Proposition 2.5 we get
T u Y δ,s c ψ u kY+δ,1s + c ψ ϕ G δ,s (T) cψ
=
k+1
1 1
1
3c ψk (k + 1) k c ψ · 3k k +1 k
3k+1 c ψ (k + 1)
k +1 k
+ cψ
3k − 1 k +1
3k+1 c ψk (k + 1) 1
1 k
1
3c ψ (k + 1) k
k +1 k
= r.
Thus T maps B(0, r ) into B(0, r ). Also, it is a contraction, since
T u − T v Y δ,s c ψ cψ
k
u kY− v Y δ,s δ,s
=0 k
u − v Y δ,s
r
k−
r
u − v Y δ,s
=0
= c ψ r k (k + 1)u − v Y δ,s k 1 = cψ (k + 1)u − v Y δ,s 1 1 3c ψk (k + 1) k k =
1 3
u − v Y δ,s .
2
End of the proof of the existence. By Proposition 2.6 we see that for ϕ G δ,s (T) sufficiently small, the operator T is a contraction on a small ball centered at the origin in · Y δ,s . Hence the transformation T has a unique fixed point u in a · Y δ,s -neighborhood of 0. Since ψ(t ) = 1, |t | 1 it follows that u (x, t ) solves the gKdV initial value problem (1.1). Finally, thanks to Lemma 2.2, with T = 1, we have proved the existence of a solution to our Cauchy problem which belongs to the space C ([−1, 1], G δ,s (T)). 2
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2.2. Uniqueness Uniqueness of the solution in C ([−1, 1], G δ,s (T)) can be proved by the following standard argument. Lemma 2.7. Suppose that u , v ∈ C ([−1, 1], G δ,s (T)) are solutions to the gKdV Cauchy problem (1.1) with u (·, 0) = v (·, 0) in G δ,s (T) and s 1/2. Then u = v. Proof. Setting w = u − v, we see that w solves the Cauchy problem
∂t w + ∂x3 w +
1 k+1
∂ x ( f w ) = 0,
w (0) = 0,
(2.25)
where
f = uk + uk−1 v + · · · + uv k−1 v + v k .
(2.26)
Then using Eq. (2.25) we form the following identity for the L 2 -energy of w
1 d 2 dt
w (t )22
w ∂x3 w dx −
=−
L (T)
T
1 k+1
w ∂x ( f w ) dx.
(2.27)
T
Integrating by parts we obtain that T w ∂x3 w dx = 0. Using this and again integrating by parts, from Eq. (2.27) we get
d dt
w (t )22 L
=− (T)
1
(k + 1)
∂x f · w 2 dx, T
from which we deduce the inequality
d 2 1 w (t )22 ∂x f L ∞ w (t ) L 2 (T) . dt L (T) (k + 1)
(2.28)
Since u , v ∈ C ([−1, 1], G δ,s (T)) we have that u and v are continuous in t on the compact set [−1, 1] and are C ∞ in x on the compact torus. Thus the L ∞ norm of both u and v is finite. This implies that the L ∞ norm of ∂x f is finite, that is
∂x f (t ) ∞ c 0 < ∞. L
(2.29)
Therefore, from (2.28) and (2.29) we obtain the differential inequality
d w (t )22 c w (t )22 , dt L (T) L (T)
|t | 1,
(2.30)
c
where c = k+01 . Solving it gives
w (t )22
L (T)
2 ec w (0) L 2 (T) ,
|t | 1.
Since w (0) L 2 (T) = 0, from (2.31) we obtain that w (t ) = 0 or u = v.
(2.31)
2
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2.3. Continuous dependence of the initial data The next result implies the continuity of the data-to-solution map. Lemma 2.8. Suppose that u and v are solutions to (1.1) corresponding to initial data ϕ and θ respectively with the norms ϕ G δ,s (T) , θG δ,s (T) small and s 1/2. Then
|u − v |C 1 ,δ,s
3c ψ 2
ϕ − θG δ,s (T) .
Proof. We have
|u − v |C 1 ,δ,s = sup (u − v )(·, t )G δ,s (T) t ∈[−1,1]
u − v Y δ,s = T u − T v Y δ,s ,
(2.32)
taking u and v in a small · Y δ,s -neighborhood of 0 we have
1
u − v Y δ,s = T u − T v Y δ,s u − v Y δ,s + c ψ ϕ − θG δ,s (T) . 3
Thus,
3
u − v Y δ,s c ψ ϕ − θG δ,s (T) . 2
(2.33)
It follows from (2.32) and (2.33) that
3
|u − v |C T ,δ,s c ψ ϕ − θG δ,s (T) 2
and therefore the proof of continuous dependence is complete.
2
This completes the proof of Theorem 1.1. 3. Uniform radius of analyticity Next we show that the radius of analyticity of the solution u (·, t ) does not change as time progresses. If ϕ ∈ G δ,s (T), then it follows from the definition of the space G δ,s (T) that there exists a positive constant L such that the following inequality holds true
ϕˆ (n) Le −δ|n| ,
∀n ∈ Z.
(3.34)
. ∞
ˆ n) = ϕˆ (n), ∀n ∈ Z. Then, from (3.34) it follows that ψ(x) = n=0 ϕˆ (n)e inx ∈ C ω (T). Furthermore, ψ( Since ϕ ∈ L 2 (T), we conclude that ϕ ∈ C ω (T). We also have the following analytic continuation result. Lemma 3.1. If ϕ ∈ G δ,s (T), then ϕ has an analytic extension in a symmetric strip around the real axis and its width is equal to δ .
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Proof. Since by (3.34) we can write
ϕ (x) =
e inx ϕˆ (n),
n∈Z
we define
ϕ˜ (x + iy ) =
e in(x+iy ) ϕˆ (n) =
n∈Z
e inx e − yn ϕˆ (n),
n∈Z
which gives ϕ˜ (x + i0) = ϕ (x). Next we show that ϕ˜ is holomorphic in the strip | y | < δ . In fact, given y such that | y | < δ there exists L > 0 such that
− yn | y ||n| −δ|n| ϕ˜ (x + iy ) ϕˆ (n) e e Le n∈Z
=L
n∈Z
e
−(δ−| y |)|n|
< ∞.
n∈Z
Differentiating the series defining ϕ˜ (x + iy ), we can show similarly that the resulting series converges absolutely. Therefore, we can apply the Cauchy–Riemann operator ∂¯ term by term to see that ∂¯ ϕ˜ = 0, which shows that ϕ˜ is analytic in | y | < δ and 2π -periodic in x. This completes the proof of Lemma 3.1. 2 Following the lines of the proof of Lemma 3.1 one can easily show that the radius of analyticity of the solution u (·, t ) does not change as time progresses. 4. Regularity in time variable It follows from Theorem 1.1 that for the initial data ϕ ∈ G δ,s (T), the solution u (x, t ) is analytic in x ∈ T and is only continuous in the time t variable. But this regularity can be improved by using the results proved in [13, Section 4]. In fact, for initial data ϕ ∈ G δ,s (T) the results in [13] guarantee that the solution u (x, t ) is Gevrey of order 3 in the time variable t near zero, i.e., u (x, ·) ∈ G 3 . In [17] non-analytic in time solutions to the Cauchy problem (1.1) with real-valued analytic initial data are constructed when k is not a multiple of four. Next, we provide a few new examples where the initial data are in G δ,s (T). If for k ∈ {1, 3, 5, . . .} we define
ϕ (x) = −Re
∞
e
−2δn inx
e
,
n =1
and for k = 4r + 2, r = 0, 1, 2, . . . we define
ψ(x) = Re i
∞
e
−2δn inx
e
,
n =1
then it is easily seen that ϕ and ψ belong to the space G δ,s (T). Furthermore, following [17] one can prove that the solutions to the Cauchy problem (1.1) with initial data ϕ and ψ , respectively, are not G σ in t, for t near zero and for 1 σ < 3.
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For k = 4 , = 1, 2, . . . the following modification of an example given in [12]
θ(x) = i 1/2
∞
e −2δn e inx
n =1
provides initial data θ ∈ G δ,s (T). Now, following [13] one can prove that the solution to the Cauchy problem (1.1) with initial data θ is not G σ in t, for t near zero and for 1 σ < 3. Acknowledgments The authors would like to thank the referee(s) for constructive comments that led to improvements in this paper. References ´ H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV [1] J. Bona, Z. Grujic, equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (6) (2005) 783–797. [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (3) (1993) 209–262. [3] P. Byers, A. Himonas, Nonanalytic solutions of the KdV equation, Abstr. Appl. Anal. 2004 (6) (2004) 453–460. [4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003) 705–749. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal. 211 (2004) 173–218. [6] W. Craig, T. Kappeler, W.A. Strauss, Gain of regularity for equations of KdV type, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1994) 147–186. [7] A. De Bouard, N. Hayashi, K. Kato, Gevrey regularizing effect for the (generalized) Korteweg–de Vries equation and nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (6) (1995) 673–725. [8] C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal. 87 (2) (1989) 359–369. [9] J. Gorsky, A. Himonas, On analyticity in space variable of solutions to the KdV equation, in: Geometric Analysis of PDE and Several Complex Variables, in: Contemp. Math., vol. 368, American Mathematical Society, 2005, pp. 233–247. [10] J. Gorsky, A. Himonas, Construction of non-analytic solutions for the generalized KdV equation, J. Math. Anal. Appl. 303 (2) (2005) 522–529. ´ H. Kalisch, Local well-posedness of the generalized Korteweg–de Vries equation in spaces of analytic functions, [11] Z. Grujic, Differential Integral Equations 15 (11) (2002) 1325–1334. [12] H. Hannah, A. Himonas, G. Petronilho, Gevrey regularity in time for generalized KdV type equations, Contemp. Math. 400 (2006) 117–127. [13] H. Hannah, A. Himonas, G. Petronilho, Gevrey regularity of the periodic gKdV equation, J. Differential Equations 250 (2011) 2581–2600. [14] N. Hayashi, Analyticity of solutions of the Korteweg–de Vries equation, SIAM J. Math. Anal. 22 (6) (1991) 1738–1743. [15] N. Hayashi, K. Kato, Analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations, Comm. Math. Phys. 184 (1997) 273–300. [16] N. Hayashi, K. Kato, Global existence of small analytic solutions to Schröodinger equations with quadratic nonlinearity, Comm. Partial Differential Equations 22 (1997) 773–798. [17] A. Himonas, G. Petronilho, Real-valued non-analytic solutions for the generalized Korteweg–de Vries equation, Proc. Amer. Math. Soc. 139 (8) (2011) 2759–2766. [18] T. Kato, On the Cauchy problem for the (generalized) Korteweg–de Vries equation, Adv. Math. Suppl. Stud. 8 (1983) 93–128. [19] T. Kato, K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (6) (1986) 455–467. [20] K. Kato, T. Ogawa, Analyticity and smoothing effect for the Korteweg–de Vries equation with a single point singularity, Math. Ann. 316 (3) (2000) 577–608. [21] C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (4) (1993) 527–620. [22] C.E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (2) (1996) 573–603. [23] F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, 2009. [24] T. Tao, Nonlinear Dispersive Equations: Local and Global Solutions, American Mathematical Society, Providence, RI, 2006. [25] S. Tarama, Analyticity of solutions of the Korteweg–de Vries equation, J. Math. Kyoto Univ. 44 (1) (2004) 1–32. [26] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (3) (1977) 321–337.