Existence and Polynomial Growth of
periodic solutions to KdV-type Equations
Yung-fu Fang
Abstract. We establish local and global existence of periodic solutions for KdV type equations, employing Fourier series and a fixed point argument. We also investigate the polynomial growth of the solutions.
1. Introduction
In this paper, we study the existence and the polynomial bound of periodic solutions for the nonlinear dispersive equation of the Korteweg-de Vries type:
(1)
½
ut + ∂xαu + uk∂xu = 0, (t, x) ∈ R+× T; u(0, x) = φ(x),
where φ is a real function, α a real number, k a positive integer and ∂α x the
fractional derivative defined by, via Fourier transform,
(2) ∂cα
x = i|ξ|αsgnξ.
The function u considered here is a real-valued and space-periodic function. The method used here is the fixed point argument applied to the correspond-ing integral equation
(3) u(t) = W (t)φ −
Z t
0
W (t − τ )w(τ )dτ, 1991 Mathematics Subject Classification. 35D05.
Key words and phrases. global existence, fixed point argument, periodic solution, KdV
type equation, Polynomial growth..
where W (t) = e−t∂xα and w = uk∂xu, see [B2] and [FG].
The original KdV equation,
(4a) ∂tu + ∂x3u + u∂xu = 0,
was derived in 1895 by Korteweg and de Vries as an approximate model of shallow water waves, see [KdV]. It also has been derived in plasma physics and in the studies of anharmonic lattices, see [MGKr]. Some generalizations of KdV equation has been used to describe certain physical problems, e.g. KdV-type equations in certain crystalline lattices, see [ABFS]. In 1975, P. Lax [L] constructed a large class of special solutions of the KdV equation which are periodic in space and almost periodic in time. In 1993, Bourgain [B2] proved existence of periodic solutions for generalized KdV equations, (4b) ∂tu + ∂x3u + uk∂xu = 0.
In 1995, Bourgain [B3] extended the result of local solutions to more general KdV equation,
(4c) ∂tu + ∂x2j+1u + F (u, lower order terms) = 0.
On the other hand, some fifth order (even 7th order) KdV-type equations, (4d) ∂tu + up∂xu + ∂x3u + ∂x5u = 0,
also has been considered, see [K]. In 1996, Bourgain [B4] obtained a polyno-mial bound of higher Sobolev norm of solutions for generalized KdV equa-tions. In 1997, Staffilani [S] improved the existence result and the polynomial bound of solution for equation (4b). In 2004, Colliander eta [CKSTT] gave multilinear estimates for for periodic case and their applications.
It is well known that the KdV equation and some KdV-type equations possess solitary waves and infinitely many conservation laws, see [L] and [MGKr]. For the equation (1), there are three quantities are conserved, namely, (5) R Tu(t)dx, R Tu2(t)dx, and R T 1 2(∂ α−1 2 x u)2(t)dx − R T 1 (k+1)(k+2)uk+2(t)dx.
In the nonperiodic case there have been some good results on questions of existence and regularity, see [KPV] and [BKPSV].
The outline of this paper is that we first show the local existence result for the initial value problem (1) with k = 1. The essence of the proof is an a priori estimate inspired by work of J. Bourgain, see [B2] and [B3]. It can be understood as a multiplier estimate on the set of R×Z. However the proof of the estimate presented here is different from those of [B2]. It essentially relies on an idea of Zygmund [Zy]. Once the local existence is proved, we invoke a conservation law to get global existence. Next we discuss the existence results for the initial value problem (1), (hereafter we write IVP), with higher order nonlinearity k ≥ 2. In section 4, we will give a straightforward proof of the a priori estimate. Finally we will discuss the polynomial bound for solution of IVP (1). The main results of this paper are the following theorems. Theorem A. Let α ≥ 3. If the initial data of (1) is in L2 for k = 1 and in Hα−12 (and small) for k ≥ 2, then the initial value problem of (1) is globally
well-posed.
Theorem B. Let α ≥ 3 and 3 2 ≤
α − 1
2 < s. If the initial data is in H
s and small, then the global solution u satisfies
(6) ku(t)kHs ≤ C|t|2s.
2. Existence Results
Throughout this paper we call
(7) A(ξ) = |ξ|αsgnξ and S = |τ − A(ξ)| + 1;
denote by eg(t, ξ) = 1 2π Z 2π 0 e−ixξg(t, x) dx and by bg(τ, ξ) = Z R e−itτeg(t, ξ) dt
the Fourier coefficients and the Fourier transforms with respect to the space variable and to both the space-time variables, respectively. First we state the local existence result for IVP (1) with k = 1:
Theorem 1. If α ≥ 3 and the initial data φ ∈ L2 (Hs, s ≥ 0), then the IVP of (1) is locally well-posed.
To prove the above theorem, we use a fixed point argument and the fol-lowing a priori estimate whose proof will be given later.
Theorem 2. If α ≥ 2, then we have the following estimates
(8a) kf kL4(R×T) ≤ C
° °S1+α
4α f kb L2(R×Z)
and its dual
(8b) k bf /S1+α4α kL2(R×Z) ≤ Ckf k
L43(R×T).
Before proving Theorem 1, we consider the corresponding linear problem: (9)
½
ut + ∂xαu + w = 0, (t, x) ∈ R+× S1; u(0, x) = φ(x).
The periodic solution of (9) can be expressed in integral form as follows. (10) u(t, x) =X ξ b φ(ξ)ei(xξ+tA)+ 2iX ξ eixξ Z eitτ − eitA τ − A w(τ, ξ)dτ.b
Call U (t, x) and V (t, x) the linear and nonlinear parts of u respectively,
(11)
(
U (t, x) =Pξφ(ξ)eb i(xξ+tA); V (t, x) = 2iPξeixξR eitτ−eitA
τ −A w(τ, ξ)dτ.b
We want to study the nonlinear part first. Choose cut-off functions ba and bb
such that ba + bb = 1, supp ba ⊂ [−2R, 2R] and suppbb ⊂ {x : |x| ≥ R}. Make
a decomposition of V (t, x) in the following way.
(12) V (t, x) = H(t, x) + Ψ1(t, x) + Ψ2(t, x), where (13) b H(τ, ξ) = bb(τ −A)τ −A w(τ, ξ),b b Ψ1(τ, ξ) = δ(τ − A) R bb(λ−A) λ−A w(λ, ξ)dλ,b b Ψ2(τ, ξ) = P kδ(k)(τ − A) bGk(ξ), b Gk(ξ) = i k(2R)k−1 k! R ¡λ−A 2R ¢k−1 b a(λ − A) bw(λ, ξ)dλ,
where δ(τ ) is the delta function and δ(k) is its k-th derivative.
Since the solution does not decay in time, it is necessary to localize it in time. We assume that ψ is a cutoff function supported in a neighborhood of 0 and denote ψδ(t) = ψ(t/δ), where δ is a small number to be determined
later. Let
(15) uδ(t, x) = ψδ(t)(Ψ1+ Ψ2)(t, x) + F (t, x).
The norm used here is defined by
(16) N (u) = kS12ukb L2(R×Z).
We want to prove the following result first.
Theorem 3. Let uδ be defined as in (15), we have the estimate
(17) N (uδ) ≤ C ° ° bw S12 ° ° L2(R×Z)+ C X ξ ¯ ¯ Z | bw| S dλ ¯ ¯2 1 2 .
Proof. For the term H, since S
2¯¯bb(τ − A)¯¯2 (τ − A)2 ≤ 1, we get kS12Hkb 2 L2 ≤ C ° ° bw S12 ° ° L2.
For the term Ψ1, since
Z S| bψδ(τ − A)|2dτ ≤ C(ψ), we have kS12( bψδ∗ bΨ1)kL2 ≤ C X ξ ¯ ¯Z | bw| S dλ ¯ ¯2 1 2 .
For the last term, using the facts that
kS12tdkψδkL2 ≤ C(ψ)(2δ)k and k bGkk L(Z) ≤ C (2R)k k! ° ° bw S12 ° ° L2, we obtain kS12( bψδ∗ bΨ2)kL2 ≤ C(ψ)e4Rδ ° ° bw S12 ° ° L2. ¤
We divide the proof for Theorem 1 into several steps. First we state and prove two lemmas. Notice that now w = u∂xu.
Lemma 4. (18) °° bw S12 ° ° L2(R×Z) ≤ Cδ α−1 8α N (u)2. Lemma 5. (19) X ξ ¯ ¯ Z | bw| S dλ ¯ ¯2 1 2 ≤ Cδα−18α N (u)2.
Proof of Lemma 4. Observe that | bw(τ, ξ)| is bounded by
(20) |ξ|X
η
Z
|bu(λ, η)||bu(τ − λ, ξ − η)|.
To cancel out the factor |ξ|, notice that ¯
¯(τ − A(ξ)) − [(λ − A(η)) + (τ − λ − A(ξ − η))]¯¯ =¯¯ − A(ξ) + A(η) + A(ξ − η)¯¯ ≥ C|ξ|α−1,
(21)
provided ξ 6= 0, η 6= 0 and ξ 6= η. Also observe that bw(τ, 0) = 0. Assume the
average of u is zero, i.e. bu(τ, 0) = 0, temporarily so that we have (21).
( This assumption will be removed later.) For the sake of convenience, we denote
(22) (
C(λ, η) = (|λ − A(η)| + 1)12|bu(λ, η)| = S12|bu(λ, η)|,
b
F (λ, η) = |bu(λ, η)| and G(λ, η) = C(λ, η) = Sb 12|bu(λ, η)|
Thus we can bound wb
S12
by
(24) Z X
η
|ξ|C(λ, η)C(τ − λ, ξ − η)
(|τ − A(ξ)| + 1)12(|λ − A(η)| + 1)12(|τ − λ − A(ξ − η)| + 1)12
dλ
From (21) one of the following cases happens.
(25) |τ − A(ξ)| ≥ C3|ξ|α−1, |λ − A(η)| ≥ C3|ξ|α−1, and |τ − λ − A(ξ − η)| ≥ C3|ξ|α−1.
For the first case of (25), we have Z X η C(λ, η)C(τ − λ, ξ − η) |ξ|α−32 (|λ − A(η)| + 1)12(|τ − λ − A(ξ − η)| + 1|12 dλ ≤ cF2(τ, ξ).
Taking L2 norm on cF2 and applying Theorem 2, we get
(26) k cF2k L2 ≤ N (u) α+1 α kuk α−1 α L2 .
Assume that u is supported by [−δ, δ] × T , since α + 1 4α < 1 2, we have (27) kukL2 ≤ δ 1 4kukL4 which implies (28) kF kL4 ≤ Cδ α−1 8α N (u).
For the second case of (25), we have Z X η C(λ, η)C(τ − λ, ξ − η) (|τ − A(ξ)| + 1)12|ξ| α−3 2 (|τ − λ − A(ξ − η)| + 1|12 dλ ≤ F G(τ, ξ)d S12 .
Taking L2 norm on dF G/S12, we have
° °F G(τ, ξ)d S12 kL2 ≤ Cδ α−1 8α N (u)2.
The proof of the last case of (25) is similar to the second one. ¤ Remark. To remove the condition that the solution is of zero average, b
u(τ, 0) = 0, we may modify the problem (1) by replacing φ by φ1 + φ0
and u by u1 + φ0, where φ0 =
Z
φ(x)dx =
Z
u(t, x)dx. All arguments go
through if A(ξ) is replaced by A(ξ) − φ0ξ.
Proof of Lemma 5. Observe that | bw|
S is bounded by
(29) Z X
η
|ξ|C(λ, η)C(τ − λ, ξ − η)
(|τ − A(ξ)| + 1)(|λ − A(η)| + 1)12(|τ − λ − A(ξ − η)| + 1|12
We use the notations denoted in the previous Lemma and distinguish again the cases in (25).
For the first case of (25), we have X ξ ¡Z | bw(τ, ξ)| |τ − A| + 1dτ ¢2 1 2 ∼ X ξ ¡Z |ξ| cF2(τ, ξ) |τ − A| + |ξ|α−1dλ ¢2 1 2 .
Let a(ξ) be a nonnegative sequence with unit l2-norm, i.e. X ξ
a2(ξ) = 1. Using the first one in (25) and
(30) Z ξ2 (|τ − A| + |ξ|α−1)2dτ ≤ C, we can estimate (31) X ξ Z a(ξ)|ξ| cF2(τ, ξ) |τ − A| + |ξ|α−1dτ ≤ Cδ α−1 4α N (u)2.
Use a duality argument, we get (18). For the second case of (25), we have
Z X η C(λ, η)C(τ − λ, ξ − η) (|τ − A(ξ)| + 1)|ξ|α−32 (|τ − λ − A(ξ − η)| + 1| 1 2 dλ ≤ F G(τ, ξ)d S .
Taking l2 norm on the integral
Z d F G/Sdτ, we get (32) X ξ ¡Z dF G(τ, ξ) S dτ ¢2 1 2 ≤ Cδα−18α N (u)2.
The proof of the last case of (25) is again similar to the second one. ¤ Here we come to the stage that we can prove Theorem 1.
Proof of Theorem 1. First we combine the results of Theorem 3 and Lemmas 1 and 2 to get , for the nonlinear part V (t, x) of the solution,
(33) kS12ubδkL2 ≤ Cδ
α−1
8α N (u)2.
Define the map by
(34) T u(t, x) = ψδ(t)U (t, x) + ψδ(t)V (t, x).
Thus the N norm of T u is bounded by
(35) C ³ kφkL2 + δ α−1 8α N (u)2 ´ .
By choosing sufficiently large M , we have, for suitable δ and R,
(36) N (u) ≤ M =⇒ N (T u) ≤ M,
provided that C(φ) + δα−18α M2 ≤ M.
Next we estimate the difference of T u and T v and get
(37) N (T u − T v) ≤ Cδα−18α ¡N (u) + N (v)¢N (u − v).
Therefore, again for suitable δ and R, we obtain
(38) N (T u − T v) ≤ 1
2N (u − v),
provided that Cδα−18α
¡
N (u) + N (v)¢≤ 1
2 which can be satisfied by choosing
δ small for given M . By Picard’s theorem, the map T is a contraction with
respect to the norm N (u), hence it has a unique fixed point. ¤ Remarks. The nonlinear term can be replaced by ∂xγu2, but 1 ≤ γ ≤ α − 1
2 . To get global existence we need a conservation law, i.e.ku(t)kL2 is constant
Theorem 6. Let α ≥ 3. If the initial data of (1) is in L2 (Hs, s ≥ 0), then there is a unique periodic solution for the IVP of (1) which exists for all time.
Remarks. The method used here can be applied to the following extension of equation (1)
(40) ut+ ∂xαu + ∂xβu + u∂xu = 0,
where 1 < β < α and 3 ≤ α. (See [K] for a particular case called the fifth order KdV-type equation.)
3. Further Results
In this section, we want to discuss the IVP of (1) for k ≥ 2. First we consider the case k = 2, then k ≥ 3.
(41)
½
ut+ ∂xαu + u2∂xu = 0, (t, x) ∈ R+× T, u(0, x) = φ(x),
where α ≥ 3.
Theorem 7. For k = 2, the IVP of (1) is locally well-posed for data in H1
(Hs, s ≥ 1), and for specified
Z
T φ2dx.
To prove the theorem 7, we need the followings. Lemma 8. (Bourgain [B2]) ³ f2− Z T f2dx ´ ∂xf = 1 3 X η+ζ6=0 ξ−η6=0 ξ−ζ6=0 ξ bf (η) bf (ζ) bf (ξ − η − ζ)eiξx−X ξ b f (ξ)2f (−ξ)eb iξx.
To estimate w we introduce the following norm and notation. |||u|||2 =X(1 + |ξ|2) Z S|bu(τ, ξ)|2dτ +X(1 + |ξ|2) ³ Z |bu(τ, ξ)|dτ ´2 ; S = 1 + |τ − B(ξ)| = 1 + |τ − A(ξ) + cξ|.
Proposition 9. For uδ, we can estimate it as follows.
|||uδ|||2 ≤ X (1 + |ξ|2) Z | bw(τ, ξ)|2 S dτ + X (1 + |ξ|2)³ Z | bw(τ, ξ)| S dτ ´2 .
This proposition can be proved in a similar manner as that in [B2]. Proof. Due to the conservation law,
Z T u2(t, x)dx = Z T φ2(x)dx, we denote (42) c = Z T φ2(x)dx
and consider the IVP
(43)
½
ut + ∂xαu + c∂xu = 0, u(0, x) = ψ(x)
for which the solution can be written as
(44) u(t, x) = Stψ(x) =
X
ξ
b
ψ(ξ)ei(ξx+(A−cξ)t).
Consider the integral equation
(45) u(t) = Stφ + Z t 0 S(t − τ )w(τ )dτ, where w = [ Z T
u2dx − u2]∂xu, which is equivalent to the IVP
(46) ( ut+ ∂xα + c∂xu = ³ R Tu2dx − u2 ´ ∂xu, u(0, x) = S0φ = φ.
We construct a sequence of functions {uk} by (47) uk+1 = X ξ b φ(ξ)ei(ξx+Bt) +X ξ eiξx Z b wk(ξ, τ ) eitτ − eiBt τ − B dτ where wk = [ Z T
u2kdx − u2k]∂xuk, and B = A − c. Observe that
¯ ¯ ¯(τ − B(ξ)) − [(λ − B(η)) + (θ − B(ζ)) + (τ − λ − θ − B(ξ − η − ζ))] ¯ ¯ ¯ ∼ ¯ ¯ ¯|ξ|α− |η|α− |ζ|α − |ξ − η − ζ|α ¯ ¯ ¯ (48)
To find a lower bound of (48), assume that η +ζ 6= 0, ξ −η 6= 0, and ξ −ζ 6= 0. Case I, if one or two of |ξ|, |η|, |ζ| are larger than the others, then
(48) ≥ ³ |η| + |ζ| + |ξ − η − ζ| ´α−1 . Case II, if |η| ∼ |ζ| ∼ |ξ − η − ζ|, (48) ≥ ³ |η| + |ζ| + |ξ − η − ζ| ´α−2 .
Apply Bourgain’s lemma and use the notation Ω(ξ) = {(η, ζ) ∈ Z2 :
η + ζ 6= 0, ξ − η 6= 0, ξ − ζ 6= 0}, we can rewrite (51) b w(τ, ξ) =1 3 X η,ζ∈Ω(ξ) ξ Z b
u(η, λ)bu(ζ, θ)bu(ξ − η − ζ, τ − λ − θ)dλdθ
− ξ
Z b
u(ξ, λ)bu(ξ, θ)bu(−ξ, τ − λ − θ)dλdθ.
Call b w1(τ, ξ) = |ξ| 3 X η,ζ∈Ω(ξ) Z
|bu|(η, λ)|bu|(ζ, θ)|bu|(ξ − η − ζ, τ − λ − θ)dλdθ
b
w2(τ, ξ) = |ξ|
Z
So it is sufficient to estimate, for j = 1, 2, X (1 + |ξ|2) Z | bwj(τ, ξ)|2 S dτ and X (1 + |ξ|2)³ Z | bwj(τ, ξ)| S dτ ´2 ,
For the case I, we distinguish four cases,
(53) |τ − B(ξ)| ≥ |ξ|α−1, |λ − B(η)| ≥ |ξ|α−1, |θ − B(ζ)| ≥ |ξ|α−1, |τ − λ − θ − B(ξ − η − ζ)| ≥ |ξ|α−1.
For the case II, we employ the inequality (1 + |ξ|)|ξ| < C|η||ζ|. We can control the solution u by the norm ||| · ||| and get (54) |||T u||| ≤ Cδα−14α |||u|||3.
Fixed point argument ensures the existence and uniqueness of the
solu-tion. ¤
To get global existence we need conservation laws. We first discuss briefly how to derive those conserved quantities given in (5). For the first one, it is straightforward to get that RTu(t)dx = RTφdx. The second one can be
proved as follows. Multiplying the equation (1) by u and integrating by parts, we get Z 1 2∂t(u 2)dx + Z u∂xαudx = 0.
Using the identity |eu(t, −ξ)| = |eu(t, ξ)|, we can prove that the second integral
above is 0 which implies that ku(t)kL2 is conserved. For the last one, we first
take the integral operator ∂x−1 on the equation, multiply by ut and then
integrate by parts.
Next we apply those conservation laws to obtain the boundedness of Hα−12
norm of solution. We first use interpolation inequalities to bound H1-norm
of u, cf [L]. Let us assume that u is a smooth periodic function temporarily and denote by (55) R Tu2dx = F0, max |u(x)| = M, R T ³ ∂α−12 x u ´2 − 2uk+2 (k+1)(k+2)dx = F1, and R Tu2xdx = S.
Since u is smooth, there exists a point x0 such that
(56) u2(x0) =
Z
T
u2(x)dx = F0,
we apply the identity u(x) = u(x0) +
Z x x0 uxdx to get u2(x) ≤ 2u2(x0) + 2 Z u2xdx ≤ 2F0+ 2S.
This implies that M2 ≤ 2F0 + 2S. On the other hand, we can bound S as
follows. S = Z T u2xdx ≤ C µZ T ³ ∂α−12 x u ´2 dx + Z T u2dx ¶ ≤ C ³ F1+ F0+ 2M k (k + 1)(k + 2)F0dx ´ . Hence we have (57) M2 ≤ 2F0+ 2C(F1+ F0) + 2CF0 (k + 1)(k + 2)M k.
Thus we can deduce that M is bounded by some constant C = C(F0, F1),
provided that F0 and F1 are small. Also we have
(58) Z T ³ ∂α−12 x u ´2 dx ≤ F1+ 2C(F0, F1)k (k + 1)(k + 2)F0.
Another approach to bound the Hα−12 -norm of solution u is that we
inter-polate between the L2 and Hα−1
2 -norms, see [B1]. Using H¨older and Sobolev
inequalities, we have Z T ³ ∂α−12 x u ´2 dx = F1+ Z T 2uk+2 (k + 1)(k + 2)dx ≤ F1+ CkukL2kukk+1 L2(k+1) ≤ F1+ CkφkL2kuk k+1 Hα−12 .
This implies that if kφk
Hα−12 is small, then we have
(60) ku(t)k
Hα−12 ≤ C for all t .
Theorem 10. For k = 2, the IVP of (1) is globally well-posed for small
data in Hα−12 (HS, s ≥ α − 1
2 ), and for specified Z
T φ2dx.
For the case k ≥ 3, besides ideas in [B2], we use those in [S] as well. Definition. i) The space Ys,b, s, b ≥ 0, is the closure of the Schwartz
func-tions S(T × R), with respect to the norm
(61) kf kYs,b = max i=1,2,3ν (s,b) i (f ), where (62) ν1(s,b)(f )2 =P ξ(1 + |ξ|)2s ³ R R| bf |(τ, ξ)dτ ´2 ν2(s,b)(f )2 =P ξ(1 + |ξ|)2s R R| bf |2(τ, ξ)(1 + |τ − A(ξ)|)2bdτ ν3(s,b)(f )2 =P ξ(1 + |ξ|)2s−2 R R| bf |2(τ, ξ)(1 + |τ − A(ξ)|)2b+1dτ.
Denote the space Ys,b[−δ, δ] of functions defined on T × [−δ, δ] with the
restriction norm
(63) kf kYs,b[−δ,δ] = inf kF kYs,b,
where the infimum is taken over all the extensions F of f on T × R.
ii) The space Ys,b, s, b ≥ 0, is the closure of the Schwartz functions S(T × R), with respect to the norm
(64) kf kYs,b = max i=1,2,3,4µ (s,b) i (f ), where (65) µ(s,b)1 (f )2 =P ξ(1 + |ξ|)2s ³ R |τ −A(ξ)>|ξ|| bf |(τ, ξ)dτ ´2 µ(s,b)2 (f )2 =P ξ(1 + |ξ|)2s R |τ −A(ξ)>|ξ|| bf |2(1 + |τ − A(ξ)|)2bdτ µ(s,b)3 (f )2 =P ξ(1 + |ξ|)2s−2 R R| bf |2(τ, ξ)(1 + |τ − A(ξ)|)4bdτ µ(s,b)4 (f ) = k∂s xf kL∞ t L2x.
As in i), we have the space Ys,b[−δ, δ].
iii) Let f and g be functions on T × [−δ, δ] and F and G be the extensions on T × R. Denote (66) ( βF(t) = R TFk(t, x)dx F(F )(τ, ξ) =RRe−itτeiξR0tβF(σ)dσF (t, ξ)dte (67) ds 1(F, G)2 = P ξ(1 + |ξ|)2s ³ R R|F(F ) − F(G)|(τ, ξ)dτ ´2 ds 2(F, G)2 = P ξ(1 + |ξ|)2s R R|F(F ) − F(G)|2(1 + |τ − A|)2bdτ ds 3(F, G)2 = P ξ(1 + |ξ|)2s R R|F(F ) − F(G)|2(1 + |τ − A|)4bdτ.
Then denote the metric space by Xks,b[−δ, δ] with respect to the metric
(68) ds∗(F, G) = inf F G n X i dsi(F, G) o .
The space Xks,b[−δ, δ] is a complete metric space, s ≥ 12, b ≥ 0, see [S]. Theorem 11. Consider the IVP (1) for k ≥ 3. If φ ∈ Hs, s ≥ α−1
2 , then there exists δ = δ(kφk
Hα−12 ) and a unique solution u in the space X
s,b k [−δ, δ] such that
(70) ds
∗(u, 0) ≤ CkφkHs.
To prove Theorem 11, we consider the associated problem of IVP (1),
(71) ½ vt+ ∂xαv + (vk− R Tvkdx)∂xv = 0, (t, x) ∈ R+× T; v(0, x) = φ(x), Consider e u(t, ξ) = eiξR0tβv(σ)dσev(t, ξ).
The importance of the IVP (71) is that if v is a solution for the problem, then u given by above is a solution for IVP (1).
Proposition 12. Let φ ∈ Hs and s ≥ α − 1
2 . Then there exists δ =
δ(kφk
Hα−12 ) such that the problem (71) is well posed in Y
s,1
2[−δ, δ] and the
solution satisfies the bound
(72) kvk
Ys, 12[−δ,δ] ≤ CkφkHs.
The proof relies on Bourgain’s ideas and following lemma. Lemma 13. (Bourgain, [B2]) If w ∈ Ys,1 2, s ≥ 1 and denote (73) P (τ, ξ) = [ψδ(wk− βw)∂xw]∼(τ, ξ), (74) ³ P ξ R R(1 + |ξ|)2s−2|P (τ, ξ)|2dτ ´1 2 ≤ Cδ²kwk Y1, 12kwk k Ys, 12 ³ P ξ R τ −A(ξ)≤|ξ|2200(1 + |ξ|) 2s|P |2dτ´ 1 2 ≤ Cδ²kwk Y1, 12kwk k Ys, 12, for some ² > 0.
Proof of Proposition 12. Define the map T on Ys,1
2[−δ, δ] such that ] T (v)(t, ξ) = ψ(t)e−itA(ξ)φ(ξ)+b ψδ(t) Z t 0 e−i(t−s)A(ξ)F¡(vk− βv)∂xv ¢ (s, ξ)ds. (75)
We want to show that the map T is a contraction.
As in Theorem 1, we first split T (v) into linear and nonlinear parts and denoted by U and V respectively.
(76) ( e U (t, ξ) = ψ(t)e−itA(ξ)φ(ξ)b e V (t, ξ) = ψδ(t) Rt 0 e−i(t−s)A(ξ)F ¡ (vk− β v)∂xv ¢ (s, ξ)ds.
To estimate U , we follow arguments in [KPV1] and [S] obtain, for j = 1, 2, 3,
To estimate V , we follow Bourgain’s argument, and use Lemma (13) and k∂Xs wk2L∞ t L2x ≤ X ξ ³ Z R (1 + |ξ|)s| ew(τ, ξ)dτ ´2 . We have, for j = 1, 2, 3, (78) νjs(V )2 ≤ Cδ2γkvk2 Y1, 12kvk 2k Ys, 12. Hence we obtain (80) kT (v)kYs, 12[−δ,δ] ≤ CkφkHs + δγkvkY1, 12[−δ,δ]kvkk Ys, 12[−δ,δ].
Thus if δ = δ(kφkH1) is small, then, for R = C(kφkHs), T is a contraction
from a ball BR into itself.
Next we observe that
(81) F(T u − T v)(t, ξ) = ψδ(t)
Z t
0
e−i(t−s)A(ξ)·
F([(u − v)Pk−1(u, v) − (βu − βv)]∂xv)F((uk− βu)∂x(u − v)) ds
which suggest that we can consider the integral equation
(82) w(t, ξ) = ψe δ(t)
Z t
0
e−i(t−s)A(ξ)·
F([wPk−1(u, v) − θ(s)]∂xv)F(uk− βu)∂xw) ds,
where Pk−1(u, v) is a polynomial of degree k−1 and θ(t) =
R
TwPk−1(u, v)dx.
Let Φ be the operator defined on Ys,1
2[−δ1, δ1], δ1 < δ, by the above integral
equation. We can show that there exists δ1 = δ1(kukYs, 12, kvkYs, 12) such that
Φ is a contraction from a ball Bρ into itself, for arbitrary ρ. By uniqueness,
we have u = v almost everywhere on [−δ1, δ1]. Repeating the argument finite
times, we conclude the proof. ¤
Theorem 14. For k > 2, the IVP of (1) is globally well-posed for data in
Hα−12 (Hs, s ≥ α − 1
2 ), with sufficiently small H
α−1
2 -norm.
4. Proof of A priori Estimate
In this part, we want to prove Theorem 2.
Theorem 2. If α ≥ 2, then we have the following estimates
(83) kf kL4(R×T)≤ C ° °S1+α 4α fb ° ° L2(R×Z), ° ° fb S1+α4α ° ° L2(R×Z)≤ Ckf kL43(R×T).
Proof. First we split the function f into positive and negative parts with respect to the dual of space variable and denote
(84) f = f++ f− =X
ξ≥0
eixξf (t, ξ) +e X ξ<0
eixξf (t, ξ).e
It suffices to prove that f+ and f− both satisfy the estimate. We will only
prove the case of f+ since the proof for f− is similar. Hence we replace f+
by f and decompose the function in the following way. Choose a smooth function ba with support in [2−1, 2]. Let ba
j(τ ) = ba(2−jτ ) and ba0 = 1 − X b aj. Consider (86) f (t, x) = X j fj(t, x), where fbj(τ, ξ) = baj(τ − |ξ|α) bf (τ, ξ). Thus we have (88) kf k2L4 ≤ X j,k kfjfkkL2.
Observe that (fjfk)(t, x) can be written as
(90) Z Z X
ξ1ξ2
ei(t(τ1+τ2)+x(ξ1+ξ2))fb
We choose a change of variables (91) ½ τ = τ1+ τ2, ξ = ξ1+ ξ2, p = p1+ p2, q = p2, where (92) ½ p1 = τ1− |ξ1|α ∈ ∆j = [2j−1, 2j+1], p2 = τ2− |ξ2|α ∈ ∆k = [2k−1, 2k+1].
(Without loss of generality, we assume that p1 and p2 are both positive. The
case of negative p1 and p2 can be treated in the same manner.) Thus, fjfk
can be rewritten as follows. (93) (fjfk)(t, x) = Z X ξ ei(tτ +xξ)Gbjk(τ, ξ)dτ, where (94) ( b Gjk(τ, ξ) = R ∆k P p∈Λj( bfj b fk)(τ, ξ, q, p)dq and Λj(τ, ξ, q) = © p ∈ ∆j+ q : ξ1, ξ2 ∈ Z+ ª .
Applying Plancherel’s Theorem, we have (95) kfjfkkL2 = k bGjkkL2.
Without loss of generality we may assume that j > k. Observe that
(96) k bGjkk2L2 = Z X ξ ¯ ¯ ¯ ¯ ¯ ¯ Z ∆k X p∈Λj ( bfjfbk)(τ, ξ, q, p)dq ¯ ¯ ¯ ¯ ¯ ¯ 2 dτ. Claim: (97) sup τ,ξ,q ¯ ¯Λj ¯ ¯ ≤ C2j α.
Assuming the claim, we get
(98) k bGjkk2L2 ≤ 1 2α−12α (j−k) ° °S1+α 4α fbj ° °2 L2 ° °S1+α 4α fbk ° °2 L2.
The case of j < k can be treated in a similar fashion. Thus we have (100) kf+k2L4 ≤ X jk 1 2α−12α |j−k| ° °S1+α 4α fb ° °2 L2. Therefore we obtain (101) kf k2L4 ≤ ° °S1+α 4α fb ° °2 L2 X jk 1 2α−12α |j−k|
which implies that f satisfies the estimate. ¤
Proof of the Claim:. Since (102) Λj(τ, ξ, q) =
©
p ∈ ∆j + q : ξ1, ξ2 ∈ Z+
ª
,
we can deduce that
(103) τ − q − 2j+1 ≤ ξ1α+ ξ2α ≤ τ − q − 2j−1.
Denote
(104) ½
A = τ − q − 2j+1, M = 3 · 2j−1,
d(a, b) = |a − b| : the distance between point a and point b.
Thus we can rewrite the above inequality as (105) A ≤ ξ1α+ ξ2α ≤ A + M,
and distinguish the cases, A À M , A ∼ M and A ¿ M .
Let C1 and C2 be the graphs of level curves of |ξ1|α+|ξ2|α at A and A+M
respectively.
(106)
½
C1 = {(ξ1, ξ2) : |ξ1|α+ |ξ2|α = A}; C2 = {(ξ1, ξ2) : |ξ1|α+ |ξ2|α = A + M }.
Notice that we can only consider the first quadrant. It can be shown easily that, along each level curve, the farthest point to the origin is on the line
a b l 1 2 3 2 1 C C C FIGURE 1. 1 l C2 b a C 2 1 FIGURE 2.
ξ
ξ
ξ
ξ
can be interpreted as the number of lattice points which lie on the straight line ξ1+ ξ2 = ξ and fall into the region between curves C1 and C2.
For the case A À M , let C3 be a circumscribed circle to the curve C2,
(107) C3 = {(ξ1, ξ2) : ξ12+ ξ22 = 2
α
r
(A + M )2
4 },
then the largest possible line segment in the region is on the line l,
(108) l = {(ξ1, ξ2) : ξ1+ ξ2 = 2α
r
A
2},
which is tangent to the curve C1, see Fig. 1. Let a and b be the intersections
of the line l and the circle C3, then we get
(110) d2 ∼ α r A2 4 − Ã 2α r A2 4 − α r (A + M )2 4 ! ≤ C√αM2.
For the case A ∼ M , the previous argument goes through.
For the case A ¿ M , since C1 is small, we can take the line segment l
between two intercepts of C2,
(111) l = {(ξ1, ξ2) : ξ1+ ξ2 = α √
A + M },
see Fig. 2, and estimate
(112) d ∼ √α
¤ Remark. It is known that the L6-norm estimate is not true. In fact,
Bour-gain proved the estimate (113) ° ° ° X |n|<N anei(nx+n 3t)°° ° 6 ¿ N ²¡ X|a n|2 ¢1 2
in his paper [B2]. The optimal estimate should be a Lp estimate for 4 ≤ p <
6, see [B] and [FG].
5. The Polynomial Bound
In the final part, we discuss a polynomial bound for Hs-norm of the global
solution. First we recall two technical lemmas.
Lemma 15. (Kenig-Ponce-Vega, [KPV3]) Assume that 0 ≤ ρ ≤ 12 and ² > 0 is small. Assume also that ν(−ρ+²,12)
2 (u) and ν
(−ρ+²,1 2)
2 (v) are bounded and RTu(t, x)dx = RTv(t, x)dx = 0. Then
(114) ν(−ρ−²,−12 +²) 2 (∂x(uv)) ≤ Cν(−ρ+², 1 2) 2 (u)ν (−ρ+²,1 2) 2 (v).
Lemma 16. (Staffilani, [S]) Assume that ρ ≥ 0, ² > 0 is small and k ≥ 3.
Then
(115) ν(ρ+²,12−²)
2 (X[0,1]uk) ≤ CkukY1+ρ, 12kuk
k−1 Y1, 12.
Instead of proving Theorem B, we state and prove a more general result. Theorem 17. Consider IVP (1) and assume that there exists an a-priori
bound for the Hα−12 -norm of u. Then if φ ∈ Hs and s ≥ α−1
2 , the global solution satisfies the bounds
(116) ( ku(t)kHs ≤ C|t| s ρ provided ρ + 1 ≤ α−1 2 < S; ku(t)kHs ≤ C|t| 4s 2ρ+(α−3) provided α−1 2 < ρ + 1 < S.
Proof. It is sufficient to show that, for all t ∈ [0,δ2], (117) k∂xsu(t)kL2 x ≤ k∂ s xφkL2 x + Ck∂ s xφk1−δL2 x ,
where δ−1 is the exponents in Theorem 9. Since
k∂s xu(t)k2L2 x = k∂ s xφk2L2 x + Z t 0 d dσk∂ s xu(σ)k2L2 xdσ = k∂xsφk2L2 x − Z R Z T
X[0,t]uk∂x(∂xsu)2dxdσ + lower order terms.
Call (118) J = Z R Z T X[0,t]uk∂x(∂xsu)2dxdσ and set (120) w(t, ξ) = ee iξR0tβu(σ)dσu(t, ξ).e
Taking Fourier transform with respect to space variable, multiplying by
eiξR0tβudσ, then taking Fourier transform with respect to time variable, we
have J =X ξ Z R \ X[0,t]wk(τ, ξ) \∂x(∂xsw)2(τ, ξ)dτ ≤X ξ Z R | \X[0,t]wk|(τ, ξ)(1 + |ξ|)ρ+²(1 + |τ − A(ξ)) 1 2−²· | \∂x(∂xsw)2|(τ, ξ)(1 + |ξ|)−ρ−²(1 + |τ − A(ξ))− 1 2+²dτ ≤ν(ρ+²,12−²) 2 (X[0,1]uk)ν (−ρ−²,−12 +²) 2 (∂x(∂xsu)2).
Employing Lemmas, Theorem 7, and interpolation between the Hα−12 and
the Hs norms, we obtain J ≤ Ckwk Y1+ρ, 12kwk k−1 Y1, 12[ν (−ρ+²,1 2) 2 (∂xsw)]2 ≤ Cd1+ρ∗ (u, 0)d1∗(u, 0)k−1ds−ρ+²∗ (u, 0)2
For ρ + 1 ≤ α−12 < S, interpolation gives (121) J ≤ C(kφk Hα−12 )kφk 2(1−2s−(α−1)2(ρ−²) ) Hs . If we choose ² = ρα−1 2s , we have (122) ku(t)kHs ≤ C|t| s ρ.
For α−12 < ρ + 1 < S, interpolation gives
(123) J ≤ C(kφk Hα−12 )kφk 2(1−ρ+ α−32 −2² 2s−(α−1) ) Hs . Choose ² = (α−1)(ρ+α−32 ) 4s , we have (124) ku(t)kHs ≤ C|t| 4s 2ρ+(α−3). ¤
Acknowledgment: I want to express my gratitude toward Professor M. Grillakis for his inspiring and helpful conversations.
References
[ABFS] L. Abdelouhab, J. Bona, M. Felland & J. Saut, Nonlocal Models for Nonlinear,
Dispersive Waves, Physica D 40 (1989), 360-392.
[B1] J. Bourgain, Fourier transform restriction phenomena for certain lattice
sub-sets and applications to nonlinear evolution equations. Part I Schr¨odinger equa-tions, Geometric and Functional Anal. 3 (1993), 107-155.
[B2] , Fourier transform restriction phenomena for certain lattice subsets
and applications to nonlinear evolution equations. Part II The KdV equations,
Geometric and Functional Anal. 3 (1993), 209-262.
[B3] , On the Cauchy problem for periodic KdV-type equations, J. Fourier Anal. Appl. (1995), Proceedings of the Conference in Honor of Jean-Pierre Kahane ( Orsay 1993 ), 17-86 (Special Issue).
[B4] , On the growth in time of higher nnorms of smooth solutions of
Hamil-tonian PDE, Int. Math. Res. Note 6 (1996), 277-304.
[BKPSV] B. Birnir, C. Kenig, G. Ponce, N. Svanstedt & L. Vega, On the Ill-posedness
of the IVP for the generalized Korteweg-de Vries and Nonlinear Schr¨odinger Equations, J. London Math. Soc. (2) 53 (1996), 551-559.
[CGX] Z. Chen, B. Guo & L. Xiang, Complete Integrability and Analytic solutions of
a KdV-type Equation, J. Math. Phys. 31 (1990), 2851-2855.
[CNP] H. Capel, F. Nijhoff & V. Papageorgiou, Complete Integrability of Lagrangian
Mappings and Lattices of KdV type, Phys. Lett. A 155 (1991), 377-387.
[CKSTT] 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.
[FG] Y.F. Fang & M. Grillakis, Existence and Uniqueness for Boussinesq Type
Equa-tions on a Circle, Comm. PDE 21 (1996), 1253-1277.
[K] V. Karpman, Stationary Solitary Waves of the Fifth Order KdV-type
Equa-tions, Phys. Lett. A 186 (1994), 300-302.
[KdV] D. Korteweg & G. de Vries, On the Change of Form of Long Waves Advancing
in a Rectangular Canal, and on a New Type of Long Stationary Waves, Phil.
Mag. 39 (1895), 422-443.
[KPV1] C. Kenig, V. Ponce & L. Vega, The Cauchy Problem for the KdV Equation in
Sobolev Spaces of Negative Indices, Duke Math. J. 71 (1993), 1-21.
[KPV2] , Higher Order Non-linear Dispersive Equations, Proc. Amer. Math. Soc. 122 (1994), 157-166.
[KPV3] , A Bilinear Estimate with Applications to the KdV Equation, J. Amer. Math. Soc. 9 (1996), 573-603.
[L] P. Lax, Periodic Solutions of the KdV Equation, Comm. Pure Appl. Math. 28 (1975), 141-188.
[MGKr] R. Miura, C. Gardner, & M. Kruskal, Korteweg- de Vries Equations and
Gen-eralizations, I, II, J. Math. Phys. 9 (1968), 1202-1209.
[S] G. Staffilani, On solutions for periodic generalized KdV equations, Internat. Math. Res. Notices (1997), 899-917.
[Z] A. Zygmund, On Fourier Coefficients and Transforms of Functions of Two
Variables, Studia Math. 50 (1974), 189-201.
Department of Mathematics, National Cheng Kung Univ, Tainan, Taiwan
E-mail address: fang@math.ncku.edu.tw
. U RL : http://math.ncku.edu.tw/∼fang
. P hone : 886-06-275-7575 ext 65131
