Analycity and smoothing effect for the coupled system of equations of
  Korteweg - de Vries type with a single point singularity
Journal of Nonlinear Mathematical Physics Volume *, Number * (20**), 1–?? Article
Analycity and smoothing effect for the coupled
system of equations of Korteweg - de Vries type
with a single point singularity
Mauricio Sepúlveda a and Octavio Vera Villagrán b
a Departamento de Ingenieŕıa Matemática, Universidad de Concepción, Casilla 160-C,
Concepción, Chile.
E-mail: mauricio@ing-mat.udec.cl
b Departamento de Matemática, Universidad del Bı́o-Bı́o, Collao 1202, Casilla 5-C,
Concepción, Chile.
E-mail: overa@ubiobio.cl
Received Month *, 200*; Accepted in Revised Form Month *, 200*
Abstract
We study that a solution of the initial value problem associated for the coupled system
of equations of Korteweg - de Vries type which appears as a model to describe the
strong interaction of weakly nonlinear long waves, has analyticity in time and smooth-
ing effect up to real analyticity if the initial data only has a single point singularity
at x = 0.
Keywords and phrases: Evolution equations, Gevrey class, Bourgain space, smoothing ef-
fect.
Mathematics Subject Classification: 35Q53
1 Introduction
We consider the following coupled system of equations of Korteweg - de Vries type
ũt + ũxxx + a3 ṽxxx + ũ ũx + a1 ṽ ṽx + a2 (ũ ṽ)x = 0, x, t ∈ R (1.1)
b1 ṽt + ṽxxx + b2 a3 ũxxx + ṽ ṽx + b2 a2 ũ ũx + b2 a1 (ũ ṽ)x = 0, (1.2)
ũ(x, 0) = ũ0(x), ṽ(x, 0) = ṽ0(x). (1.3)
where ũ = ũ(x, t), ṽ = ṽ(x, t) are real-valued functions of the variables x and t and
a1, a2, a3, b1, b2 are real constants with b1 > 0 and b2 > 0. The original coupled system is
ũt + ũxxx + a3 ṽxxx + ũ
p ũx + a1 ṽ
p ṽx + a2 (ũ
p ṽ)x = 0, x, t ∈ R (1.4)
b1 ṽt + ṽxxx + b2 a3 ũxxx + ṽ
p ṽx + b2 a2 ũ
p ũx + b2 a1 (ũ ṽ
p)x = 0 (1.5)
ũ(x, 0) = ũ0(x), ṽ(x, 0) = ṽ0(x) (1.6)
Copyright c© 200* by M Sepúlveda and O Vera
http://arxiv.org/abs/0704.1861v1
2 M Sepúlveda and O Vera
where ũ = ũ(x, t), ṽ = ṽ(x, t) are real-valued functions of the variables x and t and
a1, a2, a3, b1, b2 are real constants with b1 > 0 and b2 > 0. The power p is an integer
larger than or equal to one. The system (1.4)-(1.6) has the structure of a pair of Korteweg
- de Vries equations coupled through both dispersive and nonlinear effects. In the case
p = 1, the system (1.4)-(1.6) was derived by Gear and Grimshaw [9] as a model to describe
the strong interaction of weakly nonlinear, long waves. Mathematical results on the system
(1.4)-(1.6) were given by J. Bona et al. [5]. They proved that (1.4)-(1.6) is globally well
posed in Hs(R)×Hs(R) for any s ≥ 1 provided |a3| < 1/
b2. The system (1.4)-(1.6) has
been intensively studied by several authors (see [2, 3, 5, 7, 23] and the references therein).
We have the following conservation laws
E1(ũ) =
ũ dx , E2(ṽ) =
ṽ dx , E3(ũ, ṽ) =
(b2ũ
2 + b1ṽ
2)dx (1.7)
The time-invariance of the functionals E1 and E2 expresses the property that the mass of
each mode separately is conserved during interaction, while that of E3 is an expression of
the conservation of energy for the system of two models taken as a whole. The solutions of
(1.4)-(1.6) satisfy an additional conservation law which is revealed by the time-invariance
of the functional
b2 ũ
x + ṽ
x + 2b2a3ũxṽx − b2
− b2a2ũ2ṽ − b2a2ũ2ṽ − b2a1ũṽ2 −
The functional E4 is a Hamiltonian for the system (1.4)-(1.6) and if b2a
3 < 1, φ4 will be
seen to provide an a priori estimate for the solutions (ũ, ṽ) of (1.4)-(1.6) in the space
H1(R)×H1(R). Furthermore, the linearization of (1.1)-(1.3) about the rest state can be
reduced to two, linear Korteweg - de Vries equations by a process of diagonalization. Using
this remark and the smoothing properties (in both the temporal and spatial variables) for
the linear Korteweg - de Vries derived by Kato [13, 15], Kenig, Ponce and Vega [18, 19]
it will be shown that (1.4)-(1.6) is locally well-posed in Hs(R) × Hs(R) for any s ≥ 1
whenever
b2a3 6= 1. This result was improved by J. M. Ash et al. [1] showing that the
system (1.1)-(1.3) is globally well-posed in L2(R) × L2(R) provided that
b2a3 6= 1. In
2004, F. Linares and M. Panthee [21] improve this result showing that the system (1.1)-
(1.3) is locally well-posed in Hs(R) × Hs(R) for s > −3/4 and globally well-posed in
Hs(R)×Hs(R) for s > −3/10 under some conditions on the coefficients, indeed for a3 = 0
and b1 = b2. Following the idea W. Craig et al. [6], it is shown in [23] that C
∞ solutions
(ũ( · , t), ṽ( · , t)) to (1.1)-(1.3) are obtained for t > 0 if the initial data (ũ(x, 0), ṽ(x, 0))
belong to a suitable Sobolev space satisfying resonable conditions as |x| → ∞. Since (1.1)-
(1.3) is a coupled system of Korteweg-de Vries equations, it is natural to ask whether it
has a smoothing effect up to real analyticity if the initial data only has a single point
singularity at x = 0 as the known results for the scalar case of a single Korteweg -de
Vries equation. Using the scaling argument we can have an insight to this question.
In this paper our purpose is to prove the analyticity in time of solutions to (1.1)-(1.3)
without regularity assumption on the initial data improving those obtained in [23]. Our
main tool is the generator of dilation P = 3 t ∂t + x ∂x. which almost commutes with the
linear Korteweg-de Vries operator L = ∂t + ∂x. Indeed [L, P ] = 3L. A typical example
of initial data satisfying the assumption of the above theorem is the Dirac delta measure,
since (xk ∂x)
kδ(x) = (−1)k k! δ(x). The other example of the data is p. v. 1
, where
Analycity for the coupled system of KdV equations 3
p. v. denotes the Cauchy principal value. Linear combination of those distributions
with analytic Hs data satisfying the assumption is also possible. In this sense, the Dirac
delta measure adding the soliton initial data can be taken as an initial datum. Using the
operator K = x · ▽ + 2 i t ∂t it was proved the Gevrey smoothing effect in space variable
[8]. Indeed, it was shown that, if the initial data belongs to a Gevrey class of order 2, then
solutions of some nonlinear Schrödinger equations become analytic in the space variable
for t 6= 0. For the Korteweg-de Vries equations version of the generator of dilation is also
useful to study the analyticity in time and the Gevrey effect in the space variables for
solutions [8].
This paper is organized as follows: In section 2 we have the reduction of the problem and
we outline briefly the notation, terminology to be used subsequently and results that will
be used several times. In section 3 we prove a theorem of existence and well-posedness of
the solutions. In section 4 we prove the following theorem:
Theorem 1.1. Suppose that the initial data (ũ0, ṽ0) ∈ Hs(R) × Hs(R), s > −3/4 and
A0, A1 > 0 such that
||(x ∂x)kũ0||Hs(R) < +∞ :
||(x ∂x)kũ0||Hs(R) < +∞. (1.8)
Then for some b ∈ (1/2, 7/12), there exist T = T (||ũ0||Hs(R), ||ṽ0||Hs(R)) and a unique
solution of (1.1)-(1.3) in a certain time (−T, T ) and the solution (ũ, ṽ) is time locally
well-posed, i. e., the solution continuously depends on the initial data. Moreover, the
solution (ũ, ṽ) is analytic at any point (x, t) ∈ R× {(−T, 0) ∪ (0, T )}.
Corollary 1.1. Let s > −3/4, b ∈ (1/2, 7/12). Suppose that the initial data (ũ0, ṽ0) ∈
Hs(R)×Hs(R), and A0, A1 > 0 such that
(k!)3
||(x ∂x)kũ0||Hs(R) < +∞ :
(k!)3
||(x ∂x)kũ0||Hs(R) < +∞.(1.9)
Then there exists a unique solution (ũ, ṽ) ∈ C((−T, T ), Hs(R))∩Xsb×C((−T, T ), Hs(R))∩
Xsb to the coupled system of Korteweg- de Vries equation (1.1)-(1.3) for a certain (−T, T )
and for any t ∈ (−T, 0)∪(0, T ), the pair (ũ, ṽ) are analytic functions in the space variable
and for x ∈ R, ũ(x, · ) and ṽ(x, · ) are Gevrey 3 as function of the time variable.
Remark 1.1. In Theorem 1.1 and Corollary 1.2, the assumption on the initial data implies
analyticity and Gevrey 3 regularity except at the origin respectively. In this sense, those
results state that the singularity at the origin immediately disappears after t > 0 or t < 0,
up to analyticity.
Remark 1.2. The crucial part for obtaining a full regularity is to gain the L2(R2) regular-
ity of the solutions (uk, vk) from the negative order Sobolev space. This part is obtained
in Proposition 4.1 in Section 4. We utilize a three steps recurrence argument for treating
the nonlinearity appearing in the right hand side of
t ∂3xuk = −
Puk +
x ∂xuk + tB
k(u, u) + tB
k(v, v) + tB
k(u, v) (1.10)
t ∂3xvk = −
Pvk +
x ∂xvk + t C
k(u, u) + t C
k(v, v) + t C
k(u, v). (1.11)
4 M Sepúlveda and O Vera
Then step by step, we obtain the pointwise analytic estimates
t∈[t0−ǫ, t0+ǫ]
||∂mt ∂lxu||H1(x0−ǫ, x0+ǫ) ≤ cA
1 (m+ l)!, l, m = 0, 1, 2, . . . (1.12)
t∈[t0−ǫ, t0+ǫ]
||∂mt ∂lxv||H1(x0−ǫ, x0+ǫ) ≤ cA
2 (m+ l)!, l, m = 0, 1, 2, . . . (1.13)
Since initially we do not know whether the solution belong to even L2(R2) we should men-
tion that the local well-posedness is essentially important for our argument and therefore
it merely satisfies the coupled system equations in the sense of distribution.
2 Reduction of the Problem and Preliminary Results
As mentioned in the introduction we consider the following coupled system of equations
of Korteweg - de Vries type (1.1)-(1.3). If a3 = 0 there is no coupling in the dispersive
terms. Let us assume that a3 6= 0. We are interested in decoupling the dispersive terms in
the system (1.1)-(1.3). For this, let a23 b2 6= 1. We consider the associated linear system
Wt +AWxxx = 0, W (x, 0) =W0(x) (2.1)
where
, A =
a3 b2
The eigenvalues of A are given by
4 b2 a
 (2.2)
4 b2 a
 (2.3)
which are distinct since b1 > 0, b2 > 0 and a3 6= 0. Our assumption a23 b2 6= 1 guarantees
that α± 6= 0. Thus we can write the system (1.1)-(1.3) in a matrix form as in [21]. After
we make the change of scale
ũ(x, t) = u(α
+ x, t) and ṽ(x, t) = v(α
− x, t).
Then we obtain the system of equations
ut + uxxx + a uux + b v vx + c (u v)x = 0, x, t ∈ R (2.4)
vt + vxxx + ã u ux + b̃ v vx + c̃ (u v)x = 0, (2.5)
u(x, 0) = u0(x), v(x, 0) = v0(x) (2.6)
where a, b, c and ã, b̃, c̃ are constant.
Remark 2.1. Notice that the nonlinear terms involving the functions u and v are not
evaluated at the same point. Therefore those terms are not local anymore.
Analycity for the coupled system of KdV equations 5
For s, b ∈ R define the spaces Xsb and Xsb−1 to be the completion of the Schwartz space
S(R2) with respect to the norms
||u||Xs
(1 + |τ − ξ3|)2b (1 + |ξ|)2s |û(ξ, τ)|2 dξ dτ
||u||Xs
(1 + |τ − ξ3|)2(b−1) (1 + |ξ|)2s |û(ξ, τ)|2 dξ dτ
where Xsb = {u ∈ S ′(R2) : ||u||Xsb <∞}. Let Fx and Fx, t be the Fourier transform in the
x and (x, t) variables respectively. The Riesz operator Dx is defined by Dx = F−1ξ |ξ| Fx.
The fractional derivative is defined by
< Dx >
s = F−1ξ < ξ >
s Fx = F−1ξ (1 + |ξ|
2)s/2 Fx
< Dx, t >
s = F−1
< |ξ|+ |τ | >s Fx, t
For < · >= (1 + | · |2)1/2, we have
i) || · ||Hb(R:Hr(R)) = || < Dt >b< Dx >r · ||L2x, t(R2).
ii) Hs(R) = {u ∈ S ′(R) : < Dx >s u ∈ L2(R)}.
iii) || · ||Hs(R) = || < Dx >s · ||L2(R).
Remark 2.2. With the above notation we obtain
a) ||u||Hsx(R) = || < ξ >
s û ||L2(R).
b) ||u||L2t (R:Hrx(R)) = || < ξ >
r û ||L2(R2).
c) || < Dx >s u||L2(R) = ||u||Hs(R).
d) || < Dt >b< Dx >r u||L2x, t(R2) = ||u||Hbt (R:Hrx(R)).
e) || < Dx, t >s u||L2t (R:Hrx(R)) = || < ξ >
r< |ξ|+ |τ | >s û(ξ, τ)||L2(R2).
We consider the following operators: L = ∂t + ∂
x and J = x − 3 t ∂2x then [L, J ] ≡
LJ−J L = 0.We introduce the ”generator of dilation” P = 3 t ∂t+x ∂x for the linear part
of the coupled system (2.4)-(2.6) and the ”localized dilation operator” P0 = 3 t0 ∂t+x0 ∂x.
By employing a localization argument, we look at the operator P as a vector field P0 =
3 t0 ∂t+x0 ∂x near a fixed point (x0, t0) ∈ R×{(−T, 0)∪ (0, T )}. Since P0 is a directional
derivative toward to (x0, t), we introduce another operator L30 = t0 ∂3x which plays the role
of a non-tangential vector field to P0. Since P0 and L0 are linearly independent, the space
and time derivative can be covered by those operator. The main reason why we choose
L0 is because the corresponding variable coefficients operator L3 = t ∂3x can be treated via
the equations (1.10)-(1.11) and a cut-off procedure enables us to handle the right hand
side of those.
Remark 2.3. For L and P we have the following properties:
a) [L, P ] ≡ LP = (P + 3)L.
b) LP k = (P + 3)kL.
6 M Sepúlveda and O Vera
c) (P + 3)k∂x = ∂x(P + 2)
d) (P + 3)k∂3x = ∂
e) P0 P = P P0 + 3P0 − 2x0 ∂x.
Notation. The summation
k=k1+k2+k3
0≤k1, k2, k3≤k
is simply abbreviated by
k=k1+k2+k3
Let P ku = uk, then
ku) + ∂3x(P
ku) = LP ku = (P + 3)kLu = (P + 3)k(∂tu+ ∂
= −(P + 3)k
2) + c ∂x(u v)
(P + 3)k∂x(u
(P + 3)k∂x(v
2)− c (P + 3)k∂x(u v)
∂x(P + 2)
k(u2)−
∂x(P + 2)
k(v2)− c ∂x(P + 2)k(u v).
Noting that (P + 2)ku =
2k−jP ju. Hence
B1k(u, u) = −
∂x(P + 2)
k(u2)
(P + 2)mu · P k−mu
2m−j P ju · P k−mu
(m− j)! j! (k −m)!
2m−j P ju · P k−mu
k=k1+k2+k3
k1! k2! k3!
2k1 ∂x (uk2 · uk3) . (2.7)
In a similar way
B2k(v, v) = −
∂x(P + 2)
k(v2) = − b
k′1! k
· vk′
. (2.8)
B3k(u, v) = c ∂x(P + 2)
k(u v) = − c
k=k′′
k′′1 ! k
2 ! k
· vk′′
. (2.9)
Analycity for the coupled system of KdV equations 7
Therefore
ku) + ∂3x(P
k=k1+k2+k3
k1! k2! k3!
2k1 ∂x (uk2 · uk3)−
k′1! k
· vk′
k=k′′
k′′1 ! k
2 ! k
· vk′′
= B1k(u, u) +B
k(v, v) +B
k(u, v). (2.10)
Performing similar calculations as above we obtain
kv) + ∂3x(P
k=k1+k2+k3
k1! k2! k3!
2k1 ∂x (uk2 · uk3)−
k′1! k
· vk′
k=k′′
k′′1 ! k
2 ! k
· vk′′
= C1k(u, u) + C
k(v, v) + C
k(u, v). (2.11)
The above nonlinear terms maintain the bilinear structure like that of the original coupled
system of equations of KdV type, since Leibniz’s rule can be applied for operations of P.
Now, each uk and vk satisfies the following system of equations
∂tuk + ∂
xuk = B
k(u, u) +B
k(v, v) +B
k(u, v) ≡ Bk (2.12)
∂tvk + ∂
xvk = C
k(u, u) + C
k(v, v) + C
k(u, v) ≡ Ck (2.13)
uk(x, 0) = (x ∂x)
ku0(x) ≡ uk0(x), vk(x, 0) = (x ∂x)kv0(x) ≡ vk0 (x). (2.14)
In order to obtain a well-posedness result for the system (2.12)-(2.14) we use Duhamel’s
principle and we study the following system of integral equations equivalent to the system
(2.12)-(2.14)
ψ(t)uk = ψ(t)V (t)u
0 − ψ(t)
V (t− t′)ψT (t′)Bk(t′) dt′ (2.15)
ψ(t) vk = ψ(t)V (t) v
0 − ψ(t)
V (t− t′)ψT (t′)Ck(t′) dt′ (2.16)
where V (t) = e−t ∂
x is the unitary group associated with the linear problem and ψ(t) ∈
C∞0 (R), 0 ≤ ψ ≤ 1 is a cut-off function such that
ψ(t) =
1, if |t| < 1
0, if |t| > 2 and ψT (t) = ψ(t/T )
The following results are going to be used several times in the rest of this paper.
8 M Sepúlveda and O Vera
Lemma 2.1 ([16]). . Let s ∈ R, a, a′ ∈ (0, 1/2), b ∈ (1/2, 1) and δ < 1. Then for any
k = 0, 1, 2, . . . , we have
||ψδφk||Xs−a ≤ c δ
(a−a′)/4(1−a′) ||φk||Xs
, (2.17)
||ψδ V (t)φk||Xs
≤ c δ1/2−b||φk||Hs(R), (2.18)∣∣∣∣
∣∣∣∣ψδ
V (t− t′)Fk(t′) dt′
≤ c δ1/2−b ||Fk||Xs
. (2.19)
Lemma 2.2 ([16]). . Let s > −3/4, b, b′ ∈ (1/2, 7/12) with b < b′. Then for any
k, l = 0, 1, 2, . . . we have
||∂x(uk vl)||Xs
≤ c ||vk||Xs
||vl||Xs
. (2.20)
Lemma 2.3 ([12]). . Let s < 0, b ∈ (1/2, 7/12) and ψ = ψ(x, t) be a smooth cut-
off function such that the support of ψ is in B2(0) and ψ = 1 on B1(0). We set ψǫ =
ψ((x− x0)/ǫ, (t− t0)/ǫ). Then for f ∈ Xsb , we have
||ψǫ f ||Xs
≤ c ǫ−|s|−5|b|||ψǫ||X|s|+2 |b|
||f ||
s+2 |b|
, (2.21)
where the constant c is independent of ǫ and f.
Lemma 2.4 ([12]). . Let P be the generator of the dilation and Dx, t be an operator defined
by F−1
< |τ | + |ξ| > Fx, t. We fix an arbitrary point (x0, t0) ∈ R × {(−T, 0) ∪ (0, T )}.
1) Suppose that b ∈ (0, 1], r ∈ (−∞, 0] and g ∈ Xrb−1 with supp g ⊂ B2ǫ(x0, t0) and t∂3xg,
P 3g ∈ Xrb−1. If ǫ > 0 is sufficiently small, then we have
|| < Dx, t >3b g||L2(R:Hr(R)) ≤ c
||g||Xr
+ ||t∂3xg||Xrb−1 + ||P
3g||Xr
(2.22)
where the constant c = c(x0, t0, ǫ).
2) If g ∈ Hµ−3(R2) with supp g ⊂ B2ǫ(x0, t0) and t∂3xg, P 3g ∈ Hµ−3(R2). Then for small
ǫ, we have
|| < Dx, t >µ g||L2(R2) ≤ c
||g||Hµ−3(R2) + ||t∂3xg||Hµ−3(R2) + ||P 3g||Hµ−3(R2)
(2.23)
where the constant c = c(x0, t0, ǫ).
Lemma 2.5 ([12]). . Let 0 ≤ s, r ≤ n/2 with n/2 ≤ s + r and suppose that f ∈ Hs(Rn)
and g ∈ Hr(Rn). Then for any σ < s+ r − n/2, we have f g ∈ Hσ(Rn) and
||f g||Hσ(Rn) ≤ c(ǫ) ||f ||Hs(Rn)||g||Hr(Rn), (2.24)
where ǫ = s+ r − n/2− σ.
Corollary 2.1 ([12]). . For 1/2 < b < 1 and −3/4 < s < 0, we have
||ψ f ||
≤ c ||f ||Xs
(2.25)
where ψ ∈ C∞0 (R2) and c is independent of f.
Analycity for the coupled system of KdV equations 9
Lemma 2.6 ([12]). . Let ψ(x) be a smooth cut-off function in C∞0 ((−2, 2)) with ψ(x) = 1
on (−1, 1). We set ψǫ = ψ(x/ǫ) for 0 < ǫ < 1. Then for r ≤ 0, and f ∈ Hr, we have
||ψǫ f ||Hr(R) ≤
c ǫ−δ||f ||Hr(R) if − 1/2 ≤ r ≤ 0
c ǫ1/2+r||f ||Hr(R) if r < −1/2
where δ > 0 is an arbitrary small constant and c is independent of ǫ.
Throughout this paper c is a generic constant, not necessarily the same at each occasion
(it will change from line to line), which depends in an increasing way on the indicated
quantities.
3 Existence and Well-Posedness
We firstly solve the following (slightly general) system of equations
∂tuk + ∂
xuk = B
k(u, u) +B
k(v, v) +B
k(u, v) ≡ Bk (3.1)
∂tvk + ∂
xvk = C
k(u, u) + C
k(v, v) + C
k(u, v) ≡ Ck (3.2)
uk(x, 0) = (x ∂x)
ku0(x) ≡ uk0(x) , vk(x, 0) = (x ∂x)kv0(x) ≡ vk0 (x) (3.3)
where Bk and Ck are as above.
Definition 3.1. Let f = (f0, f1, . . . , fk) denotes the infinity series of distributions and
define
AA0(Xsb ) ≡
f = (f0, f1, . . . , fk), fi ∈ Xsb , (i = 0, 1, 2 . . .) such that ||f ||AA0 (Xsb ) < +∞
where
||f ||AA0 (Xsb ) ≡
||fk||Xs
Similarly, for u0 = {u00, u10, . . . , uk0, . . . } and v0 = {v00 , v10 , . . . , vk0 , . . . } we set
||u0||AA0 (Hs(R)) ≡
||uk0 ||Hs(R) and ||v0||AA0 (Hs(R)) ≡
||vk0 ||Hs(R)
respectively.
Remark 3.1. Each solution of the coupled system of Korteweg de Vries equations is
accompanied by the following estimate
||P ku||Xs
≤ cAk0 k!, and ||P kv||Xsb ≤ cA
1 k!, k = 0, 1, 2, . . .
Theorem 3.1. Let −3/4 < s, b ∈ (1/2, 7/12). Suppose that uk0 , vk0 ∈ Hs(R)(k =
0, 1, 2, . . .) and satisfies
||u0||AA0 (Xsb ) =
||uk0 ||Hs(R) < +∞ and ||v0||AA0 (Xsb ) =
||vk0 ||Hs(R) < +∞.
10 M Sepúlveda and O Vera
Then there exist T = T (||uk0 ||Hs(R), ||vk0 ||Hs(R)) and a unique solution u = (u0, u1, . . .) and
v = v(v0, v1, . . .) of the system (3.1)-(3.3) with uk, vk ∈ C((−T, T ) : Hs(R)) ∩Xsb and
||uk||Xs
(R) < +∞,
||vk||Xs
(R) < +∞.
Moreover, the map (uk0 , v
0 ) → (u(t), v(t)) is Lipschitz continuous, i. e.,
||u(t)− ũ(t)||AA0 (Xsb ) + ||u(t)− ũ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||u0 − ũ0||AA0 (Hs(R))
||v(t) − ṽ(t)||AA0 (Xsb ) + ||v(t) − ṽ(t)||C((−T, T ): Hs(R)) ≤ c(T ) ||v0 − ṽ0||AA0 (Hs(R)).
Proof. For given (u0, v0) ∈ AA0(Hs(R))×AA0(Hs(R)) and b > 1/2, let us define,
HR1, R2 =
(u, v) ∈ AA0(Xsb )×AA0(Xsb ) : ||u||AA0 (Xsb ) ≤ R1, ||v||AA0 (Xsb ) ≤ R2
where R1 = 2 c0 ||u0||AA0 (Hs(R)) and R2 = 2 c0 ||v0||AA0 (Hs(R)). Then HR1, R2 is a complete
metric space with norm
||(u, v)||HR1, R2 = ||u||AA0 (Xsb ) + ||v||AA0 (Xsb ).
Without loss of generality, we may assume that that R1 > 1 and R2 > 1. For (u, v) ∈
HR1, R2 , let us define the maps,
Φku0(u, v) = ψ(t)V (t)u
0 − ψ(t)
V (t− t′)ψT (t′)Bk(t′) dt′ (3.4)
Ψkv0(u, v) = ψ(t)V (t) v
0 − ψ(t)
V (t− t′)ψT (t′)Ck(t′) dt′. (3.5)
We prove that Φ × Ψ maps HR1, R2 into HR1, R2 and it is a contraction. In fact, using
lemma 2.1 and lemma 2.2 we have
||Φku0(u, v)||Xsb = ||ψ(t)V (t)u
0 ||Xsb +
∣∣∣∣ψ(t)
V (t− t′)ψT (t′)Bk(t′) dt′
≤ c0 ||uk0 ||Hs(R) + c1 T µ ||Bk||Xs
≤ c0 ||uk0 ||Hs(R) + c1 T µ
k=k1+k2+k3
k1! k2! k3!
2k1 ||uk2 ||Xsb ||uk3 ||Xsb
+ c1 T
k′1! k
1 ||vk′
||vk′
+ c1 T
k=k′′
k′′1 ! k
2 ! k
1 ||uk′′
||vk′′
Analycity for the coupled system of KdV equations 11
Applying a sum over k we have
||Φku0(u, v)||Xsb
||uk0 ||Hs(R) + c1 T µ
k=k1+k2+k3
k1! k2! k3!
2k1 ||uk2 ||Xsb ||uk3 ||Xsb
+ c1 T
k′1! k
1 ||vk′
||vk′
+ c1 T
k=k′′
k′′1 ! k
2 ! k
1 ||uk′′
||vk′′
≤ c0 ||u0||AA0 (Hs(R)) + c1 T
k=k1+k2+k3
||uk2 ||Xsb
||uk3 ||Xsb
+ c1 T
||vk′
||vk′
+ c1 T
k=k′′
k′′2 !
||uk′′
k′′3 !
||vk′′
≤ c0 ||u0||AA0 (Hs(R)) + c1 T
||uk2 ||Xsb
||uk3 ||Xsb
+ c1 T
||vk′
||vk′
+ c1 T
k′′1 !
k′′2 !
||uk′′
k′′3 !
||vk′′
= c0 ||u0||AA0 (Hs(R)) + c1 T
e2A0 ||u||2AA0 (Xsb )
+ c1 T
e2A0 ||v||2AA0 (Xsb ) + c1 T
µ c e2A0 ||u||AA0 (Xsb ) ||v||AA0 (Xsb ).
Hence, choosing d = max{a/2, b/2, c} we have
||Φu0(u, v)||AA0 (Xsb ) ≤ c0 ||u0||AA0 (Hs(R))
+ c1 T
µ d e2A0
||u||2AA0 (Xsb ) + ||v||
AA0 (X
) + ||u||AA0 (Xsb ) ||v||AA0 (Xsb )
≤ c0 ||u0||AA0 (Hs(R)) +
c1 d T
µ e2A0
||u||2AA0 (Xsb ) + ||v||
AA0 (X
. (3.6)
In a similar way, choosing d̃ = max{ã/2, b̃/2, c̃} we have
||Ψv0(u, v)||AA0 (Xsb ) ≤ c0 ||v0||AA0 (Hs(R)) +
c2 d̃ T
µ e2A0
||u||2AA0 (Xsb ) + ||v||
AA0 (X
.(3.7)
12 M Sepúlveda and O Vera
If we choose T such that
T µ ≤
3 max{c1, c2} (R1 +R2)2
Then we obtain in (3.6) and (3.7)
||Φu0(u, v)||AA0 (Xsb ) ≤ R1 and ||Ψv0(u, v)||AA0 (Xsb ) ≤ R2.
Therefore, (Φu0 , Ψv0) ∈ HR1, R2 .We show that Φu0×Ψv0 : (u, v) → (Φu0(u, v), Ψv0(u, v))
is a contraction.
Let (u, v), (ũ, ṽ) ∈ HR1, R2 , then as above we get for d = max{a/2, b/2, c}
||Φu0(u, v)− Φu0(ũ, ṽ)||AA0 (Xsb )
c1 d T
µ e2A0 (R1 +R2)
||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb )
. (3.8)
In a similar way, choosing d̃ = max{ã/2, b̃/2, c̃} we have
||Ψv0(u, v)−Ψv0(ũ, ṽ)||AA0 (Xsb )
c2 d̃ T
µ e2A0 (R1 +R2)
||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb )
. (3.9)
Choosing T µ small enough, such that
T µ ≤ 1
6 max{c1, c2} (R1 +R2)2
we obtain
||Φu0(u, v)− Φu0(ũ, ṽ)||AA0 (Xsb ) ≤
||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb )
. (3.10)
In a similar way
||Ψv0(u, v)−Ψv0(ũ, ṽ)||AA0 (Xsb ) ≤
||u− ũ||AA0 (Xsb ) + ||v − ṽ||AA0 (Xsb )
. (3.11)
Therefore the map Φu0 × Ψv0 is a contraction and we obtain a unique fixed point (u, v)
which solves the initial value problem (3.1)-(3.3) for T < T µ. The rest of the proof follows
a standard argument.
Corollary 3.1. Let −3/4 < s, b ∈ (1/2, 7/12). Suppose that (x ∂x)ku0, (x ∂x)kv0 ∈
Hs(R)(k = 0, 1, 2, . . .) and that
||uk0 ||Hs(R) < +∞ and
||vk0 ||Hs(R) < +∞.
Then there exist T = T (||uk0 ||Hs(R), ||vk0 ||Hs(R)) and a unique solution (u, v) of the coupled
system equations KdV type (1.1)-(1.3) with u, v ∈ C((−T, T ) : Hs(R)) ∩Xsb and
||P ku||Xs
(R) < +∞,
||P kv||Xs
(R) < +∞.
Analycity for the coupled system of KdV equations 13
Moreover, the map (u0, v0) → (u(t), v(t)) is Lipschitz continuous in the following sense:
||P ku(t)− P kũ(t)||Xs
+ ||P ku(t)− P kũ(t)||C((−T, T ): Hs(R)) ≤ c(T )
||(x ∂x)k(u0 − ũ0)||Hs(R)
||v(t) − ṽ(t)||Xs
+ ||v(t)− ṽ(t)||C((−T, T ): Hs(R)) ≤ c(T )
||(x ∂x)k(v0 − ṽ0)||Hs(R).
4 The main result
In this section we prove the analyticity of the solution obtained in the previous section.
We treat the solution uk ≡ P ku and vk ≡ P kv as if they satisfy the coupled system of
equations (3.1)-(3.3) in the classical sense. This can be justified by a proper approximation
procedure. The following results are going to be used in this section. Let (x0, t0) be
arbitrarily taken in R×{(−T, 0)∪(0, T )}. By ψ(x, t) we denote a smooth cut-off function
in C∞0 (B1(0)) and ψǫ = ψ((x − x0)/ǫ, (t− t0)/ǫ).
Let ψ be a smooth cut-off function around the freezing point (x0, t0) with suppψ ⊂
C∞0 (Bǫ(x0, t0)).
Proposition 4.1. For the cut-off function ψ defined above, there exists a positive constant
c and A such that
||ψ P ku||L2x, t(R2) ≤ cA
k (k!)2, k = 0, 1, 2, . . . (4.1)
||ψ P kv||L2x, t(R2) ≤ cA
k (k!)2, k = 0, 1, 2, . . . (4.2)
Proof. Using (2.22) with r = s− 1, we obtain
|| < Dx, t >3b ψP ku||L2t (R:Hs−1x (R)) ≤ c
||ψuk||Xs−1
+ ||t ∂3x(ψuk)||Xs−1
+ ||P 3(ψuk)||Xs−1
.(4.3)
Each term in (4.3) is estimated separately. For the first term in the right hand side we
use Lemma 2.3. Indeed,
||ψ uk||Xs−1
≤ ||ψ uk||Xs
≤ c ||ψ||
|s|+2|b−1|
|b−1|
||uk||Xs
≤ c(ψ)Ak1 k!. k = 0, 1, 2, . . .(4.4)
The third term is estimated again using Corollary 2.6.
||P 3(ψ uk)||Xs−1
l (l − 3)!
||(P 3−lψ)P luk||Xsb−1
≤ c(ψ)
l (l − 3)!
||P luk||Xs
l (l − 3)!
||P k+lu||Xs
Ak+l1 (k + l)!
≤ cAk2 k!. k = 0, 1, 2, . . . (4.5)
14 M Sepúlveda and O Vera
For the second term, we use (3.1) to reduce the third derivative in space to the dilation
operator P. Since the generator of dilation is Puk = 3 t ∂tuk + x ∂xuk we obtain
t ∂tuk =
Puk −
x ∂xuk. (4.6)
Multiplying (3.1) by ψ t, we have
ψ t ∂tuk + ψ t ∂
xuk = ψ tBk. (4.7)
Replacing (4.6) in (4.7) we obtain
ψ t ∂3xuk = −
ψ Puk +
ψ x∂xuk + ψ tBk. (4.8)
hence
||ψ t ∂3xuk||Xs−1
||ψ Puk||Xs−1
||ψ x∂xuk||Xs−1
+ ||ψ tBk||Xs−1
= F1 + F2 + F3. (4.9)
Using the assumption in the Theorem, we have
||ψ Puk||Xs−1
≤ c ||ψ||
||P k+1u||Xs
≤ c ||P k+1u||Xs
≤ cAk+13 (k + 1)! ≤ cA
4 k!. (4.10)
Similarly, we obtain
||ψ x∂xuk||Xs−1
||∂x(ψ xuk)||Xs−1
||∂x(ψ x)uk)||Xs−1
||∂x(ψ x vk)||Xs
+ c ||∂x(ψ x)||X−s
||uk||Xs
||ψ x||Xs
||uk||Xsb + c ||∂x(ψ x)||X−s1−b ||uk||X
||ψ x||Xs
+ ||∂x(ψ x)||X−s
Ak5k! ≤ cAk6k!. (4.11)
Using Lemma 2.3 and 2.2, we have
F3 = ||ψ tBk||Xs−1
≤ c ||ψ||
||B1k +B2k +B3k||Xsb−1
||B1k ||Xsb−1 + ||B
k ||Xsb−1 + ||B
k||Xsb−1
Analycity for the coupled system of KdV equations 15
Then replacing B1k, B
k and B
k in (2.7), (2.8) and (2.9) we deduce
F3 ≤ c
k=k1+k2+k3
k1! k2! k3!
2k1 ||uk2 ||Xsb ||uk3 ||Xsb + c
k′1! k
1 ||vk′
||vk′
k=k′′
k′′1 ! k
2 ! k
1 ||uk′′
||vk′′
k=k1+k2+k3
k1! k2! k3!
2k1 Ak27 · k2! A
7 · k3! + c
k′1! k
8 · k
8 · k
k=k′′
k′′1 ! k
2 ! k
9 · k
2 ! A
10 · k
k=k1+k2+k3
2k1 A
k2+k3
7 + c
k=k′′
k′′1 !
9 · A
≤ c k! Ak7
k−k1∑
2k1 A− k17 + c k! A
+ c k!
k=k′′
k′′1 !
≤ c k! Ak7
k−k1∑
+ c k! Ak8
+ c k!
k=k′′
k′′1 !
9 · A
≤ c e2/A7 Ak7 · k! + c e2/A8 Ak8 · k! + c k!
k=k′′
k′′1 !
e2/A7 + e2/A8
A11 · k! + c k!
k=k′′
k′′1 !
9 · A
10 . k = 0, 1, 2, . . . (4.12)
Hence, from (4.10), (4.11) and (4.12) in (4.9) we obtain that there exists a positive constant
c and A11 such that
||ψ t ∂3xuk||Xs−1
≤ cA11 · k! + c k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . .(4.13)
On the other hand, using ∂3x(ψ · f) = ψ · ∂3xf + 3 ∂2x(∂xψ · f)− 3 ∂x(∂2xψ · f) + ∂3xψ · f we
have that
||t ∂3x(ψ · uk)||Xs−1
≤ ||t ψ · ∂3xuk||Xs−1
+ 3 ||∂2x(t ∂xψ · uk)||Xs−1
+ 3 ||∂x(t ∂2xψ · uk)||Xs−1
+ ||t ∂3xψ · uk||Xs−1
. (4.14)
16 M Sepúlveda and O Vera
Using Lemma 2.2 and Lemma 2.3 we obtain
||∂2x(t ∂xψ · uk)||Xs−1
≤ ||∂x(t ∂xψ · uk)||Xsb−1 ≤ c ||t ∂xψ||Xsb ||uk||Xsb
≤ cAk10 k! (4.15)
||∂x(t ∂2xψ · uk)||Xs−1
≤ ||∂x(t ∂2xψ · uk)||Xsb−1 ≤ c ||t ∂
xψ||Xsb ||uk||Xsb
≤ cAk11 k! (4.16)
||t ∂3xψ · uk||Xs−1
≤ c || < Dx, t >3/2 t ∂3xψ||X|s|+2|b−1|
||uk||Xsb−1 ≤ c ||uk||Xsb
≤ cAk12 k!. (4.17)
Hence, replacing (4.13), (4.15),(4.16) and (4.17) in (4.14) we obtain that there exists a
constant c and A14 such that
||t ∂3x(ψuk)||Xs−1
≤ cAk14 · k! + c k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . .(4.18)
Therefore, replacing (4.4), (4.5) and (4.18) in (4.3) we obtain that there exists a constant
c and A15 such that
|| < Dx, t >3b ψ uk||L2t (R:Hs−1x (R))
≤ cAk15 · k! + c k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . . (4.19)
In a similar way, we obtain that there exists a constant c and A16 such that
|| < Dx, t >3b ψ vk||L2t (R:Hs−1x (R))
≤ cAk16 · k! + c k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . . (4.20)
Adding (4.19) and (4.20) we have
|| < Dx, t >3b ψ uk||L2t (R:Hs−1x (R)) + || < Dx, t >
3b ψ vk||L2t (R:Hs−1x (R))
≤ cAk15 · k! + cAk16 · k! + c k!
k=k′′
k′′1 !
1 2 ·Ak
≤ c (Ak15 +Ak16) · k! + c k!
k=k′′
k′′1 !
1 2 ·Ak
≤ cAk17 · k! + c k!
k=k′′
k′′1 !
1 2 ·Ak
10 . (4.21)
Analycity for the coupled system of KdV equations 17
We estimate the last term on the right hand side of (4.21)
k=k′′
k′′1 !
1 2 · Ak
9 · A
(m− j)!
2(m−j) 2 ·Aj9 · A
≤ Ak10
(m− j)!
≤ Ak10
≤ Ak10
+Ak10
≤ Ak10 k!
+Ak10 k!
≤ eA29/4 Ak10 k! + e4/A
10 Ak10 k!
≤ c Ak10 k!. (4.22)
Replacing (4.22) in (4.21) we obtain
|| < Dx, t >3b ψ uk||L2t (R:Hs−1x (R)) + || < Dx, t >
3b ψ vk||L2t (R:Hs−1x (R))
≤ cAk17 · k! + cAk19 · (k!)2
≤ cAk17 · (k!)2 + cAk19 · (k!)2
≤ cAk20 · (k!)2 (4.23)
and the result follows.
Remark 4.1. a) For simplicity, we only illustrate the conclusion for the case s ≥ −1/2−δ
with b = 1/2 + δ/3 (for small δ > 0) and the case s = −3/4 + δ and b = 7/12 − δ/3. If
s = −1/2− δ with b = 1/2+ δ/3, the initial data can involve Dirac’s delta measure δ0 and
the latter is the critical case of the local well-posedness.
b) The following inequality is simple to verify in both cases,
||ψ uk||L2x, t(R2) ≤ || < Dx >
3 b (ψ uk)||L2t (R: Hs−1x (R)) ≤ c || < Dx, t >
3 b (ψ uk)||L2t (R: Hs−1x (R)).
Proposition 4.2. Under the same assumptions as in Proposition 4.1, there exist positive
constants c and A such that
||ψ P ku||H7/2(R2) ≤ cA
k (k!)2, k = 0, 1, 2, . . . (4.24)
||ψ P kv||H7/2(R2) ≤ cA
k (k!)2, k = 0, 1, 2, . . . (4.25)
Proof. We apply Lemma 2.4 to ψ uk ≡ ψ P ku with b = 1 and r = 0.
|| < Dx, t >3 ψ P ku||L2(R:L2x(R))
||ψ uk||L2(R:L2x(R)) + ||t∂
x(ψ uk)||L2(R:L2x(R)) + ||P
3(ψ uk)||L2(R:L2x(R))
.(4.26)
18 M Sepúlveda and O Vera
Therefore, if we wish to estimate the second term in the right hand side of (4.26) with the
aid of the equation (2.12)
ψ t ∂3xuk = −
ψ Puk +
ψ x∂xuk + t ψ Bk
it is necessary to estimate ||ψ uk||L2t (R:H1x(R)) which is not yet obtained. Hence, we start
from the lower regularity setting, i. e., applying (2.23) in Lemma 2.4 to ψ uk with µ = 1/2.
Let ψ1 be a smaller size of smooth cut-off function with ψ1 ≤ ψ and ψ1 = 1 around (x0, t0).
Applying (2.23) a ψ uk = ψ P
ku with µ = 1/2 we have
|| < Dx, t >3 ψ1 P ku||H− 5/2(R2) ≤ c || < Dx, t >
3 ψ1 P
ku||L2(R2)
||ψ1uk||H− 5/2(R2) + ||t∂
x(ψ1uk)||H− 5/2(R2) + ||P
3(ψ1uk)||H− 5/2(R2)
. (4.27)
The first term on the right hand side of (4.27) has already been estimated. For the third
term we have
||P 3(ψ1uk)||H− 5/2(R2) ≤ ||P
3(ψ1uk)||L2x, t(R2)
l!(3− l)!
||(P 3−lψ1)(P luk)||L2x, t(R2)
l!(3− l)!
||P 3−lψ1||L∞x, t(R2)||P
luk||L2x, t(R2)
l!(3− l)!
||P k+lu||L2x, t(R2)
Ak+l1 k! ≤ cA
2k! ≤ cAk2(k!)2. (4.28)
For the second term on the right side hand we use the same idea of the remark above,
using the dilation operator P. Indeed,
||t ∂3x(ψ1 uk)||H−5/2 ≤ ||ψ1 t ∂
xuk||H−5/2(R2) + 3 ||∂
x(t ∂xψ1 · uk)||H−5/2(R2)
+ 3 ||∂x(t ∂2xψ1 · uk)||H−5/2(R2) + ||t (∂
xψ1)uk||H−5/2(R2). (4.29)
The last three term are bounded by the following:
||∂xψ1||L∞x, t(R2) + ||∂
xψ1||L∞x, t(R2) + ||∂
xψ1||L∞x, t(R2)
||ψ uk||L2x, t(R2)
≤ cAk3 k! ≤ cAk3 (k!)2. (4.30)
On the other hand, using
||ψ1 t ∂3xuk||H−5/2(R2) ≤
||ψ1 Puk||L2(R:L2x(R)) +
||xψ1 ∂xuk||H−5/2(R2) + ||t ψ1 Bk||H−5/2(R2)
= F1 + F2 + F3. (4.31)
F1 ≤ c ||ψ1||L∞x, t(R2) ||ψ P
k+1u||L2x, t(R2) ≤ c ||ψ P
k+1v||L2x, t(R2)
≤ cAk+14 (k + 1)! ≤ cA
5 k! ≤ cAk5 (k!)2, (4.32)
Analycity for the coupled system of KdV equations 19
F2 ≤ ||xψ1 ∂xvk||L2(R:H−1x (R))
≤ ||∂x(xψ1 vk)||L2(R:H−1x (R)) + ||∂x(xψ1)ψ vk||L2(R:H−1x (R))
≤ ||xψ1 vk||L2x, t(R2) + ||∂x(xψ1)||L∞x, t(R2)||ψ vk||L2x, t(R2)
||xψ1||L∞x, t(R2) + ||∂x(xψ1))||L∞x, t(R2)
||ψ vk||L2x, t(R2)
≤ cAk6 k! ≤ cAk6 (k!)2. (4.33)
Using Lemma 2.5(case σ = −5/2, s = 5, r = −5/2)
F3 = ||t ψ1 Bk||H−5/2(R2) ≤ c1 ||ψ1||H5(R2)||ψ
2 Bx||H−5/2(R2)
and replacing Bk by (2.10), we have
F3 ≤ c1
k=k1+k2+k3
k1! k2! k3!
2k1 ||ψ uk2ψ uk3 ||H−3/2(R2)
k′1! k
1 ||ψ vk′
ψ vk′
||H−3/2(R2)
+ c1 |c|
k=k′′
k′′1 ! k
2 ! k
1 ||ψ uk′′
ψ vk′′
||H−3/2(R2)
k=k1+k2+k3
k1! k2! k3!
2k1 ||ψ uk2 ||L2(R2)||ψ uk3 ||L2(R2)
k′1! k
1 ||ψ vk′
||L2(R2)||ψ vk′
||L2(R2)
+ c1 |c|
k=k′′
k′′1 ! k
2 ! k
1 ||ψ uk′′
||L2(R2)||ψ vk′′
||L2(R2)
k=k1+k2+k3
k1! k2! k3!
2k1 Ak27 k2!A
7 k3!
k′1! k
+ c1 |c|
k=k′′
k′′1 ! k
2 ! k
k=k1+k2+k3
Ak2+k37 + c1
+ c1 |c| k!
k=k′′
k′′1 !
20 M Sepúlveda and O Vera
and then
F3 ≤ c1
k!Ak7
k−k1∑
A− k17 + c1
k!Ak8
+ c1 |c| k!
k=k′′
k′′1 !
e2/A7 Ak7 (k + 1)! + c1
e3/A8 Ak8 (k + 1)!
+ c1 |c| k!
k=k′′
k′′1 !
10 . (4.34)
Replacing (4.30), (4.35) and (4.29) in (4.31) we obtain
||ψ1 t ∂3xuk||H−5/2(R2)
≤ c2Ak11 k! + c1 |c| k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . . (4.35)
Replacing (4.30) and (4.35) in (4.29)
||t ∂3x(ψ1 uk)||H−5/2(R2)
≤ c3Ak12 k! + c1 |c| k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . . (4.36)
Now replacing (4.28) and (4.36) in (4.27) we obtain
|| < Dx, t >3 ψ uk||H− 5/2(R2)
≤ c4Ak13 k! + c1 |c| k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . . (4.37)
In particular
||ψ uk||H1/2(R2)
≤ c5Ak14 k! + c1 |c| k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . . (4.38)
Using a similar argument as above for || < Dx, t >3 ψ P ku||H− 3/2(R2) with µ = 3/2 in
(2.23) and replacing the support of the cut-off function ψǫ we obtain
||ψ uk||H3/2(R2)
≤ c5Ak14 k! + c1 |c| k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . . (4.39)
Analycity for the coupled system of KdV equations 21
In a similar way we have
||ψ vk||H3/2(R2)
≤ c5Ak15 k! + c1 |c̃| k!
k=k′′
k′′1 !
10 , k = 0, 1, 2, . . . (4.40)
Adding (4.39) with (4.40) and performing straightforward calculations as (4.22) we obtain
||ψ uk||H3/2(R2) + ||ψ vk||H3/2(R2) ≤ C A
k (k!)2, k = 0, 1, 2, . . . (4.41)
To obtain the estimate for ||ψ P ku||H7/2(R2) and ||ψ P kv||H7/2(R2) we repeat the above
method with µ = 7/2.
Proposition 4.3. Suppose that
||ψ uk||H7/2(R2) ≤ cA
1 (k!)
2, k = 0, 1, 2, . . . (4.42)
||ψ vk||H7/2(R2) ≤ cA
2 (k!)
2, k = 0, 1, 2, . . . (4.43)
then we have
t∈[t0−ǫ, t0+ǫ]
||(t1/3∂x)P ku||H1(x0−ǫ, x0+ǫ) ≤ c1A
3 [ (k + l)!]
2, k, l = 0, 1, 2, . . .(4.44)
t∈[t0−ǫ, t0+ǫ]
||(t1/3∂x)P kv||H1(x0−ǫ, x0+ǫ) ≤ c1A
4 [ (k + l)! ]
2, k, l = 0, 1, 2, . . .(4.45)
where ǫ > 0 is so small that ψ ≡ 1 near I = (x0 − ǫ, x0 + ǫ)× (t0 − ǫ, t0 + ǫ).
Proof. Let It0 = (t0 − ǫ, t0 + ǫ) and Ix0 = (x0 − ǫ, x0 + ǫ), then we have I = Ix0 × It0 . For
any fixed t ∈ Ix0 , let L = t1/3∂x. We show that for some positive constants c and A0 the
following inequality holds
||LlP ku||H1x(Ix0 ) ≤ cA
0 [ (k + l)! ]
2, ∀ k, ∀ l = 0, 1, 2, . . . (4.46)
Now, let use induction over l. By the trace theorem, we have
||LlP ku||H1x(Ix0 ) ≤ ||t
l/3 ∂lxP
ku(t)||H1x(Ix0) ≤ (t0 + ǫ)
l/3||∂lxP ku||H3/2(Ix0×It0)
≤ (t0 + ǫ)l/3||P ku||H7/2(Ix0×It0) ≤ (t0 + ǫ)
l/3||ψ P ku||H7/2(R2)
≤ (t0 + ǫ)l/3 c1Ak1 k! ≤ (t0 + ǫ)l/3 c1Ak+l0 (k + l)
≤ (t0 + ǫ)l/3 c1Ak+l0 [ (k + l)! ]
2. (4.47)
where we take c = (t0 + ǫ)
l/3c1 and A0 = max{1, A1}. Hence, in the case l = 0, 1, 2, it is
easy to show that (4.46) follows directly from the assumption.
22 M Sepúlveda and O Vera
Now, we assume that (4.46) is true to l ≥ 2. Applying P k to the equation (2.4), we have
ku) + ∂3x(P
ku) = LP ku
= (P + 3)kLu
= (P + 3)k(∂tu+ ∂
= −(P + 3)k
2) + c ∂x(u v)
(P + 3)k∂x(u
(P + 3)k∂x(v
2)− c (P + 3)k∂x(u v)
∂x(P + 2)
k(u2)−
∂x(P + 2)
k(v2)− c ∂x(P + 2)k(u v)
such that
t ∂t(P
ku) + t ∂3x(P
ku) = − a
t ∂x(P + 2)
k(u2)− b
t ∂x(P + 2)
k(v2)− c t ∂x(P + 2)k(u v).(4.48)
Moreover, P = 3 t ∂t + x ∂x. Then
t ∂t(P
ku) =
P k+1u−
x ∂x(P
ku). (4.49)
Replacing (4.49) in (4.48) we obtain
L3P ku = t ∂3x(P ku) = −
P k+1u+
x ∂x(P
t ∂x(P + 2)
k(u2)−
t ∂x(P + 2)
k(v2)− c t ∂x(P + 2)k(u v). (4.50)
Hence, applying Ll−2 we have
||Ll+1P ku||H1x(Ix0 ) = ||L
l−2L3P ku||H1x(Ix0)
||Ll−2 P k+1u||H1x(Ix0 ) +
||Ll−2 x ∂x(P ku)||H1x(Ix0 )
||tLl−2 ∂x(P + 2)k(u2)||H1x(Ix0) +
||tLl−2 ∂x(P + 2)k(v2)||H1x(Ix0)
+ |c| ||tLl−2 ∂x(P + 2)k(u v)||H1x(Ix0 )
= F1 + F2 + F3 + F4 + F5. (4.51)
Using the induction assumption, we obtain
k+l+1
14 (k + l + 1)!. (4.52)
We estimate the term Ll−2(x ∂x) for l ≥ 3. Let r = l − 2, then we estimate Lr(x ∂x) for
r ≥ 1.
∂rx(x ∂x) =
∂r−kx (x ) · ∂kx( ∂x ). (4.53)
∂r−kx (x ) =
1 if k = r − 1
0 if k ≤ r − 2
Analycity for the coupled system of KdV equations 23
then in (4.53) we obtain
∂rx(x ∂x) = r ∂
x ( ∂x ) + x ∂
x( ∂x ) = r ∂
x + x ∂x( ∂
= (l − 2) ∂(l−2)x + x ∂x( ∂(l−2)x ),
that is, Ll−2(x ∂x) = x ∂xLl−2 + (l − 2)Ll−2, for l ≥ 3. For F2 we have
F2 ≤ ||x ∂xLl−2P ku||H1x(Ix0 ) + (l − 2) ||L
l−2P ku||H1x(Ix0 )
≤ ||x t−1/3Ll−1P ku||H1x(Ix0 ) + (l − 2) ||L
l−2P ku||H1x(Ix0)
≤ c (t0 − ǫ) (|x0|+ ǫ+ 1) ||Ll−1P ku||H1x(Ix0 ) + (l − 2) ||L
l−2P ku||H1x(Ix0 )
≤ (t0 − ǫ)−1/3(|x0 + ǫ+ 1) c1 Ak+l−114 (k + l − 1)! + c1A
k+l−1
14 (l − 2) (k + l − 1)!
k+l+1
14 (k + l + 1)! (4.54)
where we take A14 larger than (t0 − ǫ)−1/3(|x0|+ ǫ+ 1) and 3. Using that (L = t1/3 ∂3x)
tLl−2∂x = t t(l−2)/3∂(l−2)x ∂x = t t−1/3 t(l−1)/3∂(l−1)x = t2/3 Ll−1,
we have
||t2/3 Ll−1 (P + 2)k(u2)||H1x(Ix0 )
(t0 + ǫ)
l−1=l1+l2
k=k1+k2+k3
(l − 1)!
l1!l2!
k1!k2!k3!
× c2 ||Ll1P k1u||H1x(Ix0) ||L
l2P k2u||H1x(Ix0).
Using the induction assumption
(t0 + ǫ)
l−1=l1+l2
k=k1+k2+k3
1 k! (l − 1)!
(l1 + k1)!
l1! k1!
(l2 + k2)!
l2! k2!
Ak+l−114
(t0 + ǫ)
2/3 c2 c
1 (l + k − 1)!Ak+l−114
l−1=l1+l2
k=k1+k2+k3
× (l1 + k1)!
l1! k1!
(l2 + k2)!
l2! k2!
k! (l − 1)!
(l + k − 1)!
Using that
l−1=l1+l2
k=k1+k2+k3
(l1 + k1)!
l1! k1!
(l2 + k2)!
l2! k2!
k! (l − 1)!
(l + k − 1)!
≤ e2(l + k)!
we obtain
F3 ≤ (t0 + ǫ)2/3 c2 c31 e2 (l + k)!Ak+l−114 ≤
k+l+1
14 (k + l + 1)! (4.55)
where we take A14 larger than (t0 − ǫ)−1/3 c2 c21 e2, and 3. In a similar way
k+l+1
15 (k + l + 1)! (4.56)
24 M Sepúlveda and O Vera
where we take A15 larger than (t0 − ǫ)−1/3 c4 c23 e2, and 3. Finally, in a similar way
k+l+1
16 (k + l + 1)! (4.57)
where we take A16 larger than (t0 − ǫ)−1/3 c6 c25 e2, and 3. Therefore, from (4.52), (4.54),
(4.55), (4.56) and (4.57) we obtain
||Ll+1P ku||H1x(Ix0) ≤ c7A
k+l+1
17 (k + l + 1)!. (4.58)
In a similar way, we obtain
||Ll+1P kv||H1x(Ix0) ≤ c7A
k+l+1
17 (k + l + 1)!, (4.59)
and the result follows.
Proposition 4.4. Suppose that there exists a positive constants c1, c2 and A14, A15 such
t∈[t0−ǫ, t0+ǫ]
||∂lxP ku||H1x(x0−ǫ, x0+ǫ) ≤ c1A
14 [ (k + l)! ]
2, k, l = 0, 1, 2, . . . (4.60)
t∈[t0−ǫ, t0+ǫ]
||∂lxP kv||H1x(x0−ǫ, x0+ǫ) ≤ c2A
15 [ (k + l)! ]
2, k, l = 0, 1, 2, . . . (4.61)
Then we have respectively
t∈[t0−ǫ, t0+ǫ]
||∂mt ∂lxu||H1x(x0−ǫ, x0+ǫ) ≤ c3A
16 [ (m+ l)! ]
2, m, l = 0, 1, 2, . . . (4.62)
t∈[t0−ǫ, t0+ǫ]
||∂mt ∂lxv||H1x(x0−ǫ, x0+ǫ) ≤ c4A
17 [ (m+ l)! ]
2, m, l = 0, 1, 2, . . . (4.63)
where c3, c4 and A16, A17 only depend on c1, c2 and A14, A15, respectively and ǫ, (x0, t0).
Proof. Using the idea of Proposition 4.3, we fix t ∈ Ix0 . First we show that for some
positive constants c3, A16 and B16
||(x ∂x)m ∂lxP kv||H1x(Ix0) ≤ c3A
k+m+l
16(k +m+ l)!, k, m, l = 0, 1, 2, . . . (4.64)
We use induction. Suppose that (4.64) is true for m.
||(x ∂x)m+1 ∂lxP kv||H1x(Ix0)
= ||(x ∂x) (x ∂x)m ∂lxP kv||H1x(Ix0)
≤ (|x0|+ ǫ+ 1) ||(x ∂x + I)m ∂l+1x P kv||H1x(Ix0 )
≤ c(|x0|, ǫ)
||(x ∂x)j ∂l+1x P kv||H1x(Ix0 )
k+l+j+1
16(k + l + j + 1)!
≤ c3Ak+l+m+116 B
16(k + l +m+ 1)!
(A16 B16)
−(m−j)
(m− j)!
(k + l + j + 1)!
(k + l +m+ 1)!
≤ e−A16 B16c3Ak+l+m+116 B
16(k + l +m+ 1)! (4.65)
Analycity for the coupled system of KdV equations 25
where we take B16 so large that B16 ≥ max{|x0|+ǫ+1, 1}.We show that for some positive
constants c4, A17 we have
||(t ∂t)m ∂lxu||H1x(Ix0 ) ≤ c4A
17 (l +m)!, l, m = 0, 1, 2, . . .
Using that t ∂t =
(P − x ∂x), we obtain
||(t ∂t)m ∂lxu||H1x(Ix0 ) = 3
−m ||(P − x ∂x)m ∂lxu||H1x(Ix0 )
≤ 3−m
m=j1+j2
j1! j2!
||(x ∂x)j1 P j2∂lxu||H1x(Ix0 )
≤ 3−m
m=j1+j2
j1! j2!
||(x ∂x)j1 ∂lx(P − l)j2u||H1x(Ix0)
≤ 3−m
m=j1+j2+j3
j1! j2! j3!
lj3 ||(x ∂x)j1 ∂lxP j2u||H1x(Ix0).
where we replace j2 into j2 + j3. Now, using the induction hypothesis we have (with
B17 ≥ A16B16)
||(t ∂t)m ∂lxu||H1x(Ix0 )
≤ 3−m
m=j1+j2+j3
j1! j2! j3!
lj3 c3B
j1+j2+l
17 (j1 + j2 + l)!
≤ 3−m c3Bm+l17 (m+ l)!
m=j1+j2+j3
j1! j2! j3!
(j1 + j2 + l)!
(m+ l)!
, (4.66)
Observing that lj3
(j1 + j2 + l)!
(m+ l)!
≤ 1, we obtain in (4.66)
||(t ∂t)m ∂lxu||H1x(Ix0 ) ≤ 3
−m c3 (2 +B
mBl+m17 (l +m)!
≤ c4Al+m17 (l +m)!
where we take A17 = max{B17, 3−1B17 (2 +B−117 )}. We show that for some positive con-
stants c4, A18 and B18 we have
||(t ∂t)j ∂mt ∂lxu||H1x(Ix0 ) ≤ c4A
j+m+l
18 B18(j +m+ l)!, j, l, m = 0, 1, 2, . . . (4.67)
Induction in m.
||(t ∂t)j ∂m+1t ∂lxu||H1x(Ix0 ) ≤ ||∂t(t ∂t − I)
m ∂mt ∂
xu||H1x(Ix0)
= t−1 ||t ∂t(t ∂t − I)j ∂mt ∂lxu||H1x(Ix0)
≤ (t0 − ǫ)−1
||(t ∂t)j1+1 ∂mt ∂lxu||H1x(Ix0 ).
26 M Sepúlveda and O Vera
Using the induction hypothesis
||(t ∂t)j ∂m+1t ∂lxu||H1x(Ix0 )
≤ (t0 − ǫ)−1
j1+l+m+1
18 (j1 + l +m+ 1)!
= c4 (t0 − ǫ)−1Aj1+l+m+118 B
18 (j1 + l +m+ 1)!
−(j−j1
(j − j1)!
(j1 +m+ l + 1)! (j − j1)!
(j +m+ l + 1)!
= c4 (t0 − ǫ)−1 e−A18 Aj+l+m+118 B
18 (j + l +m+ 1)!
≤ c4 Aj+l+m+118 B
18 (j + l +m+ 1)!
where we take B18 larger than (t0− ǫ)−1 e−A18 . Finally, we choose j = 0 in (4.67) and take
c2 = c4 and A15 = A18B18. The result of analyticity follows.
Acknowledgments
This work has been supported by Fondap in Applied Mathematics (Project # 15000001),
CNPq/CONICYT Project, # 490987/2005-2 (Brazil) and # 2005-075 (Chile).
References
[1] Ash J M, Cohen J and Wang G, On Strongly Interacting Internal Solitary Waves.
The Journal of Fourier Analysis and Applications 2 (1996), 507–517.
[2] Bisognin E, Bisognin V and Perla Menzala G, Asymptotic Behaviour in Time
of the Solutions of a Coupled System of KdV Equations, Funkcialaj Ekvacioj 40
(1997), 353–370.
[3] Bisognin E, Bisognin V, Sepúlveda and Vera O, Coupled system of Korteweg de
Vries equations type in domains with moving boundaries. Technical Report, 2006-33,
Departamento de Ingenieria Matemática, U. de Concepción, 2006.
[4] Bona J, Ponce G, Saut G C and Tom M M, A model system for strong interaction
between internal solitary waves, Comm. Math. Phys. Appl. Math. 143 (1992) 287–313.
[5] Bourgain J, Fourier restriction phenomena for certain lattice subsets and applica-
tions to nonlinear evolution equations, I Schrödinger equation, Geom. and Functional
Anal. 3 (1993), 107–156.
[6] Craig W, Kappeler T and Strauss W, Gain of regularity for equations of Ko-
rteweg - de Vries type, Ann. Inst. Henri Poincaré 2 (1992), 147–186.
[7] Dávila M, Continuação única para um Sistema Acoplado de Equações do tipo Ko-
rteweg - de Vries e para as Equações de Benjamin-Bona-Mahony e de Boussinesq,
Tese de Doutorado. IM-UFRJ. Brasil. 1995.
Analycity for the coupled system of KdV equations 27
[8] de Bouard A, Hayashi N and Kato K, Gevrey regularizing effect for the (gener-
alized) Korteweg-de Vries equation and nonlinear Schrödinger equations, Ann. Inst.
Henri Poincaré, Anal. non-linéaire 9 (1995), 673–725.
[9] Gear J A andGrimshaw R, Weak and strong interactions between internal solitary,
waves, Comm. Math. 70 (1984), 235–258.
[10] Kajitani K and Wakabayashi S, Analytically smoothing effect for Schrödinger
type equations with variable coefficients, Direct and inverse problems of mathematical
physics (Newark, DE, 1997), Int. Soc. Anal. Appl. Comput., 5, Kluwer Acad. Publ.,
Dordrecht, 2000, 185–219.
[11] Kato K and Taniguchi K, Gevrey regularizing effect for nonlinear Schrödinger
equations, Osaka J. Math. 33 (1996), 863–880.
[12] Kato K and Ogawa T, Analyticity and Smoothing effect for the Korteweg - de
Vries equation with a single point singularity, Math. Ann. 316 (2000), 577–608.
[13] Kato T, On the Cauchy problem for the ( generalized ) Korteweg - de Vries equations,
Studies in applied mathematics, Adv. Math. Suppl. Stud., 8, Academic Press, New
York, 1983, 93–128.
[14] Kato T and Masuda K, Nonlinear evolution equations and analyticity, I Ann. Inst.
Henri Poincaré. Anal. non-linéaire 3 (1986) 455–467.
[15] Kato T and Ponce G, Commutator estimates and the Euler and Navier-Stokes
equations, Comm. Pure Applied Math., 41(1988) 891–907.
[16] Kenig C, Ponce G and Vega L, A bilinear estimate with applications to the KdV
equation, J. Amer. Math. Soc.9 (1996), 573–603.
[17] Kenig C, Ponce G andVega L, Well-posedness and scattering results for the gener-
alized Korteweg - de Vries equation via the contraction mapping principle, Commun.
Pure Appl. Math. 46 (1993), 527–620.
[18] Kenig C, Ponce G and Vega L, On the (generalized) Korteweg - de Vries equation,
Duke Math. J.59 (1989), 585–610.
[19] Kenig C, Ponce G and Vega L, Oscillatory integrals and regularity equations,
Indiana Univ. Math. J. 40 (1991), 33–69.
[20] Kumano-go H, Pseudo differential operators, Iwanami, Japan 1979, (English trans-
lation) MIT Press, cambridge, Massachusetts, 1981.
[21] Linares F and Panthee M, On the Cauchy problem for a coupled system of KdV
equations. Comm. on Pure and Appl. An. 3 (2004), 417–431.
[22] Taylor M, Pseudo Differential Operators, Prinston Univ. Press 1978.
[23] Vera O, Gain of regularity for a coupled system of nonlinear evolution dispersive
equations type, Ph. D. Thesis, UFRJ, Rio de Janeiro, Brazil, 2001.
	Introduction
	Reduction of the Problem and Preliminary Results
	Existence and Well-Posedness
	The main result
