Electronic Journal of Differential Equations, Vol. 2006(2006), No. 07, pp. 1–20.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
GLOBAL WELL-POSEDNESS OF NLS-KDV SYSTEMS FOR
PERIODIC FUNCTIONS
CARLOS MATHEUS
Abstract. We prove that the Cauchy problem of the Schrödinger-KortewegdeVries (NLS-KdV) system for periodic functions is globally well-posed for
initial data in the energy space H 1 ×H 1 . More precisely, we show that the nonresonant NLS-KdV system is globally well-posed for initial data in H s (T) ×
H s (T) with s > 11/13 and the resonant NLS-KdV system is globally wellposed with s > 8/9. The strategy is to apply the I-method used by Colliander,
Keel, Staffilani, Takaoka and Tao. By doing this, we improve the results by
Arbieto, Corcho and Matheus concerning the global well-posedness of NLSKdV systems.
1. Introduction
We consider the Cauchy problem of the Schrödinger-Korteweg-deVries (NLSKdV) system
i∂t u + ∂x2 u = αuv + β|u|2 u,
∂t v + ∂x3 v + 21 ∂x (v 2 ) = γ∂x (|u|2 ),
(1.1)
u(x, 0) = u0 (x), v(x, 0) = v0 (x), t ∈ R.
This system appears naturally in fluid mechanics and plasma physics as a model of
interaction between a short-wave u = u(x, t) and a long-wave v = v(x, t).
In this paper we are interested in global solutions of the NLS-KdV system for
rough initial data. Before stating our main results, let us recall some of the recent
theorems of local and global well-posedness theory of the Cauchy problem (1.1).
For continuous spatial variable (i.e., x ∈ R), Corcho and Linares [5] recently
proved that the NLS-KdV system is locally well-posed for initial data (u0 , v0 ) ∈
H k (R) × H s (R) with k ≥ 0, s > −3/4 and
k − 1 ≤ s ≤ 2k − 1/2
if k ≤ 1/2,
k − 1 ≤ s < k + 1/2
if k > 1/2.
Furthermore, they prove the global well-posedness of the NLS-KdV system in the
energy H 1 × H 1 using three conserved quantities discovered by Tsutsumi [7], whenever αγ > 0.
2000 Mathematics Subject Classification. 35Q55.
Key words and phrases. Global well-posedness; Schrödinger-Korteweg-de Vries system;
I-method.
c
2006
Texas State University - San Marcos.
Submitted November 13, 2006. Published January 2, 2007.
1
2
C. MATHEUS
EJDE-2006/07
Also, Pecher [6] improved this global well-posedness result by an application of
the I-method of Colliander, Keel, Stafillani, Takaoka and Tao (see for instance [3])
combined with some refined bilinear estimates. In particular, Pecher proved that,
if αγ > 0, the NLS-KdV system is globally well-posed for initial data (u0 , v0 ) ∈
H s × H s with s > 3/5 in the resonant case β = 0 and s > 2/3 in the non-resonant
case β 6= 0.
On the other hand, in the periodic setting (i.e., x isn the space of periodic
functions T), Arbieto, Corcho and Matheus [1] proved the local well-posedness of
the NLS-KdV system for initial data (u0 , v0 ) ∈ H k × H s with 0 ≤ s ≤ 4k − 1 and
−1/2 ≤ k − s ≤ 3/2. Also, using the same three conserved quantities discovered by
Tsutsumi, one obtains the global well-posedness of NLS-KdV on T in the energy
space H 1 × H 1 whenever αγ > 0.
Motivated by this scenario, we combine the new bilinear estimates of Arbieto,
Corcho and Matheus [1] with the I-method of Tao and his collaborators to prove
the following result.
Theorem 1.1. The NLS-KdV system (1.1) on T is globally well-posed for initial
data (u0 , v0 ) ∈ H s (T) × H s (T) with s > 11/13 in the non-resonant case β 6= 0 and
s > 8/9 in the resonant case β = 0, whenever αγ > 0.
The paper is organized as follows. In the section 2, we discuss the preliminaries for the proof of the theorem 1.1: Bourgain spaces and its properties, linear
estimates, standard estimates for the non-linear terms |u|2 u and ∂x (v 2 ), the bilinear estimates of Arbieto, Corcho and Matheus [1] for the coupling terms uv and
∂x (|u|2 ), the I-operator and its properties. In the section 3, we apply the results of
the section 2 to get a variant of the local well-posedness result of [1]. In the section
4, we recall some conserved quantities of (1.1) and its modification by the introduction of the I-operator; moreover, we prove that two of these modified energies
are almost conserved. Finally, in the section 5, we combine the almost conservation
results in section 4 with the local well-posedness result in section 3 to conclude the
proof of the theorem 1.1.
2. Preliminaries
A successful procedure to solve some dispersive equations (such as the nonlinear
Schrödinger and KdV equations) is to use the Picard’s fixed point method in the
following spaces:
Z X
1/2
kf kX k,b :=
hni2k hτ + n2 i2b |fb(n, τ )|dτ
= kU (−t)f kHtb (R,Hxk ) ,
n∈Z
kgkY s,b :=
Z X
hni2s hτ − n3 i2b |b
g (n, τ )|dτ
1/2
= kV (−t)f kHtb (R,Hxs ) ,
n∈Z
2
3
where h·i := 1 + | · |, U (t) = eit∂x and V (t) = e−t∂x . These spaces are called
Bourgain spaces. Also, we introduce the restriction in time norms
kf kX k,b (I) := inf kfekX k,b
fe|I =f
and kgkY s,b (I) := inf ke
g kY s,b
g
e|I =g
where I is a time interval.
The interaction of the Picard method has been based around the spaces Y s,1/2 .
Because we are interested in the continuity of the flow associated to (1.1) and the
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GLOBAL PERIODIC SOLUTIONS FOR NLS-KDV SYSTEMS
3
s
Y s,1/2 norm do not control the L∞
t Hx norm, we modify the Bourgain spaces as
follows:
kukX k := kukX k,1/2 + khnik u
b(n, τ )kL2n L1τ ,
kvkY s := kvkY s,1/2 + khnis vb(n, τ )kL2n L1τ
and, given a time interval I, we consider the restriction in time of the X k and Y s
norms
kukX k (I) := inf ke
ukX k and kvkY s (I) := inf ke
v kY s
u
e|I =u
v
e|I =v
Furthermore, the mapping properties of U (t) and V (t) naturally leads one to
consider the companion spaces
hnik u
b(n, τ ) ,
kukZ k := kukX k,−1/2 + hτ + n2 i L2n L1τ
hnis vb(n, τ ) kvkW s := kvkY s,−1/2 + hτ − n3 i L2n L1τ
In the sequel, ψ denotes a non-negative smooth bump function supported on [−2, 2]
with ψ = 1 on [−1, 1] and ψδ (t) := ψ(t/δ) for any δ > 0.
Notation. Fix (k, s) a pair of indices such that the local well-posedness of the
periodic NLS-KdV system holds. Given two non-negative real numbers A and B, we
write A . B whenever A ≤ C·B, where C = C(k, s) is a constant which may depend
only on (k, s). Also, we write A & B if A ≥ c · B, where c = c(k, s) is sufficiently
small (depending only on (k, s)), and A ∼ B if A . B . A. Furthermore, we use
A B to mean A ≤ cḂ where c = c(k, s) is a small constant (depending only on
(k, s)), and A B to denote A ≥ C · B with C = C(k, s) a large constant. Finally,
given, for instance, a function ψ and a number b, we put also A .ψ,b B to mean
A ≤ C · B where C = C(k, s, ψ, b) is a constant depending also on the specified
function ψ and number b (besides (k, s)).
Next, we recall some properties of the Bourgain spaces:
Lemma 2.1. X 0,3/8 ([0, 1]), Y 0,1/3 ([0, 1]) ⊂ L4 (T × [0, 1]). More precisely,
kψ(t)f kL4xt . kf kX 0,3/8
and
kψ(t)gkL4xt . kgkY 0,1/3 .
For the proof of the above lemma see [2]. Another basic property of these spaces
are their stability under time localization:
b
s b
2
Lemma 2.2. Let Xτs,b
=h(ξ) := {f : hτ − h(ξ)i hξi |f (τ, ξ)| ∈ L }. Then
kψ(t)f kX s,b
τ =h(ξ)
.ψ,b kf kX s,b
τ =h(ξ)
0
for any s, b ∈ R. Moreover, if −1/2 < b ≤ b < 1/2, then for any 0 < T < 1, we
have
0
kψT (t)f kX s,b0 .ψ,b0 ,b T b−b kf kX s,b .
τ =h(ξ)
τ =h(ξ)
Proof. First of all, note that hτ − τ0 − h(ξ)ib .b hτ0 i|b| hτ − h(ξ)ib , from which we
obtain
keitτ0 f kX s,b
.b hτ0 i|b| kf kX s,b .
τ =h(ξ)
τ =h(ξ)
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C. MATHEUS
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b 0 )eitτ0 dτ0 , we conclude
ψ(τ
Z
b 0 )|hτ0 i|b| kf k s,b
kψ(t)f kX s,b
.b
|ψ(τ
X
Using that ψ(t) =
R
τ =h(ξ)
.
τ =h(ξ)
Since ψ is smooth with compact support, the first estimate follows.
Next we prove the second estimate. By conjugation we may assume s = 0 and,
by composition it suffices to treat the cases 0 ≤ b0 ≤ b or ≤ b0 ≤ b ≤ 0. By duality,
we may take 0 ≤ b0 ≤ b. Finally, by interpolation with the trivial case b0 = b, we
may consider b0 = 0. This reduces matters to show that
kψT (t)f kL2 .ψ,b T b kf kX 0,b
τ =h(ξ)
for 0 < b < 1/2. Partitioning the frequency spaces into the cases hτ − h(ξ)i ≥ 1/T
and hτ − h(ξ) ≤ 1/T , we see that in the former case we’ll have
kf kX 0,0
τ =h(ξ)
≤ T b kf kX 0,b
τ =h(ξ)
and the desired estimate follows because the multiplication by ψ is a bounded
operation in Bourgain’s spaces. In the latter case, by Plancherel and CauchySchwarz
kf (t)kL2x . kfd
(t)(ξ)kL2ξ
Z
.
|fb(τ, ξ)|dτ )L2
ξ
hτ −h(ξ)i≤1/T
Z
.b T b−1/2 hτ − h(ξ)i2b |fb(τ, ξ)|2 dτ )1/2 L2
ξ
=T
b−1/2
kf kX s,b
.
τ =h(ξ)
Integrating this against ψT concludes the proof of the lemma.
Also, we have the following duality relationship between X k (resp., Y s ) and Z k
(resp., W s ):
Lemma 2.3. We have
Z
χ[0,1] (t)f (x, t)g(x, t) dx dt . kf kX s kgkZ −s ,
Z
χ[0,1] (t)f (x, t)g(x, t) dx dt . kf kY s kgkW −s
for any s and any f, g on T × R.
Proof. See [4, p. 182–183] (note that, although this result is stated only for the
spaces Y s and W s , the same proof adapts for the spaces X k and Z k ).
Now, we recall some linear estimates related to the semigroups U (t) and V (t):
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5
Lemma 2.4 (Linear estimates). It holds
kψ(t)U (t)u0 kZ k . ku0 kH k ,
kψ(t)V (t)v0 kW s . kv0 kH s ;
Z t
kψT (t)
U (t − t0 )F (t0 )dt0 kX k . kF kZ k ,
0
Z t
kψT (t)
V (t − t0 )G(t0 )dt0 kY s . kGkW s .
0
For a proof of the above lemma, see [3], [4] or [1]. Furthermore, we have the
following well-known multiinear estimates for the cubic term |u|2 u of the nonlinear
Schrödinger equation and the nonlinear term ∂x (v 2 ) of the KdV equation:
Lemma 2.5. kuvwkZ k . kuk
3
X k, 8
kvk
3
X k, 8
kwk
3
X k, 8
for any k ≥ 0.
For the proof of the above lemma, see See [2] and [1].
Lemma 2.6. k∂x (v1 v2 )kW s . kv1 k s, 13 kv2 k s, 21 + kv1 k s, 12 kv2 k s, 13 for any s ≥
Y
Y
Y
Y
−1/2, if v1 = v1 (x, t) and v2 = v2 (x, t) are x-periodic functions having zero x-mean
for all t.
The proof of the above lemma can be found in [2], [3] and [1]. Next, we revisit the
bilinear estimates of mixed Schrödinger-Airy type of Arbieto, Corcho and Matheus
[1] for the coupling terms uv and ∂x (|u|2 ) of the NLS-KdV system.
Lemma 2.7. kuvkZ k . kuk k, 38 kvk s, 12 + kuk k, 12 kvk s, 31 whenever s ≥ 0 and
X
Y
X
Y
k − s ≤ 3/2.
Lemma 2.8. k∂x (u1 u2 )kW s . ku1 kX k,3/8 ku2 kX k,1/2 + ku1 kX k,1/2 ku2 kX k,3/8 whenever 1 + s ≤ 4k and k − s ≥ −1/2.
Remark 2.9. Although the lemmas 2.7 and 2.8 are not stated as above in [1], it
is not hard to obtain them from the calculations of Arbieto, Corcho and Matheus.
Finally, we introduce the I-operator: let m(ξ) be a smooth non-negative symbol
on R which equals 1 for |ξ| ≤ 1 and equals |ξ|−1 for |ξ| ≥ 2. For any N ≥ 1 and
α
the spatial Fourier multiplier
α ∈ R, denote by IN
ξ α b
α
f (ξ).
Id
N f (ξ) = m
N
For latter use, we recall the following general interpolation lemma.
Lemma 2.10 ([4, Lemma 12.1]). Let α0 > 0 and n ≥ 1. Suppose Z, X1 , . . . , Xn
are translation-invariant Banach spaces and T is a translation invariant n-linear
operator such that
n
Y
α
kI1 T (u1 , . . . , un )kZ .
kI1α uj kXj ,
j=1
for all u1 , . . . , un and 0 ≤ α ≤ α0 . Then
α
kIN
T (u1 , . . . , un )kZ .
n
Y
α
kIN
uj kXj
j=1
for all u1 , . . . , un , 0 ≤ α ≤ α0 and N ≥ 1. Here the implicit constant is independent
of N .
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C. MATHEUS
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After these preliminaries, we can proceed to the next section where a variant of
the local well-posedness of Arbieto, Corcho and Matheus is obtained. In the sequel
1−s
we take N 1 a large integer and denote by I the operator I = IN
for a given
s ∈ R.
3. A variant local well-posedness result
This section is devoted to the proof of the following proposition.
R
Proposition 3.1. For any (u0 , v0 ) ∈ H s (T) × H s (T) with T v0 = 0 and s ≥ 1/3,
the periodic NLS-KdV system (1.1) has a unique local-in-time solution on the time
interval [0, δ] for some δ ≤ 1 and
(
16
(kIu0 kX 1 + kIv0 kY 1 )− 3 − , if β 6= 0,
(3.1)
δ∼
(kIu0 kX 1 + kIv0 kY 1 )−8− ,
if β = 0.
Moreover, we have kIukX 1 + kIvkY 1 . kIu0 kX 1 + kIv0 kY 1 .
Proof. We apply the I-operator to the NLS-KdV system (1.1) so that
iIut + Iuxx = αI(uv) + βI(|u|2 u),
Ivt + Ivxxx + I(vvx ) = γI(|u|2 )x ,
Iu(0) = Iu0 ,
Iv(0) = Iv0 .
To solve this equation, we seek for some fixed point of the integral maps
Z t
Φ1 (Iu, Iv) := U (t)Iu0 − i
U (t − t0 ){αI(u(t0 )v(t0 )) + βI(|u(t0 )|2 u(t0 ))}dt0 ,
0
Z t
Φ2 (Iu, Iv) := V (t)Iv0 −
V (t − t0 ){I(v(t0 )vx (t0 )) − γI(|u(t0 )|2 )x }dt0 .
0
The interpolation lemma 2.10 applied to the linear and multilinear estimates in the
lemmas 2.4, 2.5, 2.6, 2.7 and 2.8 yields, in view of the lemma 2.2,
3
1
kΦ1 (Iu, Iv)kX 1 . kIu0 kH 1 + αδ 8 − kIukX 1 kIvkY 1 + βδ 8 − kIuk3X 1 ,
1
1
kΦ2 (Iu, Iv)kY 1 . kIv0 kH 1 + δ 6 − kIvk2Y 1 + γδ 8 − kIuk2X 1 .
3
In particular, these integrals maps are contractions provided that βδ 8 − (kIu0 kH 1 +
1
kIv0 kH 1 )2 1 and δ 8 − (kIu0 kH 1 + kIv0 kH 1 ) 1. This completes the proof. 4. Modified energies
We define the following three quantities:
M (u) := kukL2 ,
(4.1)
Z
L(u, v) := αkvk2L2 + 2γ =(uux )dx,
(4.2)
Z
Z
Z
α
α
βγ
E(u, v) := αγ v|u|2 dx + γkux k2L2 + kvx k2L2 −
v 3 dx +
|u|4 dx. (4.3)
2
6
2
In the sequel, we suppose αγ > 0. Note that
|L(u, v)| . kvk2L2 + M kux kL2 ,
kvk2L2
. |L| + M kux kL2 .
(4.4)
(4.5)
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7
Also, the Gagliardo-Nirenberg and Young inequalities implies
5
kux k2L2 + kvx k2L2 . |E| + |L| 3 + M 8 + 1,
5
3
|E| . kux k2L2 + kvx k2L2 + |L| + M 8 + 1
(4.6)
(4.7)
In particular, combining the bounds (4.4) and (4.7),
10
|E| . kux k2L2 + kvx k2L2 + kvkL32 + M 10 + 1.
(4.8)
Moreover, from the bounds (4.5) and (4.6),
kvk2L2 . |L| + M |E|1/2 + M 6 + 1
(4.9)
kuk2H 1 + kvk2H 1 . |E| + |L|5/3 + M 8 + 1
(4.10)
and hence
d
L(Iu, Iv)
dt
Z
Z
Iv(IvIvx − I(vvx ))dx + 2αγ Iv(I(|u|2 ) − |Iu|2 )x dx
Z
Z
+ 4αγ< Iux (IuIv − I(uv))dx + 4βγ< ((Iu)2 Iu − I(u2 u))Iux dx
= 2α
=:
4
X
(4.11)
Lj .
j=1
and
d
E(Iu, Iv)
dt Z
Z
α
= α (I(vvx ) − IvIvx )Ivxx dx +
(Iv)2 (I(vvx ) − IvIvx )dx+
2
Z
+ 2βγ= (I(|u|2 u)x − ((Iu)2 Iu)x )Iux dx
Z
Z
2
+ αγ |Iu| (IvIvx − I(vvx ))dx + αγ (|Iu|2 − I(|u|2 ))IvIvx dx
Z
Z
+ αγ Ivxx (|Iu|2 − I(|u|2 ))x dx − 2αγ= Iux (I(uv) − IuIv)x dx
Z
Z
2
2
2
2
2
+ αγ
(I(|u| ) − |Iu| )x |Iu| dx + 2α γ= IvIu(I(uv − IuIv))dx
Z
+ 2β 2 γ= Iu(Iu)2 (I(|u|2 u) − (Iu)2 Iu)dx
Z
Z
2
2
− 2αβγ= IvIu(I(|u| u) − Iu(Iu)) dx − 2αβγ= (Iu)2 Iu(I(uv) − IuIv)dx
=:
12
X
Ej
j=1
(4.12)
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4.1. Estimates for the modified L-functional.
Proposition 4.1. Let (u, v) be a solution of (1.1) on the time interval [0, δ]. Then,
for any N ≥ 1 and s > 1/2,
|L(Iu(δ), Iv(δ)) − L(Iu(0), Iv(0))|
19
1
. N −1+ δ 24 − (kIukX 1,1/2 + kIvkY 1,1/2 )3 + N −2+ δ 2 − kIuk4X 1,1/2 .
(4.13)
Proof. Integrating (4.11) with respect to t ∈ [0, δ], it follows that we have to bound
the (integral over [0, δ] of the) four terms on the right hand side. To simplify
the computations, we assume that the Fourier transform of the functions are nonnegative and we ignore the appearance of complex conjugates (since they are irrelevant in our subsequent arguments). Also, we make a dyadic decomposition of
the frequencies |ni | ∼ Nj in many places. In particular, it will be important to get
extra factors Nj0− everywhere in order to sum the dyadic blocks.
Rδ
We begin with the estimate of 0 L1 . It is sufficient to show that
Z δ
m(n + n ) − m(n )m(n ) X
1
2
1
2 vb1 (n1 , t)|n2 |vb2 (n2 , t)vb3 (n3 , t)
m(n
)m(n
)
1
2
0 n +n +n =0
1
2
3
(4.14)
3
Y
−1 65 −
kvj kY 1,1/2
.N δ
j=1
• |n1 | |n2 | ∼ |n3 |, |n2 | & N . In this case, note that
m(n1 + n2 ) − m(n1 )m(n2 ) . | ∇m(n2 ) · n1 | . N1 , if |n1 | ≤ N,
m(n1 )m(n2 )
m(n2 )
N2
m(n1 + n2 ) − m(n1 )m(n2 ) . N1 1/2 , if |n1 | ≥ N.
m(n1 )m(n2 )
N
Hence, using the lemmas 2.1 and 2.2, we obtain
Z
0
δ
3
Y
N1
5
0−
kv1 kL4 k(v2 )x kL4 kv3 kL2 . N −2+ δ 6 − Nmax
kvi kY 1,1/2
L1 .
N2
j=1
if |n1 | ≤ N , and
δ
Z
|
L1 | .
0
3
3
Y
Y
5
5
N1 1/2 1
0−
δ 6−
kvi kY 1,1/2 . N −2+ δ 6 − Nmax
kvi kY 1,1/2 .
N
N1 N3
j=1
j=1
• |n2 | |n1 | ∼ |n3 |, |n1 | & N . This case is similar to the previous one.
• |n1 | ∼ |n2 | & N . The multiplier is bounded by
m(n1 + n2 ) − m(n1 )m(n2 ) . N1 1− .
m(n1 )m(n2 )
N
In particular, using the lemmas 2.1 and 2.2,
Z
|
δ
L1 | .
0
3
Y
5
N1 1−
0−
kv1 kL2 k(v2 )x kL4 kv3 kL4 . N −1+ δ 6 − Nmax
kvi kY 1,1/2 .
N
j=1
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GLOBAL PERIODIC SOLUTIONS FOR NLS-KDV SYSTEMS
Now, we estimate
Z
δ
X
0 n +n +n
1
2
3
Rδ
0
9
L2 . Our task is to prove that
m(n + n ) − m(n )m(n ) 1
2
1
2 u1 (n1 , t)c
u2 (n2 , t)vb3 (n3 , t)
|n1 + n2 |c
m(n
)m(n
)
1
2
=0
19
. N −1+ δ 24 − ku1 kX 1,1/2 ku2 kX 1,1/2 kv3 kY 1,1/2
(4.15)
• |n2 | |n1 | ∼ |n3 | & N . We estimate the multiplier by
m(n + n ) − m(n )m(n ) N2
1
2
1
2 . h( )1/2 i.
m(n1 )m(n2 )
N
Thus, using L2xt L4xt L4xt Hölder inequality and the lemmas 2.1 and 2.2
Z δ
19
1
N2 1/2
i
δ 24 − ku1 kX 1,1/2 ku2 kX 1,1/2 kv3 kY 1,1/2
L2 . h
N
hN2 iN3
0
19
0−
ku1 kX 1,1/2 ku2 kX 1,1/2 kv3 kY 1,1/2 .
. N −1+ δ 24 − Nmax
• |n1 | |n2 | ∼ |n3 |. This case is similar to the previous one.
• |n1 | ∼ |n2 | & N . Estimating the multiplier by
m(n + n ) − m(n )m(n ) N2 1−
1
2
1
2 .
m(n1 )m(n2 )
N
we conclude
Z
δ
L2 .
0
19
N2 1− 1
δ 24 − ku1 kX 1,1/2 ku2 kX 1,1/2 kv3 kY 1,1/2
N
N1 N2
19
0−
ku1 kX 1,1/2 ku2 kX 1,1/2 kv3 kY 1,1/2 .
. N −2+ δ 24 − Nmax
Next, let us compute
Z
0
δ
Rδ
0
L3 . We claim that
m(n + n ) − m(n )m(n ) 1
2
1
2 u1 (n1 , t)vb2 (n2 , t)|n3 |c
u3 (n3 , t)
c
m(n
)m(n
)
1
2
(4.16)
n1 +n2 +n3 =0
X
19
. N −2+ δ 24 − ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2
• |n2 | |n1 | ∼ |n3 |, |n1 | & N . The multiplier is bounded by
( ∇m(n )·n
2
m(n1 + n2 ) − m(n1 )m(n2 ) | m(n11 ) 2 | . N
N1 , if |n2 | ≤ N,
.
1/2
N2
m(n1 )m(n2 )
,
if |n2 | ≥ N.
N
So, it is not hard to see that
Z δ
19
0−
L3 . N −2+ δ 24 − Nmax
ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2
0
• |n1 | |n2 | ∼ |n3 |, |n2 | & N . This case is completely similar to the previous one.
• |n1 | ∼ |n2 | & N . Since the multiplier is bounded by N2 /N , we get
Z δ
19
0−
L3 . N −2+ δ 24 − Nmax
ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2 .
0
10
C. MATHEUS
EJDE-2006/07
Rδ
Finally, it remains to estimate the contribution of 0 L4 . It suffices to see that
Z δ
4
m(n + n + n ) − m(n )m(n )m(n ) X
Y
1
2
3
1
2
3 ubj (nj , t)
|n4 |
m(n1 )m(n2 )m(n3 )
0 n +n +n +n =0
j=1
1
2
3
1
. N −2+ δ 2 −
4
4
Y
kuj kX 1,1/2
j=1
(4.17)
• N1 , N2 , N3 & N . Since the multiplier verifies
m(n + n + n ) − m(n )m(n )m(n ) N N N 1/2
1
1
2
3
1
2
3 2
3
,
.
m(n1 )m(n2 )m(n3 )
N N N
appliying L4xt L4xt L4xt L4xt Hölder inequality and the lemmas 2.1, 2.2, we have
1/2
Z δ
1
4
N1 N2 N3
δ 2− Y
L4 .
kuj kX 1,1/2
N N N
N1 N2 N3 j=1
0
1
0−
. N −3+ δ 2 − Nmax
4
Y
kuj kX 1,1/2 .
j=1
• N1 ∼ N2 & N and N3 , N4 N1 , N2 . Here the multiplier is bounded by
1/2
N1 N2 1/2
h NN3
i. Hence,
N N
1/2
Z δ
1
4
Y
N1 N2
N3 1/2
δ 2−
kuj kX 1,1/2
L4 .
h
i
N N
N
N1 N2 hN3 i j=1
0
1
0−
. N −2+ δ 2 − Nmax
4
Y
kuj kX 1,1/2 .
j=1
• N1 ∼ N4 & N and N2 , N3 N1 , N4 . In this case we have the following estimates
for the multiplier
m(n + n + n ) − m(n )m(n )m(n ) 1
2
3
1
2
3 m(n1 )m(n2 )m(n3 )

∇m(n1 )(n2 +n3 ) . N2 +N3 , if N2 , N3 ≤ N


m(n1 )
N1

N1 N2 1/2
N3 1/2
.
)
h(
i,
if N2 ≥ N,
N N
N


 N1 N3 1/2 h( N2 )1/2 i,
if N3 ≥ N.
N N
N
Therefore, it is not hard to see that, in any of the situations N2 , N3 ≤ N , N2 ≥ N
or N3 ≥ N , we have
Z δ
4
Y
1
0−
L4 . N −2+ δ 2 − Nmax
kuj kX 1,1/2 .
0
j=1
• N1 ∼ N2 ∼ N4 & N and N3 N1 , N2 , N4 . Here we have the following bound
1/2
Z δ
1
4
N1 N2
N3 1/2
δ 2− Y
L4 .
h
i
kuj kX 1,1/2 .
N N
N
N1 N2 N3 j=1
0
At this point, clearly the bounds (4.14), (4.15), (4.16) and (4.17) concludes the
proof of the proposition 4.1.
EJDE-2006/07
GLOBAL PERIODIC SOLUTIONS FOR NLS-KDV SYSTEMS
11
4.2. Estimates for the modified E-functional.
Proposition
4.2. Let (u, v) be a solution of (1.1) on the time interval [0, δ] such
R
that T v = 0. Then, for any N ≥ 1, s > 1/2,
|E(Iu(δ), Iv(δ)) − E(Iu(0), Iv(0))|
1
2
3
3
1
. N −1+ δ 6 − + N − 3 + δ 8 − + N − 2 + δ 8 − (kIukX 1 + kIvkY 1 )3
1
1
+ N −1+ δ 2 − (kIukX 1 + kIvkY 1 )4 + N −2+ δ 2 − kIuk4X 1 (kIuk2X 1 + kIvkY 1 ).
(4.18)
Proof. Again we integrate (4.12) with respect to t ∈ [0, δ], decompose the frequencies into dyadic blocks, etc., so that our objective is to bound the (integral over
[0, δ] of the) Ej for each j = 1, . . . , 12.
Rδ
For the expression 0 E1 , apply the lemma 2.3. We obtain
Z
|
δ
E1 | . kIvxx kY −1 kIvIvx − I(vvx )kW 1 . kIvkY 1 kIvIvx − I(vvx )kW 1
0
Writing the definition of the norm W 1 , it suffices to prove the bound
Z X
m(n1 + n2 ) − m(n1 )m(n2 )
hn3 i
v
b
(n
,
τ
)
n
v
b
(n
,
τ
)
1 1 1
2 2 2 2 2
m(n1 )m(n2 )
Ln3 ,τ3
hτ3 − n33 i1/2
hn i Z X m(n + n ) − m(n )m(n )
3
1
2
1
2
vb1 (n1 , τ1 ) n2 vb2 (n2 , τ2 )
+
hτ3 − n33 i
m(n1 )m(n2 )
L2n3 L1τ3
1
. N −1+ δ 6 − kv1 kY 1,1/2 kv2 kY 1,1/2 .
(4.19)
P3
Recall that the dispersion relation j=1 τj − n3j = −3n1 n2 n3 implies that, since
n1 n2 n3 6= 0, if we put Lj := |τj − n3j | and Lmax = max{Lj ; j = 1, 2, 3}, then
Lmax & hn1 ihn2 ihn3 i.
• |n2 | ∼ |n3 | & N , |n1 | |n2 |. The multiplier is bounded by
m(n + n ) − m(n )m(n ) 1
2
1
2 .
m(n1 )m(n2 )
(
N1
N2 ,
N1 1/2
,
N
if |n1 | ≤ N,
if |n1 | ≥ N.
Thus, if |τ3 − n33 | = Lmax , we have
Z X
m(n1 + n2 ) − m(n1 )m(n2 )
hn3 i
v
b
(n
,
τ
)
n
v
b
(n
,
τ
)
2
1
1
1
2
2
2
2
m(n1 )m(n2 )
Ln3 ,τ3
hτ3 − n33 i1/2

N
N
1
3

4
4

N2 (N1 N2 N3 )1/2 kv1 kLxt k(v2 )x kLxt


1

−1+
−
0−
. N
δ 3 Nmax kv1 kY 1,1/2 kv2 kY 1,1/2 ,
if |n1 | ≤ N,
.
1/2
N1
N3
1

4
4


N2
N1 (N1 N2 N3 )1/2 kv1 kLxt k(v2 )x kLxt


3
1
. N − 2 + δ 3 − N 0− kv1 k 1 kv2 k 1 ,
if |n1 | ≥ N.
1,
1,
max
2
2
Y
Y
12
C. MATHEUS
EJDE-2006/07
and
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
v
b
(n
,
τ
)
n
v
b
(n
,
τ
)
2 1
1
1
1
2
2
2
2
hτ3 − n33 i
m(n1 )m(n2 )
Ln3 Lτ3
N
N3
1
kv k 4 k(v ) k 4

1

 N2 (N1 N2 N3 ) 2 − 1 Lxt 2 x Lxt

1

. N −1+ δ 3 − N 0− kv1 k 1,1/2 kv2 k 1,1/2 ,
if |n1 | ≤ N,
Y
Y
max
1
.
N1 1/2 N3
δ3−

1
1

N2
N1 (N N N ) 12 − kv1 kY 1, 2 kv2 kY 1, 2


1 2 3


− 32 + 31 − 0−
.N
δ Nmax kv1 k 1, 12 kv2 k 1, 21 ,
if |n1 | ≥ N.
Y
Y
If either |τ1 − n31 | = Lmax or |τ2 − n32 | = Lmax , we have
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
v
b
(n
,
τ
)
n
v
b
(n
,
τ
)
1 1 1
2 2 2 2 2
m(n1 )m(n2 )
Ln3 ,τ3
hτ3 − n33 i1/2

1−
N1
N3
δ6

1
1

N2 (N1 N2 N3 )1/2 N1 kv1 kY 1, 2 kv2 kY 1, 2


1
. N −1+ δ 6 − N 0− kv k
if |n1 | ≤ N,
max 1 Y 1,1/2 kv2 kY 1,1/2 ,
.
1
N1 1/2 N3
1
−

6
kv1 k 1, 12 kv2 k 1, 12

N2
N1 (N1 N2 N3 )1/2 δ

Y
Y


− 23 + 31 − 0−
.N
δ Nmax kv1 k 1, 12 kv2 k 1, 12 ,
if |n1 | ≥ N.
Y
Y
and
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
v
b
(n
,
τ
)
n
v
b
(n
,
τ
)
2 1
1
1
1
2
2
2
2
hτ3 − n33 i
m(n1 )m(n2 )
Ln3 Lτ3

1−
N1
N3
δ6


1
1

N2 (N N N ) 21 − N1 kv1 kY 1, 2 kv2 kY 1, 2

1
2
3


1
. N −1+ δ 6 − N 0− kv k 1,1/2 kv k 1,1/2 ,
if |n1 | ≤ N,
2 Y
max 1 Y
.
1
N1 1/2 N3
δ6−

1
1


N1 (N N N ) 12 − kv1 kY 1, 2 kv2 kY 1, 2
 N2
1 2 3


3
1
. N − 2 + δ 6 − N 0− kv k 1 kv k 1 ,
if |n1 | ≥ N.
2
1,
1,
max 1
2
2
Y
Y
• |n1 | ∼ |n2 | & N . Estimating the multiplier by
m(n + n ) − m(n )m(n ) N1 1−
1
2
1
2 ,
.
m(n1 )m(n2 )
N
we have that, if |τ3 − n33 | = Lmax ,
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
v
b
(n
,
τ
)
n
v
b
(n
,
τ
)
2
1
1
1
2
2
2
2
m(n1 )m(n2 )
Ln ,τ
hτ3 − n33 i1/2
3 3
Z
hn i
X
m(n
+
n
)
−
m(n
)m(n
)
1
2
1
2
3
vb1 (n1 , τ1 ) n2 vb2 (n2 , τ2 )
+
hτ3 − n33 i
m(n1 )m(n2 )
L2n3 L1τ
3
1
1
N1 1−
N3
δ 3−
N1 1−
N3
δ 3− .
+
kv1 k 1, 21 kv2 k 1, 12
1
Y
Y
N
N
(N1 N2 N3 )1/2 N1
(N1 N2 N3 ) 2 − N1
3
1
0−
. N − 2 + δ 3 − Nmax
kv1 k
1
Y 1, 2
kv2 k
1
Y 1, 2
EJDE-2006/07
GLOBAL PERIODIC SOLUTIONS FOR NLS-KDV SYSTEMS
13
and, if either |τ1 − n31 | = Lmax or |τ2 − n32 | = Lmax ,
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
v
b
(n
,
τ
)
n
v
b
(n
,
τ
)
2
1
1
1
2
2
2
2
m(n1 )m(n2 )
Ln ,τ
hτ3 − n33 i1/2
3 3
hn i Z X m(n + n ) − m(n )m(n )
3
1
2
1
2
+
vb1 (n1 , τ1 ) n2 vb2 (n2 , τ2 ) 2 1
hτ3 − n33 i
m(n1 )m(n2 )
Ln3 Lτ3
1
1
N1 1−
δ 6−
N3
N3
δ 6− N1 1−
kv1 k 1, 21 kv2 k 1, 12
+
1
Y
Y
N
N
(N1 N2 N3 )1/2 N1
(N1 N2 N3 ) 2 − N1
.
1
3
0−
kv1 k
. N − 2 + δ 6 − Nmax
1
Y 1, 2
For the expression
δ
Z
Rδ
0
kv2 k
1
Y 1, 2
E2 , it suffices to prove that
X m(n3 + n4 ) − m(n3 )m(n4 )
m(n3 )m(n4 )
0
2
. N −2+ δ 3 −
4
Y
.
vb1 (n1 , t)vb2 (n2 , t)vb3 (n3 , t) n4 vb4 (n4 , t)
(4.20)
kvj kY 1,1/2 .
j=1
Since at least two of the Ni are bigger than N/3, we can assume that N1 ≥ N2 ≥ N3
and N1 & N . Hence,

2
Q4
Q4
N1 1− δ 3 −
−2+ 23 − 0−

δ Nmax j=1 kvj kY 1,1/2 ,

j=1 kvj kY 1,1/2 . N
N
N1 N2 N3




if |n3 | ∼ |n4 | & N,




Z δ
Q4
2
 N3 δ 23 − Q4
0−
kv k 1,1/2 . N −2+ δ 3 − Nmax
j=1 kvj kY 1,1/2 ,
E2 . N4 N1 N2 N3 j=1 j Y

if |n3 | |n4 |, |n3 | ≤ N |n4 | & N,
0




2
Q4
Q4

N3 1/2 δ 3 −

−2+ 32 − 0−

δ Nmax j=1 kvj k 1, 12 ,

j=1 kvj kY 1, 12 . N
N
N1 N2 N3
Y


if |n3 | |n4 |, |n3 | ≥ N, |n4 | & N.
Next, we estimate the contribution of
δ
Z
Rδ
0
E3 . We claim that
X m(n1 n2 n3 ) − m(n1 )m(n2 )m(n3 )
m(n1 )m(n2 )m(n3 )
0
1
. N −1+ δ 2 −
4
Y
c4 (n4 , t)
u
c1 (n1 , t)c
u2 (n2 , t)c
u3 (n3 , t) |n4 |2 u
kuj kX 1,1/2 .
j=1
(4.21)
• |n1 | ∼ |n2 | ∼ |n3 | ∼ |n4 | & N . Since the multiplier satisfies
m(n1 n2 n3 ) − m(n1 )m(n2 )m(n3 )
N1 32
.
m(n1 )m(n2 )m(n3 )
N
we obtain
Z
δ
E3 .
0
4
4
Y
Y
1
1
N1 23
N4
0−
δ 2−
kuj kX 1,1/2 . N −2+ δ 2 − Nmax
kuj kX 1,1/2 .
N
N1 N2 N3
j=1
j=1
14
C. MATHEUS
EJDE-2006/07
• Exactly two frequencies are bigger than N/3. We consider the most difficult case
|n4 | & N , |n1 | ∼ |n4 | and |n2 |, |n3 | |n1 |, |n4 |. The multiplier is estimated by

1/2 N2 1/2

i
, if |n2 | ≥ N,
h NN3

m(n1 n2 n3 ) − m(n1 )m(n2 )m(n3 )
1/2 NN 1/2
N
2
3
. h N
i N
, if |n3 | ≥ N,

m(n1 )m(n2 )m(n3 )
 N2 +N3
if |n2 |, |n3 | ≤ N.
N1 ,
Thus,
δ
Z
E3 . N
−1+
δ
1
2−
0−
Nmax
0
4
Y
kuj kX 1,1/2 .
j=1
• Exactly three frequencies are bigger than N/3. The most difficult case is |n1 | ∼
|n2 | ∼ |n4 | & N and |n3 | |n1 |, |n2 |, |n4 |. Here the multiplier is bounded by
1/2
N1 N2
N3 1/2
m(n1 n2 n3 ) − m(n1 )m(n2 )m(n3 )
.
h
i.
m(n1 )m(n2 )m(n3 )
N N
N
Hence,
Z
δ
E3 .
0
1
N1 N2
N N
0−
. N −1+ δ 2 − Nmax
1/2
h
4
Y
4
Y
1
N4
N3 1/2
kuj kX 1,1/2
i
δ 2−
N
N1 N2 N3
j=1
kuj kX 1,1/2 .
j=1
Rδ
The contribution of 0 E4 is controlled if we are able to show that
Z δX
m(n1 + n2 ) − m(n1 )m(n2 )
vb1 (n1 , t) |n2 | vb2 (n2 , t)c
u3 (n3 , t)c
u4 (n4 , t) .
m(n1 )m(n2 )
0
7
N −1+ δ 12 −
2
Y
kuj kX 1,1/2 kvj kY 1,1/2 .
j=1
(4.22)
We crudely bound the multiplier by
|
m(n1 + n2 ) − m(n1 )m(n2 )
Nmax 1−
|.
.
m(n1 )m(n2 )
N
The most difficult case is |n2 | ≥ N . We have two possibilities:
• Exactly two frequencies are bigger than N/3. We can assume N3 N2 . In
particular,
Z δ
7
2
Y
Nmax 1− δ 12
kuj k 1, 21 kvj k 1, 12
E4 .
X
Y
N
N1 N3 N4 j=1
0
7
0−
. N −1+ δ 12 − Nmax
2
Y
kuj k
1
X 1, 2
kvj k
1
Y 1, 2
.
j=1
• At least three frequencies are bigger than N/3. In this case,
Z δ
2
Y
7
0−
E4 . N −2+ δ 12 − Nmax
kuj k 1, 12 kvj k 1, 12 .
0
X
j=1
Y
EJDE-2006/07
GLOBAL PERIODIC SOLUTIONS FOR NLS-KDV SYSTEMS
The expression
δ
Z
Rδ
0
E5 is controlled if we are able to prove
X m(n1 + n2 ) − m(n1 )m(n2 )
m(n1 )m(n2 )
0
.N
−1+
δ
7
12 −
2
Y
15
u
c1 (n1 , t)c
u2 (n2 , t)vb3 (n3 , t) |n4 | vb4 (n4 , t)
(4.23)
kuj kX 1,1/2 kvj kY 1,1/2 .
j=1
This follows directly from the previous analysis for (4.22).
Rδ
For the term 0 E6 , we apply the lemma 2.3 to obtain
δ
Z
E6 . k(Iv)xx kY −1 k(|Iu|2 − I(|u|2 ))x kW 1 . kIvkY 1 k(|Iu|2 − I(|u|2 ))x kW 1 .
0
So, the definition of the W 1 norm means that we have to prove
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
u
c
(n
,
τ
)c
u
(n
,
τ
)
|n
|
2
1
1
1
2
2
2
3
3
1/2
m(n
)m(n
)
Ln3 ,τ3
hτ3 − n3 i
1
2
Z X
hn
i
m(n
+
n
)
−
m(n
)m(n
)
3
1
2
1
2
+
u
c
(n
,
τ
)c
u
(n
,
τ
)
|n
|
2 1
1
1
1
2
2
2
3
m(n1 )m(n2 )
Ln3 Lτ3
hτ3 − n33 i1/2
n
o
− 23 + 81 −
− 23 38 −
. N
δ
+N δ
ku1 kX 1,1/2 ku2 kX 1,1/2 .
(4.24)
P
P
Note that
τj = 0 and
nj = 0. In particular, we obtain the dispersion relation
τ3 − n33 + τ2 + n22 + τ1 + n21 = −n33 + n21 + n22 .
• |n1 | & N , |n2 | |n1 |. Denoting by L1 := |τ1 + n21 |, L2 := |τ2 + n22 | and
L3 := |τ3 − n33 |, the dispersion relation says that in the present situation Lmax :=
max{Lj } & N33 . Since the multiplier is bounded by
m(n + n ) − m(n )m(n ) 1
2
1
2 .
m(n1 )m(n2 )
( ∇m(n
1 )n2
m(n1 )
1/2
N2
,
N
.
N2
N1 ,
if |n2 | ≤ N,
if |n2 | ≥ N,
we deduce that
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
|n
|
u
c
(n
,
τ
)c
u
(n
,
τ
)
3
1 1 1 2 2 2 m(n1 )m(n2 )
L2n ,τ
hτ3 − n33 i1/2
3 3
Z X
m(n1 + n2 ) − m(n1 )m(n2 )
hn3 i
|n3 |
u
c1 (n1 , τ1 )c
u2 (n2 , τ2 ) 2 1
+
m(n1 )m(n2 )
Ln3 Lτ3
hτ3 − n33 i1/2
1
.
N32 δ 8 −
ku1 kX 1,1/2 ku2 kX 1,1/2
3
−
N 2 N N1
.N
3
− 32 +
1
0−
δ 8 − Nmax
ku1 kX 1,1/2 ku2 kX 1,1/2 .
16
C. MATHEUS
EJDE-2006/07
• |n1 | ∼ |n2 | & N , |n3 |3 |n2 |2 . In the present case the multiplier is bounded by
N1 1−
and the dispersion relation says that Lmax & N33 . Thus,
N
Z X
m(n1 + n2 ) − m(n1 )m(n2 )
hn3 i
|n
|
u
c
(n
,
τ
)c
u
(n
,
τ
)
2
3
1
1
1
2
2
2
m(n1 )m(n2 )
Ln3 ,τ3
hτ3 − n33 i1/2
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
|n
|
+
u
c
(n
,
τ
)c
u
(n
,
τ
)
3
1 1 1 2 2 2 2
m(n1 )m(n2 )
Ln L1τ
hτ3 − n33 i1/2
3
3
1
N32
.
3
2−
N3
3
N1 1− δ 8 −
ku1 kX 1,1/2 ku2 kX 1,1/2
N
N1 N2
1
0−
. N − 2 + δ 8 − Nmax
ku1 kX 1,1/2 ku2 kX 1,1/2 .
• |n1 | ∼ |n2 | & N and |n3 |3 . |n2 |2 . Here the dispersion relation does not give
1/2
, we
useful information about Lmax . Since the multiplier is estimated by NN2
obtain the crude bound
Z X
hn3 i
m(n1 + n2 ) − m(n1 )m(n2 )
|n
|
u
c
(n
,
τ
)c
u
(n
,
τ
)
3
1 1 1 2 2 2 2
m(n1 )m(n2 )
Ln3 ,τ3
hτ3 − n33 i1/2
Z X
m(n1 + n2 ) − m(n1 )m(n2 )
hn3 i
|n
|
u
c
(n
,
τ
)c
u
(n
,
τ
)
+
3
1 1 1 2 2 2 2
m(n1 )m(n2 )
Ln3 L1τ3
hτ3 − n33 i1/2
3
. N32
N2 1/2 δ 8 −
ku1 kX 1,1/2 ku2 kX 1,1/2
N
N1 N2
2
3
0−
ku1 kX 1,1/2 ku2 kX 1,1/2 .
. N − 3 + δ 8 − Nmax
Rδ
Next, the desired bound related to 0 E7 follows from
Z δ X
m(n1 + n2 ) − m(n1 )m(n2 ) u1 (n1 , t)vb2 (n2 , t)|n3 |c
u3 (n3 , t)
|n1 + n2 |c
m(n1 )m(n2 )
(4.25)
0
19
. N −1+ δ 24 − ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2
• |n1 | |n2 | & N . The multiplier is . (|n2 |/N )1/2 so that
Z δ
Z δX
1
|n1 + n2 |c
u1 (n1 , t)|n2 |1/2 vb2 (n2 , t)|n3 |c
u3 (n3 , t)
E7 . 1/2
N
0
0
19
. N −1 δ 24 − ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2 .
• |n1 | ∼ |n2 | & N . The multiplier is . |n2 |/N . Hence,
Z δ
19
E7 . N −1 δ 24 − ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2 .
0
|n1 | & N , |n2 | ≤ N . The multiplier is again . N2 /N , so that it can be estimated
as above.
Rδ
Now we turn to the term 0 E8 . The objective is to show that
Z
0
δ
4
4
m(n + n ) − m(n )m(n ) Y
Y
1
1
2
1
2 ubj (nj , t) . N −1+ δ 2 −
kuj kX 1,1/2
|n1 + n2 |
m(n1 )m(n2 )
j=1
j=1
(4.26)
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GLOBAL PERIODIC SOLUTIONS FOR NLS-KDV SYSTEMS
17
• At least three frequencies are bigger than N/3. We can assume |n1 | ≥ |n2 |. The
multiplier is bounded by Nmax /N so that
Z δ
1
4
4
Y
1
Nmax δ 2 − Y
0−
E8 .
kuj kX 1,1/2 . N −2+ δ 2 − Nmax
kuj kX 1,1/2 .
N N2 N3 N4 j=1
0
j=1
• Exactly two frequencies are bigger than N/3. Without loss of generality, we
suppose |n1 | ∼ |n2 | & N and |n3 |, |n4 | N . Since the multiplier satisfies
m(n + n ) − m(n )m(n ) Nmax 1−
1
2
1
2 ,
.
m(n1 )m(n2 )
N
we get the bound
Z δ
1
4
4
Y
1
Nmax 1− δ 2 − Y
0−
E8 .
kuj kX 1,1/2 .
kuj kX 1,1/2 . N −1+ δ 2 − Nmax
N
N2 N3 N4 j=1
0
j=1
Rδ
The contribution of 0 E9 is estimated if we prove that
Z δ
m(n1 + n2 ) − m(n1 )m(n2 ) u1 (n1 , t)vb2 (n2 , t)c
u3 (n3 , t)vb4 (n4 , t)
c
m(n1 )m(n2 )
0
.N
−2+
δ
7
12 −
(4.27)
ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2 kv4 kY 1,1/2 .
This follows since at least two frequencies are bigger than N/3 and the multiplier
is always bounded by (Nmax /N )1− , so that
Z δ
Nmax 1−
ku1 kL4 kv2 kL4 ku3 kL4 kv4 kL4
E9 .
N
0
1
1
Nmax 1− δ 4 + 3 −
.
ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2 kv4 kY 1,1/2
N
N1 N2 N3 N4
7
. N −2+ δ 12 − ku1 kX 1,1/2 kv2 kY 1,1/2 ku3 kX 1,1/2 kv4 kY 1,1/2 .
Rδ
Now, we treat the term 0 E10 . It is sufficient to prove
Z δ X
6
m(n4 + n5 + n6 ) − m(n4 )m(n5 )m(n6 ) Y
ubj (nj , t)
m(n4 )m(n5 )m(n6 )
0
j=1
.N
−2+
δ
6
Y
1
2−
(4.28)
kuj kX 1 .
j=1
This follows easily from the facts that the multiplier is bounded by (Nmax /N )3/2 ,
at least two frequencies are bigger than N/3, say |ni1 | ≥ |ni2 | & N , the Strichartz
1
bound X 0,3/8 ⊂ L4 and the inclusion1 X 2 + ⊂ L∞
xt . Indeed, if we combine these
informations, it is not hard to get
Z δ
6
Y
1
1
1
Nmax 32
−
2
δ
E10 .
kuj kX 1
N
N i 1 N i 2 N i 3 N i4
(Ni5 Ni6 )1/2− j=1
0
.N
−2+
δ
1
2−
0−
Nmax
6
Y
kuj kX 1
j=1
1This inclusion is an easy consequence of Sobolev embedding.
18
C. MATHEUS
EJDE-2006/07
Rδ
For the expression 0 E11 , we use again that the multiplier is bounded by
(Nmax /N )3/2 , at least two frequencies are bigger than N/3 (say |ni1 | ≥ |ni2 | & N ),
1
1
the Strichartz bounds in lemma 2.1 and the inclusions X 2 + , Y 2 + ⊂ L∞
xt to obtain
Z δ X
4
m(n1 + n2 + n3 ) − m(n1 )m(n2 )m(n3 ) Y
ubj (nj , t)vb5 (n5 , t)
m(n1 )m(n2 )m(n3 )
0
j=1
.
1
4
Nmax 32
1
δ 2− Y
kuj kX 1 kv5 kY 1
N
Ni1 Ni2 Ni3 Ni4 N 1/2− j=1
i5
1
. N −2+ δ 2 −
4
Y
(4.29)
kuj kX 1 kv5 kY 1 .
j=1
The analysis of
Rδ
0
E12 is similar to the
Rδ
0
E11 . This completes the proof.
5. Global well-posedness below the energy space
In this section we combine the variant local well-posedness result in proposition
3.1 with the two almost conservation results in the propositions 4.1 and 4.2 to prove
the theorem 1.1.
R
Remark 5.1. Note that the spatial mean T v(t, x)dx is preserved during the evolution (1.1). Thus, we can assume that the
R initial data v0 has zero-mean, since
otherwise we make the Rchange w = v −
R T v0 dx at the expense of two harmless
linear terms (namely, u T v0 dx and ∂x v T v0 ).
The definition of the I-operator implies that the initial data satisfies kIu0 k2H 1 +
kIv0 k2H 1 . N 2(1−s) and kIu0 k2L2 + kIv0 k2L2 . 1. By the estimates (4.4) and (4.8),
we get that |L(Iu0 , Iv0 )| . N 1−s and |E(Iu0 , Iv0 )| . N 2(1−s) .
Also, any bound for L(Iu, Iv) and E(Iu, Iv) of the form |L(Iu, Iv)| . N 1−s and
|E(Iu, Iv)| . N 2(1−s) implies that kIuk2L2 . M , kIvk2L2 . N 1−s and kIuk2H 1 +
kIvk2H 1 . N 2(1−s) .
Given a time T , if we can uniformly bound the H 1 -norms of the solution at times
t = δ, t = 2δ, etc., the local existence result in proposition 3.1 says that the solution
can be extended up to any time interval where such a uniform bound holds. On
the other hand, given a time T , if we can interact T δ −1 times the local existence
result, the solution exists in the time interval [0, T ]. So, in view of the propositions
4.1 and 4.2, it suffices to show
1
19
(N −1+ δ 24 − N 3(1−s) + N −2+ δ 2 − N 4(1−s) )T δ −1 . N 1−s
(5.1)
and
1
2
3
3
1
(N −1+ δ 6 − + N − 3 + δ 8 − + N − 2 + δ 8 − )N 3(1−s)
(5.2)
T
1
1
+ N −1+ δ 2 − N 4(1−s) + N −2+ δ 2 − N 6(1−s)
. N 2(1−s)
δ
16
At this point, we recall that the proposition 3.1 says that δ ∼ N − 3 (1−s)− if β 6= 0
and δ ∼ N −8(1−s)− if β = 0. Hence,
• β 6= 0. The condition (5.1) holds for
−1 +
5 16
(1 − s) + 3(1 − s) < (1 − s),
24 3
i.e. , s > 19/28
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GLOBAL PERIODIC SOLUTIONS FOR NLS-KDV SYSTEMS
19
and
1 16
(1 − s) + 4(1 − s) < (1 − s), i.e. , s > 11/17;
2 3
Similarly, condition (5.2) is satisfied if
5 16
−1 +
(1 − s) + 3(1 − s) < 2(1 − s), i.e. , s > 40/49;
6 3
2 5 16
− +
(1 − s) + 3(1 − s) < 2(1 − s), i.e. , s > 11/13;
3 6 3
3 7 16
− +
(1 − s) + 3(1 − s) < 2(1 − s), i.e. , s > 25/34;
2 8 3
1 16
−1 +
(1 − s) + 4(1 − s) < 2(1 − s), i.e. , s > 11/14;
2 3
1 16
(1 − s) + 6(1 − s) < 2(1 − s), i.e. , s > 7/10.
−2 +
2 3
Thus, we conclude that the non-resonant NLS-KdV system is globally well-posed
for any s > 11/13.
• β = 0. Condition (5.1) is fulfilled when
5
−1 + 8(1 − s) + 3(1 − s) < (1 − s), i.e. , s > 8/11
24
and
1
−2 + 8(1 − s) + 4(1 − s) < (1 − s), i.e. , s > 5/7;
2
Similarly, the condition (5.2) is verified for
5
−1 + 8(1 − s) + 3(1 − s) < 2(1 − s), i.e. , s > 20/23;
6
2 5
− + 8(1 − s) + 3(1 − s) < 2(1 − s), i.e. , s > 8/9;
3 6
3 7
− + 8(1 − s) + 3(1 − s) < 2(1 − s), i.e. , s > 13/16;
2 8
1
−1 + 8(1 − s) + 4(1 − s) < 2(1 − s), i.e. , s > 5/6;
2
1
−2 + 8(1 − s) + 6(1 − s) < 2(1 − s), i.e. , s > 3/4.
2
Hence, we obtain that the resonant NLS-KdV system is globally well-posed for any
s > 8/9.
−2 +
References
[1] A. Arbieto, A. Corcho and C. Matheus, Rough solutions for the periodic Schrödinger - Korteweg - deVries system, Journal of Differential Equations, 230 (2006), 295–336.
[2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Anal., 3 (1993), 107–156,
209–262.
[3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for
KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 (2003), 705–749.
[4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic
KdV equations, and applications, J. Funct. Analysis, 211 (2004), 173–218.
[5] A. J. Corcho, and F. Linares, Well-posedness for the Schrödinger - Kortweg-de Vries system,
Preprint (2005).
[6] H. Pecher, The Cauchy problem for a Schrödinger - Kortweg - de Vries system with rough
data, Preprint (2005).
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C. MATHEUS
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[7] M. Tsutsumi, Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation, Math. Sciences Appl., 2 (1993), 513–528.
Carlos Matheus
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil
E-mail address: matheus@impa.br