Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 05, pp. 1–25.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.05

GLOBAL WELL-POSEDNESS FOR CAUCHY PROBLEMS OF

ZAKHAROV-KUZNETSOV EQUATIONS ON

CYLINDRICAL SPACES

SATOSHI OSAWA, HIDEO TAKAOKA

Abstract. We study the global well-posedness of the Zakharov-Kuznetsov

equation on cylindrical spaces. Our goal is to establish the existence of global-

in-time solutions below the energy class. To prove the results, we adapt the
I-method to extend the local solutions globally in time. The main tool in our

argument is multilinear estimates in the content of Bourgain’s spaces. Using

modified energies induced by the I-method, we obtain polynomial bounds on
the Hs growth of global solutions.

1. Introduction

In this article, we study the Cauchy problem for the Zakharov-Kuznetsov equa-
tion on a cylinder,

∂tu+ ∂x∆u+ u∂xu = 0, (x, y) ∈ R× T, t ∈ R
u(x, y, 0) = u0(x, y), (x, y) ∈ R× T,

(1.1)

where u = u(x, y, t) is a real-valued function, T = R/(2πZ), and ∆ = ∂2
x + ∂2

y is
the Laplacian. This equation was introduced by Zakharov and Kuznetsov [26], as
a model for the propagation of ionic-acoustic waves in magnetized plasma (see also
[14] for the derivation of the equation). This equation is one of the two-dimensional
extensions of the Korteweg-de Vries (KdV) equation

∂tu+ ∂xxxu+ u∂xu = 0.

The KdV equation is one of the famous nonlinear dispersive equation, which de-
scribes the behavior of shallow water waves. Another two dimensional generaliza-
tions of the KdV equation is the Kadomtsev-Petviashvili (KP) equations

∂x (∂tu+ ∂xxxu+ u∂xu)± ∂yyu = 0,

where the KP-I equation corresponds to minus sign, while the KP-II equation to
plus sign.

2020 Mathematics Subject Classification. 35Q53, 42B37.
Key words and phrases. Zakharov-Kuznetsov equation; low regularity; global well-posedness;

bilinear estimate.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted January 3, 2024. Published January 22, 2024.

1



2 S. OSAWA, H. TAKAOKA EJDE-2024/05

The Zakharov-Kuznetsov equation (1.1) has at least two conserved quantities:

E[u](t) =
1

2

∫
R×T

(
|∇u(x, y, t)|2 − 1

3
u(x, y, t)

3
)
dx dy,

M [u](t) =

∫
R×T

u(x, y, t)
2
dx dy.

The conservation laws identify H1(R× T) as the energy space.
The main purpose of this article is to consider a global well-posedness result of

problem (1.1) with rough initial data below the energy space.
For the Zakharov-Kuznetsov in the R2 setting, Faminskii [7] proved the local well-

posedness in the energy space H1(R2). After this, Linares and Pastor [16] showed
the local well-posedness for s > 3/4, by proving a sharp maximal function estimates.
Grünrock and Herr [10] proved local well-posedness in Hs(R2) for s > 1/2, by
using the Fourier restriction norm method and Strichartz estimates. The Fourier
restriction norm method utilizing the Xs,b spaces was introduced by Bourgain [2, 3]
(also by Kenig, Ponce and Vega [11]), to prove the local well-posedness for nonlinear
Schrödinger and KdV equations in low regularity Sobolev spaces. At the same
time, Molinet and Pilod [19] showed the local well-posedness in Hs(R2) for s >
1/2 using bilinear Strichartz estimates. Observing proofs of global well-posedness,
Shan [24] obtained that the solution exists globally in time to data in Hs(R2)
for s > 5/7. Shan used the I-method based on H1 conservation laws. The I-
method was introduced for global well-posedness of the KdV equation by Colliander,
Keel, Staffilani, Takaoka and Tao [5]. Recently, Kinoshita [12] showed the local
well-posedness for s > −1/4, using Loomis-Whitney inequality and an orthogonal
decomposition technique. In [12], the local ill-posedness results for s < −1/4 was
also obtained. This means that the data-to-solution map from the unit ball in
Hs(R2) to C([0, T ];Hs) fails to be smooth for any T > 0. As a corollary from the
result of local well-posedness combining with the L2 conservation law, the global
well-posedness in L2(R2) follows.

In the R×T setting, Molinet and Pilod in the same paper as above [19] showed
the global well-posedness in Hs(R × T) for s ≥ 1. The strategy is similar to the
case of R2 setting, however it is more difficult to make sure the proof because of
the lack of the smoothing effects. In [21], we showed the local well-posedness for
s > 9/10 by refining the bilinear estimates. The main motivation here comes from
the corresponding question of global well-posedness for s < 1.

In the T2 setting, the local well-posedness result in Hs(T2) has been obtained
by Linares, Panthee, Robert and Tzvetkov [15]. They showed that the initial value
problem is locally well-posed in Hs(T2) for s > 5/3. The proof used short-time
Strichartz estimates. In [23], Schippa improved the local well-posedness to s > 3/2
by using short-time bilinear Strichartz estimates. Moreover, Kinoshita and Schippa
[13] obtained the local well-posedness for s > 1. They estimated the nonlinear
interaction term by short-time trilinear estimates.

Observing related results on the Zakharov-Kuznetsov equation. Yamazaki [25]
showed the stability and instability results for line solitary waves of the Zakharov-
Kuznetsov equations in R× TL, where TL is the torus of length 2πL. The solitary
waves are constructed by hyperbolic functions. The long-time behavior stability of
solitary waves were also studied in [4] and in [22].



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 3

We aim to prove that the initial value problem of the Zakharov-Kuznetsov equa-
tion in Hs(R × T) is globally well-posed for some s < 1. This proof is based on
bilinear estimates developed in [19] and I-method [5].

The main result of the paper is the following theorem.

Theorem 1.1. The initial value problem of (1.1) is globally well-posed in Hs(R×T)
for s > 29/31. In other words, for any u0 ∈ Hs(R× T), for all T > 0, there exists
a unique solution u of (1.1) such that

u ∈ C([0, T ] : Hs((R× T)) ∩Xs,1/2+
T .

Moreover, for all 0 < T ′ < T , there exists a neighborhood U of u0 in Hs(R × T)
such that the data-to-solution map

U 3 v0 7→ v(t) ∈ C([0, T ′] : Hs((R× T)) ∩Xs,1/2+
T ′

is smooth, where v(t) is a unique solution of (1.1) to the initial data v0. Here

function space Xs,b
T is defined in Section 2.

Remark 1.2. By time reversibility, we need to consider only the existence for
positive time.

This article is organized as follows. In Section 2, we recall some harmonic anal-
ysis tools including Littlewood-Paley decompositions of functions. We also reviews
some preliminary results for linear estimates in the Bourgain spaces Xs,b [2, 3, 11].
In Section 3, we follow the I-method scheme [5]. We give several lemmas of bi-
linear estimates in conjunction with rescaling argument in [6] and give the local
well-posedness results for rescaled data. In Section 4, we define modified energy
functional E[Iu] in term of the Hs-norm of the solution for s < 1. Section 5 is
devoted to the proof of main theorem. One of the key steps in the construction of
global solutions is to control the increment of the modified energy. We estimate the
growth of the modified energy and provide a priori estimates for the solutions.

2. Notation and function spaces

In this section, we will introduce some function spaces that will be used through-
out the paper. We denote the absolute value of (ξ, q) as |(ξ, q)|2 = 3ξ2 + q2. For
positive real quantities a and b, the notation a . b means that there is a constant
c such that a ≤ cb. When a . b and b . a, we write a ∼ b. Moreover, b+ means
that there exists δ > 0 such that b+ δ.

Denote Tλ = λT = R/(2πλZ) for λ > 0. Let f(x, y) be a function defined on
R × Tλ. The Fourier transform of f with respect to the variables (x, y) is denote
by

Fλxyf(ξ, q) =

∫
R

∫ 2πλ

0

e−i(xξ+yq)f(x, y) dydx,

where (ξ, q) ∈ R× Z/λ. Let inverse Fourier inverse transform be denote by

F−1,λ
ξq f(x, y) =

1

(2π)2λ

∑
q∈Z/λ

∫
R
ei(ξx+qy)f(ξ, q) dξ

for (x, y) ∈ R×Tλ. For simplicity, we abbreviate Fλxy and F−1,λ
ξq by Fλ and F−1,λ,

respectively, when no confusion is likely.



4 S. OSAWA, H. TAKAOKA EJDE-2024/05

Let u(x, y, t) be a function of R×Tλ×R, and let ûλ(ξ, q, τ) be Fourier transform
of u(x, y, t) in a similar manner as well

ûλ(ξ, q, τ) =

∫
R2

∫ 2πλ

0

e−i(tτ+ixξ+yq)u(x, y, t) dy dx dt.

Similarly, we define the inverse Fourier transform

ǔλ(x, y, t) =
1

(2π)3λ

∑
q∈Z/λ

∫
R2

ei(tτ+ξx+qy)u(ξ, q, τ) dξ dτ.

We denote σ(ξ, q) = ξ3 + ξq2. Let η ∈ C∞0 (R) satisfy 0 ≤ η ≤ 1, η[−1,1] = 1, and

supp η ⊂ [−2, 2]. For a dyadic number N = 2k with k ∈ N, we denote

φ(ξ) = η(ξ)− η(2ξ), φN (ξ, q) = φ(N−1|(ξ, q)|),

ψN (ξ, q, τ) = φ(N−1(τ − σ(ξ, q))),

where (ξ, q, τ) ∈ R×Z/λ×R. Here, we prepare notation of interval as IN = suppφN
for a dyadic number N . We define the Littlewood-Paley decomposition as

PNu = F−1,λ(φNFλu), QLu = (ψLû
λ)̌ λ.

For a, b ∈ R, we denote a ∧ b = min{a, b} and a ∨ b = max{a, b}. For s ∈ R, we
define the Sobolev spaces Hs

λ = Hs
λ(R× Tλ) equipped with the norm

‖f‖Hsλ =
( 1

2πλ

∑
q∈Z/λ

∫
R
〈|(ξ, q)|〉2s|Fλf(ξ, q)|2 dξ

)1/2

, (2.1)

where 〈x〉 = 1 + |x|. We also use L2
λ = H0

λ.
We use the restriction operator RK with respect to the x variable as

RKf(x) =

∫
R
φ(ξK−1)Fxf(ξ)eixξ dξ

for dyadic number K, where Fx is the Fourier transform with respect to the x
variable.

We note that the following properties hold [6]:∫
R

∫ 2πλ

0

f(x, y)g(x, y) dx dy =
1

λ

∑
q∈Z/λ

∫
R
Fλf(ξ, q)Fλg(ξ, q) dξ,

and
Fλ(fg)(ξ, q) =

(
Fλf ∗λ Fλg

)
(ξ, q)

=
1

λ

∑
q1∈Z/λ

∫
R
Fλf(ξ − ξ1, q − q1)Fλg(ξ1, q1) dξ1.

(2.2)

Next, we describe the Bourgain space via the Fourier transform. Following [11],

we introduce a class of function spaces related to Bourgain space Xs,b
λ . Define the

Bourgain space Xs,b
λ as follows.

Definition 2.1. For s, b ∈ R and T > 0,

‖u‖Xs,bλ =
( 1

λ

∑
q∈Z/λ

∫
R2

〈τ − σ(ξ, q)〉2b〈|(ξ, q)|〉2s|û(ξ, q, τ)|2 dξdτ
)1/2

,

‖u‖Xs,bT,λ = inf
{
‖u‖Xs,bλ | u : R× Tλ → C, u|R×Tλ×[0,T ] = u

}
.

(2.3)



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 5

When a norm of u introduced in (2.3) is bounded, we denote u ∈ Xs,b
λ . We also

use the same notation L2
λ = X0,0

λ for functions on R×Tλ ×R as defined before for
one on R× Tλ, if there is no confusion.

The space Xs,b
λ,T will be used for proof of the local well-posedness result, that is

under restriction of time on [0, T ].

If λ = 1, we denote Fλ, F−1,λ, ·̂λ, ·̌λ, Hs
λ, Xs,b

λ , Xs,b
T,λ by F , F−1, ·̂, ·̌, Hs, Xs,b,

Xs,b
T , respectively.

Remark 2.2. The space Xs,b
λ will be characterized by the formula

‖f‖Xs,bλ = ‖et∂x∆f‖HbtHsλ ,

where e−t∂x∆ is the operator associated linear Zakharov-Kuznetsov equation on
R× Tλ, described by

Fλ(e−t∂x∆f)(ξ, q) = eitσ(ξ,q)Fλf(ξ, q).

We summarize some inequalities which were stated in [8, 11, 19, 21].

Lemma 2.3. Let s ∈ R and b > 1/2. Then

‖η(t)e−t∂x∆f‖Xs,bλ . ‖f‖Hsλ ,

for all f ∈ Hs
λ.

Lemma 2.4. Let s ∈ R and b > 1/2. Then

‖η(t)

∫ t

0

e−(t−t′)∂x∆f(t′)dt′‖Xs,bλ . ‖f‖Xs,b−1
λ

,

for all f ∈ Xs,b−1
λ .

Lemma 2.5. For any T > 0, s ∈ R and for all −1/2 < b′ ≤ b < 1/2,

‖f‖
Xs,b

′
T,λ

. T b−b
′
‖f‖Xs,bT,λ ,

for all f ∈ Xs,b
λ .

The polynomial σ(ξ, q) which appeared in (2.3) plays an important role in char-
acterizing of solution. Now, we define the resonance function

H(ξ1, ξ2, q1, q2) = σ(ξ1 + ξ2, q1 + q2)− σ(ξ1, q1)− σ(ξ2, q2)

= 3ξ1ξ2(ξ1 + ξ2) + ξ2q
2
1 + ξ1q

2
2 + 2(ξ1 + ξ2)q1q2,

(2.4)

which plays an important role in the control of the frequency instability range
between nonlinear interactions.

3. Rescaled solutions

Throughout this paper, we assume that λ > 1. Recall Tλ = λT = R/(2πλZ).
By rescaling

uλ(x, y, t) =
1

λ2
u
(x
λ
,
y

λ
,
t

λ3

)
, (3.1)

we consider the Cauchy problem for the Zakharov-Kuznetsov equation on R× Tλ,

∂tu
λ + ∂x∆uλ + uλ∂xu

λ = 0, (x, y) ∈ R× Tλ, t ∈ R,

uλ(x, y, 0) = uλ0 (x, y) ∈ Hs
λ, (x, y) ∈ R× Tλ.

(3.2)



6 S. OSAWA, H. TAKAOKA EJDE-2024/05

If we construct the solution uλ(t) of (3.2) on the time interval [0, λ3T ], we have the
solution u(t) of (1.1) on [0, T ].

Let ζ be a combination of spatial variables, ζ = (ξ, q) ∈ R × Z/λ. Define the
Fourier multiplier operator I, which was originally introduced in [5] to consider
global well-posedness for KdV equation. For N ∈ 2N, define the operator I by

FλIf(ζ) = m(ζ)Fλf(ζ),

where m is a smooth, radially symmetric, non-increasing function satisfying

m(ζ) =

{
1, (|ζ| ≤ N),

( |ζ|N )s−1, (|ζ| ≥ 2N).

On the low frequency part |ζ| ≤ N , the operator I is the identity operator, while
on the high frequency, I is regarded as the integral operator. Remark that I maps
Hs
λ functions to H1

λ one.
In this section, we prove the following local well-posedness results on I−1H1

λ.

Proposition 3.1 (A variant of local well-posedness). Let s > 9/10. The Cauchy
problem (3.2) is locally well-posed in Hs

λ for data uλ0 satisfying uλ0 ∈ Hs
λ. Moreover,

the solution exists on a time interval [0, δ] with δ ∼ ‖Iuλ0‖H1
λ

, and the solution uλ(t)

satisfies the estimate

‖Iuλ‖
X

1,1/2+
δ,λ

. ‖Iuλ0‖H1
λ
.

Next we give several lemmas in conjunction with rescaling argument in [6].

Lemma 3.2 ([19, Lemma 3.7]). Denote a set Λ ⊂ R× Z/λ. Let the projection on
the q be axis contained in a set I ⊂ Z/λ. Assume that there is a positive constant
C such that for any fixed q0 ∈ I ∩ Z/λ and |Λ ∩ {(ξ, q0) | q0 ∈ Z/λ}| ≤ C. Then,
|Λ| ≤ λC(|I|+ 1).

Using this lemma and the mean value theorem, we have the estimates for con-
structing bilinear estimates.

Lemma 3.3 ([19, Lemma 3.8]). Let I and J be two intervals on the real line and
f : J → R be a smooth function. Then

|{x ∈ J | f (x ) ∈ I }| . |I |
infξ∈J |f ′(ξ)|

.

Lemma 3.4. Let a 6= 0, b, c be real numbers and I an interval on the real line.
Then

|{q ∈ Z/λ | aq2 + bq + c ∈ I}| . λ
( |I|1/2
|a|1/2

+ 1
)
.

Proof. Following [19, Lemma 3.9], we only consider the case of a > 0 and b = c = 0.
Put I = [as2, at2] for s, t ∈ R with s2 < t2 and s, t ≥ 0. From the shape of parabola
curve and distribution of Z/λ points, we can say

|{q ∈ Z/λ | aq2 + bq + c ∈ I}| ≤ 2λ(|t− s|+ 1).

Since

|t− s| =
√

(t− s)2 ≤
√

2(t2 − s2) =

√
2(at2 − as2)√

a
=

√
2|I|1/2

|a|1/2
,



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 7

we obtain

|{q ∈ Z/λ | aq2 + bq + c ∈ I}| . λ
( |I|1/2
|a|1/2

+ 1
)
,

which completes the proof. �

Using these lemmas, we obtain the following bilinear estimates.

Lemma 3.5. For u, v ∈ L2
λ, we have

‖(PN1QL1u)(PN2QL2v)‖L2
λ
. (N1 ∧N2)(L1 ∧ L2)1/2‖PN1QL1u‖L2

λ
‖PN2QL2v‖L2

λ
,

(3.3)

‖RK((PN1
QL1

u)(PN2
QL2

v))‖L2
λ

.
(N1 ∧N2)1/2

K1/4
(L1 ∧ L2)1/2(L1 ∨ L2)1/4‖PN1QL1u‖L2

λ
‖PN2QL2v‖L2

λ
.

(3.4)

Moreover, when N1 ∧N2 � N1 ∨N2, for 0 < θ < 1 we have

‖(PN1QL1u)(PN2QL2v)‖L2
λ

.
(N1 ∧N2)1/2

N1 ∨N2
(L1 ∧ L2)1/2(L1 ∨ L2)1/2‖PN1

QL1
u‖L2

λ
‖PN2

QL2
v‖L2

λ
,

(3.5)

‖(PN1
QL1

u)(PN2
QL2

v)‖L2
λ

.
(N1 ∧N2)(1+θ)/2

(N1 ∨N2)1−θ (L1 ∧ L2)1/2(L1 ∨ L2)θ/2‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
,

(3.6)

where N1, N2, L1, L2,K are dyadic numbers.

Proof. Estimate (3.6) follows from the interpolation argument with (3.3) and (3.5),
so that we prove (3.3), (3.5) and (3.4) in this order.

Using the Plancherel’s identity (2.1), convolution structure (2.2) and the Cauchy-
Schwarz inequality, we obtain

‖(PN1QL1u)(PN2QL2v)‖L2
λ

=
( 1

λ

∑
q∈Z/λ

∫
R2

∣∣(F̂λ(PN1
QL1

u) ∗λ F̂λ(PN2
QL2

v)
)
(ξ, q, τ)

∣∣2 dξdτ)1/2

.
1

λ1/2
sup

(ξ,q,τ)∈R×Z/λ×R
|Aξ,q,τ |1/2‖PN1

QL1
u‖L2

λ
‖PN2

QL2
v‖L2

λ
,

where

Aξ,q,τ =
{

(ξ1, q1, τ1) ∈ R× Z/λ× R | |(ξ1, q1)| ∈ IN1
, |(ξ − ξ1, q − q1)| ∈ IN2

,

|τ1 − σ(ξ1, q1)| ∈ IL1
, |τ − τ1 − σ(ξ − ξ1, q − q1)| ∈ IL2

}
.

By the definition of Aξ,q,τ and Lemma 3.2 with |I| ∼ λ(N1 ∧N2), we obtain

|Aξ,q,τ | . λ(N1 ∧N2)2(L1 ∧ L2),

which is (3.3).
Next we prove (3.5). By the triangle inequality

|τ1 − σ(ξ1, q1)|+ |τ − τ1 − σ(ξ − ξ1, q − q1)|
≤ |τ − σ(ξ1, q1)− σ(ξ − ξ1, q − q1)|
= |τ − σ(ξ, q)−H(ξ1, ξ − ξ1, q1, q − q1)|,



8 S. OSAWA, H. TAKAOKA EJDE-2024/05

we obtain

|Aξ,q,τ | . (L1 ∧ L2)|Bξ,q,τ |,
where

Bξ,q,τ =
{

(ξ1, q1) ∈ R× Z/λ | |(ξ1, q1)| ∈ IN1 , |(ξ − ξ1, q − q1)| ∈ IN2 ,

|τ − σ(ξ, q)−H(ξ1, ξ − ξ1, q1, q − q1)| . L1 ∨ L2

}
.

Let us focus on the resonance function H in (2.4). We have∣∣∂H
∂ξ1

(ξ1, ξ−ξ1, q1, q−q1)
∣∣ = |3ξ2

1 +q2
1−(3(ξ−ξ1)2 +(q−q1)2)| & (N1∨N2)2. (3.7)

If we define B̃ξ,q,τ (q1) = {ξ1 ∈ R | (ξ1, q1) ∈ Bξ,q,τ} for each q1, Lemma 3.3 yields

|B̃ξ,q,τ (q1)| . L1 ∨ L2

(N1 ∨N2)2
.

Hence

|Bξ,q,τ | . λ
(L1 ∨ L2)(N1 ∧N2)

(N1 ∨N2)2
.

Combining this with |Aξ,q,τ | . (L1 ∧ L2)|Bξ,q,τ |, we obtain (3.5).
From the Cauchy-Schwarz inequality and the Plancherel’s identity as before, it

holds that

‖RK((PN1
QL1

u)(PN2
QL2

v))‖L2
λ

.
1

λ1/2
sup

(ξ,q,τ)∈R×Z/λ×R
|AKξ,q,τ |1/2‖PN1

QL1
u‖L2

λ
‖PN2

QL2
v‖L2

λ
,

where

AKξ,q,τ =
{

(ξ1, q1, τ1) ∈ R× Z/λ× R | |ξ| ∼ K, |(ξ1, q1)| ∈ IN1
,

|(ξ − ξ1, q − q1)| ∈ IN2 , |τ1 − σ(ξ1, q1)| ∈ IL1 ,

|τ − τ1 − σ(ξ − ξ1, q − q1)| ∈ IL2

}
.

Using the triangle inequality, we have

|AKξ,q,τ | . (L1 ∧ L2)|BKξ,q,τ |,
where

BKξ,q,τ =
{

(ξ1, q1) ∈ R× Z/λ | |ξ| ∼ K, |(ξ1, q1)| ∈ IN1 , |(ξ − ξ1, q − q1)| ∈ IN2 ,

|τ − σ(ξ, q)−H(ξ1, ξ − ξ1, q1, q − q1)| . L1 ∨ L2

}
.

For the bound of BKξ,q,τ , we calculate the second derivative of H as∣∣∂2H
∂ξ1

2 (ξ1, ξ − ξ1, q1, q − q1)
∣∣ = 6|ξ| ∼ K, (3.8)

for (ξ1, q1) ∈ BKξ,q,τ . Let B̃Kξ,q,τ (q1) = {ξ1 ∈ R | (ξ1, q1) ∈ BKξ,q,τ} for each q1.
Combining with Lemma 3.4, we obtain

|B̃Kξ,q,τ (q1)| . (L1 ∨ L2)1/2

K1/2
,

for all q1 ∈ Z/λ. Finally,

|BKξ,q,τ | . λ
(N1 ∧N2)(L1 ∨ L2)

K1/2
(3.9)



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 9

and we obtain (3.4) by |AKξ,q,τ | . (L1 ∧ L2)|BKξ,q,τ |. �

Next we prove bilinear estimates in I−1X
1,1/2+
λ .

Lemma 3.6. For s > 9/10, we have

‖∂xI(uv)‖
X

1,−1/2+
λ

. ‖Iu‖
X

1,1/2+
λ

‖Iv‖
X

1,1/2+
λ

.

Proof. We may assume the functions ûλ and v̂λ are nonnegative, since by the

definition of Xs,b
λ norm.

Replacing 〈ζ〉〈τ − σ(ζ)〉1/2+m(ζ)û(ζ, τ) and 〈ζ〉〈τ − σ(ζ)〉1/2+m(ζ)v̂(ζ, τ) by
û(ζ, τ) and v̂(ζ, τ), respectively, and duality argument, one has to show that

J =
1

λ2

∑
q,q1∈Z/λ

∫
R4

Γξ1,q1,τ1ξ,q,τ

m(ζ)

m(ζ1)m(ζ − ζ1)

× ûλ(ζ1, τ1)v̂λ(ζ − ζ1, τ − τ1)ŵλ(ζ, τ) dξdξ1dτdτ1

. ‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
,

(3.10)

for w ∈ L2
λ whose Fourier transform is nonnegative. Here

Γξ1,q1,τ1ξ,q,τ =
|ξ|〈ζ〉

〈ζ1〉〈ζ − ζ1〉〈τ − σ(ζ)〉1/2−〈τ1 − σ(ζ1)〉1/2+〈τ − τ1 − σ(ζ − ζ1)〉1/2+
.

Using dyadic decomposition, we rewrite J as the following

J =
∑

N0,N1,N2
L0,L1,L2

JL0,L1,L2

N0,N1,N2
,

where

JL0,L1,L2

N0,N1,N2
=

1

λ2

∑
q,q1∈Z/λ

∫
R4

Γξ1,q1,τ1ξ,q,τ

m(ζ)

m(ζ1)m(ζ − ζ1)

× ̂PN1
QL1

u
λ
(ζ1, τ1) ̂PN2

QL2
v
λ
(ζ − ζ1, τ − τ1)

× ̂PN0
QL0

w
λ
(ξ, q, τ) dξdξ1 dτ dτ1.

(3.11)

We decompose again J to five parts in frequencies:

JLL→L =
∑

N1∨N2∨N0�N
L0,L1,L2

JL0,L1,L2

N0,N1,N2
, JLH→H =

∑
N1�N2∼N0

N0&N
L0,L1,L2

JL0,L1,L2

N0,N1,N2
,

JHL→H =
∑

N2�N1∼N0
N0&N

L0,L1,L2

JL0,L1,L2

N0,N1,N2
, JHH→L =

∑
N0�N1∼N2

N1&N
L0,L1,L2

JL0,L1,L2

N0,N1,N2
,

JHH→H = J − (JLL→L + JHL→H + JLH→H + JHH→L).

Estimate of JLL→L. In this case, we have

m(ζ)

m(ζ1)m(ζ − ζ1)
∼ 1, Γξ1,q1,τ1ξ,q,τ .

N1 ∨N2

(N1 ∧N2)L
1/2−
0 L

1/2+
1 L

1/2+
2

.

We split this case two cases, when N1 ∧N2 � N1 ∨N2, and N1 ∧N2 ∼ N1 ∨N2.



10 S. OSAWA, H. TAKAOKA EJDE-2024/05

First, we consider the contribution of the case when N1∧N2 � N1∨N2. In this
case, by (3.5) and the Cauchy-Schwarz inequality, we have that the contribution of
the case to JLL→L is bounded by

.
∑

N1∨N2∨N0�N
N1∧N2�N1∨N2

L0,L1,L2

N1 ∨N2

(N1 ∧N2)L
1/2−
0 L

1/2+
1 L

1/2+
2

× ‖(PN1
QL1

u)(PN2
QL2

v)‖L2
λ
‖PN0

QL0
w‖L2

λ

×
∑

N1∨N2∨N0�N
L0,L1,L2

1

(N1 ∧N2)0+L0+
0 L0+

1 L0+
2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ

. ‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
,

where we use N0 . N1 ∨N2.
Second, we consider the contribution of the case when N1 ∧ N2 ∼ N1 ∨ N2,

namely N1 ∼ N2. Note that PN = PN P̃N for P̃N = PN/2 +PN +P2N . In this case,
the contribution of the case to JLL→L is bounded by

.
∑

N1∨N2∨N0�N
N1∼N2
L0,L1,L2

N1 ∨N2

(N1 ∧N2)L
1/2−
0 L

1/2+
1 L

1/2+
2

× ‖P̃N0((PN1QL1u)(PN2QL2v))‖L2
λ
‖PN0QL0w‖L2

λ

.
∑

N1∼N2
L1,L2

1

L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖w‖L2

λ

. ‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
.

Hence

JLL→L . ‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
.

Estimate of JLH→H . We split this case into two cases that N1 � N . N2 ∼ N0

and that N . N1 � N2 ∼ N0.
If N1 � N . N2 ∼ N0, we have

m(ζ)

m(ζ1)m(ζ − ζ1)
∼ 1.

Applying the Cauchy-Schwarz inequality again with (3.5), we have that the contri-

bution of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
N0

N1L
1/2−
0 L

1/2+
1 L

1/2+
2

‖(PN1
QL1

u)(PN2
QL2

v)‖L2
λ
‖PN0

QL0
w‖L2

λ

.
1

N
1/2
1 L

1/2−
0 L0+

1 L0+
2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ

.
1

N0+
1 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
.

(3.12)

If N . N1 � N2 ∼ N0, we have

m(ζ)

m(ζ1)m(ζ − ζ1)
∼ N1−s

1

N1−s .



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 11

In a similar way to above, we have that the contribution of this case to JL0,L1,L2

N0,N1,N2

is bounded by

.
N0N

1−s
1

N1N1−sL
1/2−
0 L

1/2+
1 L

1/2+
2

‖(PN1
QL1

u)(PN2
QL2

v)‖L2
λ
‖PN0

QL0
w‖L2

λ

.
1

N
s−1/2
1 L

1/2−
0 L0+

1 L0+
2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ

.
1

N0+
1 L0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ
.

(3.13)

Therefore, by (3.12) and (3.13) in conjunction with previous estimate, we have

JLH→H . ‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
,

which is acceptable.

Estimate of JHL→H . The proof is same as for JLH→H , because of the symmetry.

Estimate of JHH→L. By symmetry, we assume N0 � N1 ≤ N2. We separate this
case into two cases that N0 � N � N1 ∼ N2 and that N . N0 � N1 ∼ N2.

If N0 � N � N1 ∼ N2, we have

m(ζ)

m(ζ1)m(ζ − ζ1)
∼ N1−s

1 N1−s
2

N2(1−s) .

Applying the L2
λ norm of functions ˜(PN1QL1u)(PN0QL0w) and PN2QL2v, we have

that then the contribution of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
N1−s

1 N1−s
2 N2

0

N2(1−s)N1N2L
1/2−
0 L

1/2+
1 L

1/2+
2

‖ ˜(PN1
QL1

u)(PN0
QL0

w)‖L2
λ
‖PN2

QL2
v‖L2

λ
,

where
̂̃
f
λ

(ζ, τ) = f̂λ(−ζ,−τ). We apply (3.6) to this and have that the contribution

of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
N1−s

1 N1−s
2 N2

0

N2(1−s)N1N2L
1/2−
0 L

1/2+
1 L

1/2+
2

N
(1+θ)/2
0 (L0 ∧ L1)1/2(L0 ∨ L1)θ/2

N1−θ
1

× ‖PN1QL1u‖L2
λ
‖PN0QL0w‖L2

λ
‖PN2QL2v‖L2

λ

.
1

N1−sN
s−1/2−3θ/2
1 L0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN0QL0w‖L2

λ
‖PN2QL2v‖L2

λ
,

(3.14)
for small θ > 0.

If N . N0 � N1 ∼ N2, we have

m(ζ)

m(ζ1)m(ζ − ζ1)
∼ N1−s

1 N1−s
2

N1−sN1−s
0

.

Similarly, the contribution of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
1

N
s−1/2−3θ/2
1 N1−sL0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN0QL0w‖L2

λ
‖PN2QL2v‖L2

λ
,

(3.15)
for small θ > 0.



12 S. OSAWA, H. TAKAOKA EJDE-2024/05

Summing with respect to N0, N1, N2, L0, L1, L2, we obtain

JHH→L . ‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
.

Estimate of JHH→H . In this case, we can assume N � N0 ∼ N1 ∼ N2 and have

m(ζ)

m(ζ1)m(ζ − ζ1)
∼ N1−s

0

N1−s , Γξ1,q1,τ1ξ,q,τ ∼ |ξ|
N0L

1/2−
0 L

1/2+
1 L

1/2+
2

.

Now, we separate this case into five subcases:

(i) |ξ| . 1,
(ii) |ξ1| ∧ |ξ − ξ1| . 1� |ξ|,
(iii) 1� |ξ1|∧|ξ−ξ1|∧|ξ| and ||ζ|2−|ζ1|2|∨||ζ1|2−|ζ−ζ1|2|∨||ζ−ζ1|2−|ζ|2| &

N
6/5
0 (L0 ∨ L1 ∨ L1)0+,

(iv) 1� |ξ1| ∧ |ξ− ξ1| ∧ |ξ|, ||ζ|2− |ζ1|2| ∧ ||ζ1|2− |ζ − ζ1|2| ∧ ||ζ − ζ1|2− |ζ|2| �
N

6/5
0 (L0 ∨ L1 ∨ L1)0+ and (|ξ| ∨ |ξ1| ∨ |ξ − ξ1|)(|ξ| ∧ |ξ1| ∧ |ξ − ξ1|) �

N
6/5
0 (L0 ∨ L1 ∨ L1)0+,

(v) 1� |ξ1| ∧ |ξ− ξ1| ∧ |ξ|, ||ζ|2− |ζ1|2| ∧ ||ζ1|2− |ζ − ζ1|2| ∧ ||ζ − ζ1|2− |ζ|2| �
N

6/5
0 (L0 ∨ L1 ∨ L1)0+ and (|ξ| ∨ |ξ1| ∨ |ξ − ξ1|)(|ξ| ∧ |ξ1| ∧ |ξ − ξ1|) .

N
6/5
0 (L0 ∨ L1 ∨ L1)0+.

Subcase (i). Denote

JL0,L1,L2

N0,N1,N2
=
∑
k∈N

JL0,L1,L2

N0,N1,N2
(k),

where

JL0,L1,L2

N0,N1,N2
(k) =

1

λ2

∑
q,q1∈Z/λ

∫
Yk

Γξ1,q1,τ1ξ,q,τ

m(ζ)

m(ζ1)m(ζ − ζ1)
̂PN1QL1u

λ
(ζ1, τ1)

× ̂PN2
QL2

v
λ
(ζ − ζ1, τ − τ1) ̂PN0

QL0
w
λ
(ζ, τ) dξdξ1dτ dτ1,

Yk = {(ξ, ξ1, τ, τ1) ∈ R4 | |ξ| ∼ 2−k}.
We apply the Cauchy-Schwarz inequality, and we have that the contribution of this

case to JL0,L1,L2

N0,N1,N2
(k) is bounded by

.
2−k

N1−sNs
0L

1/2−
0 L

1/2+
1 L

1/2+
2

‖R2−k ((PN1QL1u)(PN2QL2v)) ‖L2
λ
‖PN0QL0w‖L2

λ
.

For |ξ| ∼ 2−k, we obtain by (2.4)

|∂
2H
∂ξ2

1

(ξ1, ξ − ξ1, q1, q − q1)| = 6|ξ| ∼ 2−k.

Using (3.4), we obtain

‖R2−k((PN1QL1u)(PN2QL2v))‖L2
λ

. 2k/4N
1/2
0 (L1 ∨ L2)1/4(L1 ∧ L2)1/2‖PN1QL1u‖L2

λ
‖PN2QL2v‖L2

λ
.

Then

JL0,L1,L2

N0,N1,N2
(k)

.
2−3k/4

N1−sN
s−1/2
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
.

(3.16)



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 13

Summing with respect to dyadic numbers N0, N1, N2, L0, L1, L2 and k ∈ N, we have
the desired bound for the contribution of this case to JHH→H .

Subcase (ii). By symmetry, we may suppose |ξ−ξ1| ≤ |ξ1|∧1. First, we calculate
the resonance function (2.4) as

∂H
∂ξ1

(ξ1, ξ − ξ1, q, q − q1) = 3ξ(ξ − 2ξ1) + q(q − 2q1).

First we consider the case when |∂H/∂ξ1| & ξ2. We shall use the dyadic decompo-
sition |ξ| ∼ K. By the Cauchy-Schwarz inequality such as Case (i) above, we have

that the contribution of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
∑
K∈2N

K

N1−sNs
0L

1/2−
0 L

1/2+
1 L

1/2+
2

× ‖RK((PN1
QL1

u)(PN2
QL2

v))‖L2
λ
‖RKPN0

QL0
w‖L2

λ
.

Recall the proof of Lemma 3.5, we have

‖RK((PN1
QL1

u)(PN2
QL2

v))‖L2
λ

.
1

λ1/2
sup

(ξ,q,τ)∈R×Z/λ×R
|Aξ,q,τ (K)|1/2‖PN1QL1u‖L2

λ
‖PN2QL2v‖L2

λ
,

where

Aξ,q,τ (K) =
{

(ξ1, q1, τ1) ∈ R× Z/λ× R | |ξ| ∼ K, |ζ1| ∈ IN1
, |ζ − ζ1| ∈ IN2

,

|τ1 − σ(ζ1)| ∈ IL1
, |τ − τ1 − σ(ζ − ζ1)| ∈ IL2

}
.

From the triangle inequality,

|Aξ,q,τ (K)| . (L1 ∧ L2)|Bξ,q,τ (K)|,

where

Bξ,q,τ (K) =
{

(ξ1, q1) ∈ R× Z/λ | |ξ| ∼ K, |ζ1| ∈ IN1 , |ζ − ζ1| ∈ IN2 ,

|τ − σ(ζ)−H(ξ1, ξ − ξ1, q1, q − q1)| . L1 ∨ L2}.

Let B̃ξ,q,τ (K, q1) = {ξ1 ∈ R | ζ1 ∈ Bξ,q,τ (K)} for each q1. Then combining Lemma
3.3 and the bound |∂H/∂ξ1| & ξ2 ∼ K2, we obtain

|B̃ξ,q,τ (K, q1)| . L1 ∨ L2

K2
.

Hence, we obtain

|Bξ,q,τ | . λ
(L1 ∨ L2)(N1 ∧N2)

K2
.

Then,

‖RK((PN1
QL1

u)(PN2
QL2

v))‖L2
λ

.
N

1/2
0 (L1 ∨ L0)1/2(L1 ∧ L0)1/2

K
‖PN1

QL1
u‖L2

λ
‖PN0

QL0
w‖L2

λ
.



14 S. OSAWA, H. TAKAOKA EJDE-2024/05

Combining the Cauchy-Schwarz inequality and bilinear estimate, we obtain that

the contribution of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
∑
K.N0

1

N1−sN
s−1/2
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖RKPN0

QL0
w‖L2

λ

.
1

N1−sN
s−1/2−
0 L0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ
.

(3.17)
Again summing with respect to dyadic numbers N0, N1, N2, L0, L1, L2, we have the
desired bound for the contribution of this case to JHH→H .

Next, we consider the case when |∂H/∂ξ1| � ξ2. In this case, we have

∂H
∂ξ1

(ξ1, ξ − ξ1, q1, q − q1) = 3ξ(ξ − ξ1)− 3ξξ1 + q(q − q1)− qq1

= 6ξ(ξ − ξ1)− 3ξ2 + q(q − q1)− qq1

= O(ξ)− 3ξ2 + q(q − q1)− qq1.

To satisfy the hypothesis of resonance function |∂H/∂ξ1| � ξ2, N0 ∼ N1 ∼ N2 and
|ξ− ξ1| . 1� |ξ|, we need |q− q1| ∼ N0, |ξ| . |q| ∧ |q1| to set q(q− q1)− qq1 ∼ ξ2.
Furthermore we have

|q| ∼ |q1| ∼ |q − q1| ∼ |ξ| ∼ N0, 3ξξ1(ξ − ξ1) ∼ O(ξ2),

(ξ − ξ1)(q2 − (q − q1)2) ∼ O(ξ2).

Let us recall the resonance function H in (2.4),

H(ξ1, ξ− ξ1, q1, q− q1) = 3ξξ1(ξ− ξ1) + (ξ− ξ1)(q2− (q− q1)2) + (q− q1)ξ1(q+ q1),

If |H(ξ1, ξ − ξ1, q1, q − q1)| & ξ2, then L0 ∨ L1 ∨ L2 & ξ2 by (2.4). Hence we
have |ξ| . N0+

0 (L0 ∨L1 ∨L2)1/2−. The same estimate as in (3.13) implies that the
contribution of this case to JHH→H is bounded by

.
N0+

0 N1−s
0

N0N1−sL0+
0 L0+

1 L0+
2

‖(PN1
QL1

u)(PN2
QL2

v)‖L2
λ
‖PN0

QL0
w‖L2

λ

∼ 1

N1−sNs−
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
.

(3.18)

Summing in dyadic numbers N0, N1, N2, L0, L1, L2, we have the bound of this case
to JHH→H by

. ‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
,

as desired.
On the other hand, if |H(ξ1, ξ−ξ1, q1, q−q1)| � ξ2, it follows that |(q−q1)ξ1(q+

q1)| . O(ξ2) and then |q + q1| . 1. In this case, we use the form

q1 = [q1] + (q1 − [q1])

and then

q − q1 = [q] + (q − [q])− q1 = [q]− [q1] + (q − [q]− q1 + [q1]) ,

where [a] denotes integer part of a ∈ R. Note that q1 − [q1], q − q1 − [q − q1] ∈
Z/λ∩ [0, 1). The restriction |q+ q1| . 1 implies |2[q1] + [q]| . 1. By performing the
same calculation as [21, Proof of Proposition 3.1] and [11, Proof of Theorem 2.1],



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 15

we use the Cauchy-Schwarz inequality to have that the contribution of this case to
JHH→H is bounded by

. sup
(ζ,τ)∈R×Z/λ×R

I(ζ, τ)‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
,

where

I(ζ, τ) =
N1−s

0

λ1/2L
1/2−
0 N1−s

×
( ∑
|2[q1]+[q]|.1

∫
|ξ−ξ1|.1

dξ1
〈τ − σ(ζ)−H(ξ1, ξ − ξ1, q1, q − q1)〉1+

)1/2

,

where we assume |H(ξ1, ξ − ξ1, q1, q − q1)| � ξ2 in the integration of region. Then
it suffices to show that

sup
(ζ,τ)∈R×Z/λ×R

I(ζ, τ)2 <∞. (3.19)

The resonance function is reformulated as

H = 3ξξ1(ξ − ξ1) + (ξ − ξ1)(q2 − (q − q1)2) + (q − q1)ξ1(q + q1),

which is equivalent to a quadratic equation in ξ1,

3ξξ2
1 − ((2q − q1)q + 3ξ2)ξ1 − (q2 − (q − q1)2)ξ +H = 0.

The roots of the quadratic equation are the values of ξ1 as ξ±1 , where

ξ±1 =
(2q1 − q)q1 − 3ξ2 ±

√
((2q1 − q)q − 3ξ2)

2
+ 12ξ2 (q2 − (q − q1)2)− 12ξH

6ξ
.

Using the change of variables, we have

dξ1 = ± dH√
((2q1 − q)q − 3ξ2)

2
+ 12ξ2 (q2 − (q − q1)2)− 12ξH

.

We evaluate integrals using the change of variables as∫
|ξ−ξ1|.1

dξ1
〈τ − σ(ζ)−H(ξ1, ξ − ξ1, q1, q − q1)〉1+

.
∫
R

dH
〈τ − σ(ζ)−H〉1+| ((2q1 − q)q − 3ξ2)

2
+ 12ξ2 (q2 − (q − q1)2)− 12ξH|1/2

.
1

|ξ|1/2 + |ξ(τ − σ(ζ)) + ((2q1 − q)q − 3ξ2)
2
/12 + ξ2 (q2 − (q − q1)2) |1/2

.

Then we have the bound∫
|ξ−ξ1|.1

dξ1
〈τ − σ(ζ)−H(ξ1, ξ − ξ1, q1, q − q1)〉1+

.
1

N
1/2
0

.

Therefore,

I(ζ, τ) .
N1−s

0

λ1/2L
1/2−
0 N1−s

( ∑
|2[q1]+[q]|.1

1

N
1/2
0

)1/2

.
1

L
1/2−
0 N1−sN

s−3/4
0

. 1,

(3.20)

which is acceptable for (3.19).



16 S. OSAWA, H. TAKAOKA EJDE-2024/05

Subcase (iii). The proof follows from the same as one in [19] (also [21, Proposition

3.1]). By symmetry, we assume ||ζ|2 − |ζ1|2| & N6/5
0 L0+

0 . Then

|∂H
∂ξ1

(ξ1, ξ − ξ1, q1, q − q1)| =
∣∣|ζ|2 − |ζ1|2∣∣ & N6/5

0 L0+
0 ,

which modifies the computation appeared in (3.7) for the proof of (3.5). Hence the

contribution of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
N1−s

0

N1−sL
1/2−
0 L

1/2+
1 L

1/2+
2

N
1/2
0 (L1 ∨ L2)1/2

N
3/5
0 L0+

0

× ‖PN1
QL1

u‖L2
λ
‖PN1

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ

.
1

N1−sN
s−9/10
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN1

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
,

(3.21)

which is acceptable after taking the sum in dyadic numbers N0, N1, N2, L0, L1, L2.

Subcase (iv). We can rewrite resonance function H as

H(ξ1, ξ − ξ1, q1, q − q1) = 3ξξ1(ξ − ξ1) + ξ1q
2 − ξq2

1 − 2ξ1qq1 + 2ξqq1

= −3ξξ1(ξ − ξ1) + P (ξ, ξ1, q, q1),

where

P (ξ, ξ1, q, q1) = 6ξξ1ξ2 + ξ1q
2 − ξq2

1 − 2ξ1qq1 + 2ξqq1.

By the same argument as in [19] (also [21, Proposition 3.1]), we have in this case

|H(ξ1, ξ − ξ1, q1, q − q1)| & (|ξ| ∨ |ξ1| ∨ |ξ − ξ1|)2(|ξ| ∧ |ξ1| ∧ |ξ − ξ1|)
& (|ξ| ∨ |ξ1| ∨ |ξ − ξ1|)(|ξ| ∧ |ξ1| ∧ |ξ − ξ1|)|ξ|

� N
6/5
0 L0+

0 |ξ| & |ξ|11/5L0+
0 .

By symmetry we assume L0 = L0∨L1∨L2, which implies |H(ξ1, ξ−ξ1, q1, q−q1)| .
L0. Combining these estimates in this case, we obtain |ξ| . L

5/11−
0 . We use (3.4)

to have that the contribution of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
∑

K.L1/2−−
0

K

N1−sNs
0L

1/2−
0 L

1/2+
1 L

1/2+
2

N
1/2
0

K1/4
(L1 ∧ L2)1/2(L1 ∨ L2)1/2

× ‖PN1QL1u‖L2
λ
‖PN1QL2v‖L2

λ
‖RKPN0QL0w‖L2

λ

.
1

N1−sN
s−9/10
0 L0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN1QL2v‖L2

λ
‖RKPN0QL0w‖L2

λ
,

(3.22)
which is acceptable after taking the sum in dyadic numbers N0, N1, N2, L0, L1, L2.

Subcase (v). Finally, we consider this case. We repeat the argument in [19]
(also [21, Proposition 3.1]). Introducing dyadic numbers Kj , we have that the



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 17

contribution this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
1

N1−sNs
0L

1/2−
0 L

1/2+
1 L

1/2+
2

×
∑

K0,K1,K2

K0‖RK0((RK1PN1QL1u)(RK2PN2QL2v))‖L2
λ
‖RK0PN0QL0w‖L2

λ
.

(3.23)
In the proof of (3.4), we modify that in (3.9)

|BKξ,q,τ | . λ
(K1 ∧K2)(L1 ∨ L2)

K
1
2

,

which shows

‖RK0
((RK1

PN1
QL1

u)(RK2
PN2

QL2
v))‖L2

λ

.
(K1 ∧K2)1/2(L1 ∧ L2)1/2(L1 ∨ L2)1/4

K
1/4
0

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
.

Then the left hand-side of (3.23) is controlled by

.
∑

K0,K1,K2

K
3/4
0 (K1 ∧K2)1/2(L1 ∧ L2)1/2(L1 ∨ L2)1/4

N1−sNs
0L

1/2−
0 L

1/2+
1 L

1/2+
2

× ‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
,

where the sum is based on the region (K0 ∨K1 ∨K2)(K0 ∧K1 ∧K2) . N6/5
0 (L0 ∨

L1 ∨ L1)0+. Since K
3/4
0 (K1 ∧K2)1/2 . N

9/10
0 (L0 ∨ L1 ∨ L2)0+, we have that the

contribution of this case to JL0,L1,L2

N0,N1,N2
is bounded by

.
1

N1−sN
s−9/10
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

RK1
u‖L2

λ
‖PN2

QL2
RK2

v‖L2
λ
‖RK0

PN0
QL0

w‖L2
λ
,

(3.24)

which is acceptable.
We completed the proof of JHH→H , and hence of Lemma 3.6. �

Proof of Proposition 3.1. Define

Ψ(uλ)(t) = η(t)e−t∂x∆uλ0 +
1

2
η(t)

∫ t

0

e−(t−t′)∂x∆∂x
(
uλ(t′)2

)
dt′.

We show that the map Ψ defines a contraction in

Yδ,λ =
{
uλ ∈ I−1X

1,1/2+
δ,λ | ‖Iuλ‖

X
1,1/2+
δ,λ

≤ 2C‖Iuλ0‖H1
λ

}
,

for (small) δ > 0, where the norm on Yδ,λ is induced by ‖Iuλ‖
X

1,1/2+
δ,λ

, In fact, from

Lemmas 2.3, 2.4 and 2.5 and 3.6, it follows that

‖IΨ(uλ)‖
X

1,1/2+
δ,λ

≤ C‖Iuλ0‖H1
λ

+ C‖∂xI(uλ)2‖
X

1,−1/2+ε
δ,λ

≤ C‖Iuλ0‖H1
λ

+ Cδε‖∂xI(uλ)2‖
X

1,−1/2+2ε
δ,λ

≤ C‖Iuλ0‖H1
λ

+ Cδε‖Iuλ‖2
X

1,−1/2+
δ,λ

≤ 2C‖Iuλ0‖H1
λ
,

(3.25)



18 S. OSAWA, H. TAKAOKA EJDE-2024/05

for uλ ∈ Yδ,λ and choosing small δ > 0 which will depend on ‖Iuλ0‖Hsλ . Similarly,

‖IΨ(uλ)− IΨ(vλ)‖
X

1,1/2+
δ,λ

≤ 1

2
‖Iuλ − Ivλ‖

X
1,1/2+
δ,λ

,

for uλ, vλ ∈ Yδ,λ and small δ > 0. Then the contraction mapping theorem tells us
that there is a unique solution uλ = Ψ(uλ) ∈ Yδ,λ to the Cauchy problem (3.2).

The persistence property uλ ∈ C([0, δ] : Hs
λ) and the uniqueness in whole space

Iuλ ∈ X
1,1/2+
δ,λ follow in a similar way to [11, Proof of Theorem 1.5], by using a

variant of (3.25), therefore we omit them. �

4. Modified energy

The conserved quantities associated with (3.2) are

Eλ[uλ](t) =
1

2

∫
R×Tλ

(
|∇uλ(x, y, t)|2 − 1

3
uλ(x, y, t)

3
)
dx dy,

Mλ[uλ](t) =

∫
R×Tλ

uλ(x, y, t)
2
dx dy.

From Fλuλ0 (ξ, q) = Fu0(λξ, λq), it is easy to see that

‖Iuλ0‖L2
λ
≤ ‖uλ0‖L2

λ
≤ ‖u0‖L2

λ
= o(1) (4.1)

for λ� 1. Here the choice of the large parameter N will be made latter, but λ > 1
is chosen by

N1−s

λ1+s
= o(1), (4.2)

in which

‖∇Iuλ0‖L2
λ
.
N1−s

λ1+s
‖u0‖Hs = o(1), ‖Iuλ0‖H1

λ
= o(1). (4.3)

Moreover, we have

‖u
( t
λ3

)
‖Hs . λ1+s‖Iuλ(t)‖Hsλ ≤ λ

1+s‖Iuλ(t)‖H1
λ
. (4.4)

Remark 4.1. By Gagliardo-Nirenberg inequality, the conservation law of L2
λ-norm

and (4.1), the solution uλ(t) satisfies

‖∇Iuλ(t)‖2L2
λ
− C1‖uλ0‖4L2

λ
≤ C2E

λ[Iuλ](t), (4.5)

where constants C1 and C2 are independent of λ.

Let us introduce the modified energy for proving global well-posedness. Using
the Fundamental Theorem of Calculus, we obtain

Eλ[Iuλ](δ)− Eλ[Iuλ](0) =

∫ δ

0

dEλ[Iuλ](t)

dt
dt



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 19

for δ > 0. We continue calculation of the integrand in the right-hand side. Then

dEλ[Iuλ]

dt
(t) =

∫
R×Tλ

( d
dt
∇Iuλ · ∇Iuλ − 1

2
(Iuλ)2 d

dt
Iuλ

)
dx dy

=

∫
R×Tλ

(
− I∂tuλ∆Iuλ − 1

2
I∂tu

λ(Iuλ)2
)
dx dy

=

∫
R×Tλ

∂x

(
I∆uλ +

1

2
I(uλ)2

)(1

2
(Iuλ)2 + ∆Iuλ

)
dx dy

= −1

2

∫
R×Tλ

I∆uλ∂x
(
(Iuλ)2 − I(uλ)2

)
dx dy

− 1

4

∫
R×Tλ

I(uλ)2∂x
(
(Iuλ)2 − I(uλ)2

)
dx dy,

(4.6)

where we use the equation in (3.2). Taking the integral on [0, δ], we have

Eλ[Iuλ](δ)− Eλ[Iuλ](0)

= −1

2

∫ δ

0

∫
R×Tλ

I∆uλ∂x
(
(Iuλ)2 − I(uλ)2

)
dx dy dt

− 1

4

∫ δ

0

∫
R×Tλ

I(uλ)2∂x
(
(Iuλ)2 − I(uλ)2

)
dx dy dt

(4.7)

We will estimate two terms on the right hand of (4.7) with ‖Iuλ‖
X

1,1/2+
λ,δ

by multi-

linear estimates associated with the transition of energy in (4.7).

Lemma 4.2. For s > 9/10, we have

‖∂x(IuIv − I(uv))‖
X

1,−1/2+
λ

. N−1/10+‖Iu‖
X

1,1/2+
λ

‖Iv‖
X

1,1/2+
λ

. (4.8)

Remark 4.3. Comparing to the estimate in Lemma 3.6, we have the small factor
at the front of the right-hand, which corresponding to properties such as dispersion
or the smoothing effects.

Proof. We recast the proof of Lemma 3.6 in the form of formula (4.8). Here we
used the same symbol over as in the proof of Lemma 3.6. By Plancherel identity,
it suffices to show that

J =
1

λ2

∑
q,q1∈Z/λ

∫
R4

Γξ1,q1,τ1ξ,q,τ

m(ζ)

m(ζ1)m(ζ − ζ1)
Ξ(ζ, ζ1, ζ − ζ1)

× ûλ(ζ1, τ1)v̂λ(ζ − ζ1, τ − τ1)ŵλ(ζ, τ) dξdξ1dτdτ1

. ‖u‖L2
λ
‖v‖L2

λ
‖w‖L2

λ
,

(4.9)

for u, v, w ∈ L2
λ whose Fourier transform are nonnegative. In (4.9), we have

Γξ1,q1,τ1ξ,q,τ =
|ξ|〈ζ〉

〈ζ1〉〈ζ − ζ1〉〈τ − σ(ζ)〉1/2−〈τ1 − σ(ζ1)〉1/2+〈τ − τ1 − σ(ζ − ζ1)〉1/2+
,

Ξ(ζ, ζ1, ζ − ζ1) =
N1/10−

m(ζ)
|m(ζ1)m(ζ − ζ1)−m(ζ)| .

Note the trivial bound

Ξ(ζ, ζ1, ζ − ζ1) .
N1/10−

m(ζ)
. (4.10)



20 S. OSAWA, H. TAKAOKA EJDE-2024/05

Using dyadic decomposition in a similar way to the proof of Lemma 3.6, we rewrite
J as the following

J =
∑

N0,N1,N2
L0,L1,L2

JL0,L1,L2

N0,N1,N2
,

where

JL0,L1,L2

N0,N1,N2
=

1

λ2

∑
q,q1∈Z/λ

∫
R4

Γξ1,q1,τ1ξ,q,τ

m(ζ)

m(ζ1)m(ζ − ζ1)
Ξ(ζ, ζ1, ζ − ζ1)

× ̂PN1
QL1

u
λ
(ζ1, τ1) ̂PN2

QL2
v
λ
(ζ − ζ1, τ − τ1)

× ̂PN0
QL0

w
λ
(ξ, q, τ) dξdξ1dτ dτ1.

(4.11)

We repeat the same procedure as one of Lemma 3.6. Decompose again J to five
parts in frequencies:

JLL→L =
∑

N1∨N2∨N0�N
L0,L1,L2

JL0,L1,L2

N0,N1,N2
, JLH→H =

∑
N1�N2∼N0

N0&N
L0,L1,L2

JL0,L1,L2

N0,N1,N2
,

JHL→H =
∑

N2�N1∼N0
N0&N

L0,L1,L2

JL0,L1,L2

N0,N1,N2
, JHH→L =

∑
N0�N1∼N2

N1&N
L0,L1,L2

JL0,L1,L2

N0,N1,N2
,

JHH→H = J − (JLL→L + JHL→H + JLH→H + JHH→L).

We estimate each of them by case analysis.

Estimate of JLL→L. In this region, we have Ξ(ζ, ζ1, ζ − ζ1) = 0, since by m(ζ) =
m(ζ1) = m(ζ − ζ1) = 1. Hence JLL→L = 0.

Estimate of JLH→H . We split the case into two cases, when N1 & N and when
N1 � N . When N1 & N , we use the bound (4.10). On the other hand, when
N1 � N , the mean value theorem gives

|m(ζ1)m(ζ − ζ1)−m(ζ)| = |m(ζ − ζ1)−m(ζ)| . m′(N0)N1 ∼
m(N0)N1

N0
. (4.12)

By making use of the estimates in (3.12) and (3.13), we easily have the desired
bounds for the contribution of this case to JLH→H . Indeed, when N1 � N , the
computation in (3.12) with (4.10) leads the bound

.
N1

N0

N1/10−

N
1/2
1 L

1/2−
0 L0+

1 L0+
2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ

.
1

N0+
1 L

1/2−
0 L0+

1 L0+
2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
.

While N1 & N , we have |m(ζ1)m(ζ−ζ1)−m(ζ)| . m(N0) and then Ξ(ζ, ζ1, ζ−ζ1) .
N1/10−. By the computation in (3.13) with (4.12) we estimate this contribution by

. N1/10− 1

N
s−1/2
1 L

1/2−
0 L0+

1 L0+
2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ

.
1

N0+
1 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
.

Then the desired estimate follows.



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 21

Estimate of JHL→H . We use symmetry arguments as for JLH→H to conclude the
proof.

Estimate of JHH→L. In this case, we use the bound (4.10). By employing the
same estimates as in (3.14) and (3.15), we have the desired bounds. Actually, the
computation in (3.14) will be changed into

.
N1/10−

N1−sN
s−1/2−3θ/2
1 L0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN0QL0w‖L2

λ
‖PN2QL2v‖L2

λ

.
1

N
2/5−3θ/2
1 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN0

QL0
w‖L2

λ
‖PN2

QL2
v‖L2

λ
,

for small θ > 0, while the formula in (3.15) is changed by the same.

Estimate of JHH→H : In this case, we can assume N � N0 ∼ N1 ∼ N2 and then

Ξ(ζ, ζ1, ζ − ζ1) . N1/10−.

We divide the region into five regions as in the proof of Lemma 3.6. On each case,
we recall the estimates in (3.16), (3.17), (3.18), (3.20), (3.21), (3.22) and (3.24),
by multiplying Ξ(ζ, ζ1, ζ − ζ1). We see that JHH→H . ‖u‖L2

λ
‖v‖L2

λ
‖w‖L2

λ
holds

provided s > 9/10.
In fact, in subcase (i), the estimate corresponding to (3.16) will be

. N1/10− 2−3k/4

N1−sN
s−1/2
0 L0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ

.
2−3k/4

N
2/5−
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
,

which is acceptable after summing over N0, N1, N2, L0, L1, L2.
In subcase (ii), we revisit the estimates in (3.17), (3.18) and (3.20). We require

the following estimate corresponding to (3.17)

. N1/10− 1

N1−sN
s−1/2−
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ

.
1

N
2/5−
0 L0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ
.

The corresponding estimate to (3.18) is

. N1/10− 1

N1−sNs−
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN2

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ

∼ 1

N
9/10−
0 L0+

0 L0+
1 L0+

2

‖PN1QL1u‖L2
λ
‖PN2QL2v‖L2

λ
‖PN0QL0w‖L2

λ
.

The estimate corresponding to (3.20) is

I(ζ, τ) . N1/10− 1

L
1/2−
0 N1−sN

s−3/4
0

. 1.

By a similar proof to one of Lemma 3.6, we have a desired bound.
In subcase (iii), it suffices to show the following estimate corresponding to (3.21),

. N1/10− N1−s
0

N1−sL
1/2−
0 L

1/2+
1 L

1/2+
2

N
1/2
0 (L1 ∨ L2)1/2

N
3/5
0 L0+

0



22 S. OSAWA, H. TAKAOKA EJDE-2024/05

× ‖PN1
QL1

u‖L2
λ
‖PN1

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ

.
1

N0+
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN1

QL2
v‖L2

λ
‖PN0

QL0
w‖L2

λ
,

which holds for s > 9/10.
In subcase (iv), the corresponding estimate to (3.22) is

. N1/10− 1

N1−sN
s−9/10
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN1

QL2
v‖L2

λ
‖RKPN0

QL0
w‖L2

λ

.
1

N0+
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

u‖L2
λ
‖PN1

QL2
v‖L2

λ
‖RKPN0

QL0
w‖L2

λ
,

which is acceptable for s > 9/10.
Finally, in subcase (v), we need the following estimate corresponding estimate

to (3.24)

. N1/10− 1

N1−sN
s−9/10
0 L0+

0 L0+
1 L0+

2

× ‖PN1
QL1

RK1
u‖L2

λ
‖PN2

QL2
RK2

v‖L2
λ
‖RK0

PN0
QL0

w‖L2
λ

.
1

N0+
0 L0+

0 L0+
1 L0+

2

‖PN1
QL1

RK1
u‖L2

λ
‖PN2

QL2
RK2

v‖L2
λ
‖RK0

PN0
QL0

w‖L2
λ
,

which is also acceptable for s > 9/10. Hence the proof is complete �

By Lemma 4.2, we have the following lemma.

Lemma 4.4. For s > 9/10, we have∣∣ ∫ δ

0

∫
R×Tλ

I∆u∂x
(
(Iu)2 − I(u)2

)
dx dy dt

∣∣ . 1

N1/10− ‖Iu‖
3

X
1,1/2+
δ,λ

.

Proof. We apply the Parseval’s identity and the Cauchy-Schwartz inequality to get
the bound of the left-hand side by

. ‖I∆u‖
X
−1,1/2+
δ,λ

‖∂x(Iu)2 − I(u)2)‖
X

1,1/2−
δ,λ

. ‖Iu‖
X

1,1/2+
δ,λ

‖∂x(Iu)2 − I(u)2)‖
X

1,1/2−
δ,λ

.

Employing Lemma 4.2, we have the desired bound. �

Lemma 4.5. For s > 31/40, we have∣∣ ∫
R×Tλ×R

Iu2∂x
(
(Iu)2 − I(u)2

)
dx dy dt

∣∣ . 1

N1/10+
‖Iu‖4

X
1,1/2+
λ

.

Proof. We apply Parseval’s identity to the left-hand side and use the same argument
as in the proof of Lemma 3.6.

It suffices to show that

J =
1

λ3

∑
∗

∫
∗
|ξ3 + ξ4|Γτ1,τ2,τ3,τ4ζ1,ζ2,ζ3,ζ4

Ξ(ζ1, ζ2, ζ3, ζ4)

4∏
j=1

ûλ(ζj , τj) . ‖u‖4L2
λ
, (4.13)

for u ∈ L2
λ whose Fourier transform is nonnegative. In (4.13),

∑
∗
∫
∗ stands for

the sum-integral of (ζj , τj) ∈ R × Z/λ × R (1 ≤ j ≤ 4) over ξ1 + ξ2 + ξ3 + ξ4 =



EJDE-2024/05 ZAKHAROV-KUZNETSOV EQUATION ON CYLINDRICAL SPACES 23

η1 + η2 + η3 + η4 = τ1 + τ2 + τ3 + τ4 = 0, and

Γτ1,τ2,τ3,τ4ζ1,ζ2,ζ3,ζ4
=

4∏
j=1

1

m(ζj)〈ζj〉〈τj − σ(ζj)〉1/2+
,

Ξ(ζ1, ζ2, ζ3, ζ4) = m(ζ1 + ζ2) |m(ζ3)m(ζ4)−m(ζ3 + ζ4)|N1/10+.

Note the trivial bound

Ξ(ζ1, ζ2, ζ3, ζ4) . N1/10+. (4.14)

By dyadic decomposition in a similar way to the proof of Lemma 3.6, we use
〈ζj〉 ∼ Nj and 〈τj − σ(ζj)〉 ∼ Lj , as before,

J =
∑

N1,N2,N3,N4
L1,L2,L3,L4

JL1,L2,L3,L4

N1,N2,N3,N4
,

where

JL1,L2,L3,L4

N1,N2,N3,N4
=

1

λ3

∑
∗

∫
∗
|ξ3 + ξ4|Γτ1,τ2,τ3,τ4ζ1,ζ2,ζ3,ζ4

Ξ(ζ1, ζ2, ζ3, ζ4)

4∏
j=1

̂PNjQLju
λ
(ζj , τj).

We will show (4.13) using a the case-by-case analysis.
In case N3 ∨ N4 � N , we have Ξ(ζ1, ζ2, ζ3, ζ4) = 0, which is canceled out

when evaluating (4.13). Then we may assume N3 ∨ N4 & N in (4.13). Note that
|ξ3 + ξ4| = |ξ1 + ξ2| . (N1 ∨N2) ∧ (N3 ∨N4). We take (PN1

QL1
u)(PN2

QL2
u) and

(PN3QL3u)(PN4QL4u) in L2
λ by using (3.3), respectively. By (4.14), the contribu-

tion of this case to JL1,L2,L3,L4

N1,N2,N3,N4
is bounded by

.
(N1 ∧N2)(N1 ∨N2)11/20

N1m(N1)N2m(N2)L0+
1 L0+

2

(N3 ∧N4)(N3 ∨N4)9/20

N3m(N3)N4m(N4)L0+
3 L0+

4

N1/10+
4∏
j=1

‖PNjQLju‖L2
λ

.
4∏
j=1

1

N
9/40−
j m(Nj)L

0+
j

‖PNjQLju‖L2
λ

.
4∏
j=1

1

N0+
j L0+

j

‖PNjQLju‖L2
λ
,

since the function f(x) = x9/40−m(x) is non-decreasing on [1,∞) provided s >
31/40. This is acceptable after taking the dyadic sum on Nj and Lj . �

The following lemma is not difficult to prove using Lemma 4.5 as in [5, 6].

Lemma 4.6. For s > 31/40(< 29/31),∣∣ ∫ δ

0

∫
R×Tλ

I(u)2∂x
(
(Iu)2 − I(u)2

)
dx dy dt

∣∣ . 1

N1/10+
‖Iu‖4

X
1,1/2+
δ,λ

.

5. Proof of main theorem

In this section, we will give the proof of Theorem 1.1, by the local well-posedness
results of Proposition 3.1 in conjunction with Lemmas 4.4 and 4.6. The proof
follows the strategy described in [5]. We prove the global well-posedness by the
local well-posedness theory for solution Iu obtained in Proposition 3.1 and by the
upper bound of increment of modified energy from (4.7).



24 S. OSAWA, H. TAKAOKA EJDE-2024/05

Proof of Theorem 1.1. We consider the Cauchy problem (1.1) on an arbitrary time
interval [0, T ]. We rescale solutions by scaling (3.1) and consider the rescaled prob-
lem in (3.2) with initial data

uλ0 (x, y) =
1

λ2
u0

(x
λ
,
y

λ

)
.

The goal is to construct rescaled solution uλ(t) to the Cauchy problem (3.2) on the
time interval [0, λ3T ].

By the local well-posedness result of Proposition 3.1 along with (4.3), our solution
uλ(t) satisfies ‖Iuλ‖

X
1,1/2+
δ,λ

� 1. Applying Lemmas 4.4 and 4.6 to the computation

of (4.7), we have

Eλ[Iuλ](δ) ≤ Eλ[Iuλ](0) +
o(1)

N1/10− = o(1).

By using (4.1) and (4.5), we have

‖Iuλ(δ)‖H1
λ

= o(1).

We can iterate this process up to N1/10− times before doubling ‖Iuλ(δ)‖H1
λ
.

This process extends the local solution obtained by Proposition 3.1 to the time
O(N1/10−δ). We choose N such that by (4.2)

N1/10−δ > λ3T ∼ N3(1−s)/(1+s)T,

which may be done for s > 29/31, hence complete the claim of Theorem 1.1.
Furthermore, replacing λ ∼ N (1−s)/(1+s) and N ∼ T 10(1+s)/(31s−29)+ in (4.4),

the growth of Sobolev norm of the solution u(t) to the Cauchy problem (1.1) has a
bound at least

‖u(T )‖Hs . λ1+s‖Iuλ(λ3T )‖H1
λ
. T 10(1−s)/(31s−29)+.

This completes the proof. �

Acknowledgments. The authors want to thank Jean-Claude Saut and Robert
Schippa for providing valuable advice according to their areas of expertise. We
sincerely thank the referee for useful comments on our manuscript. The first au-
thor was supported by JST SPRING, Grant Number JPMJSP2148 and the second
author was supported by JSPS KAKENHI, Grant Number 18H01129.

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Satoshi Osawa

Department of Mathematics, Kobe University, Kobe, 657-8501, Japan

Email address: sohsawa@math.kobe-u.ac.jp

Hideo Takaoka

Department of Mathematics, Kobe University, Kobe, 657-8501, Japan
Email address: takaoka@math.kobe-u.ac.jp


	1. Introduction
	2. Notation and function spaces
	3. Rescaled solutions
	Estimate of JLL L
	Estimate of JLH H
	Estimate of JHL H
	Estimate of JHH L
	Estimate of JHH H
	Subcase (i)
	Subcase (ii)
	Subcase (iii)
	Subcase (iv)
	Subcase (v)

	4. Modified energy
	Estimate of JLL L
	Estimate of JLH H
	Estimate of JHL H
	Estimate of JHH L
	Estimate of JHH H:

	5. Proof of main theorem
	Acknowledgments

	References