On Global Solution to the Klein-Gordon-Hartree Equation
below Energy Space
Changxing Miao1 and Junyong Zhang2
1Institute of Applied Physics and Computational Mathematics,
P.O. Box 8009, Beijing 100088, P.R. China.
E-mail: miao changxing@iapcm.ac.cn
2The Graduate School of China Academy of Engineering Physics
P. O. Box 2101, Beijing, China, 100088
E-mail: zhangjunyong111@sohu.com
Abstract
In this paper, we consider the Cauchy problem for Klein-Gordon equation with
a cubic convolution nonlinearity in R3. By making use of Bourgain’s method in
conjunction with a precise Strichartz estimate of S.Klainerman and D.Tataru, we
establish the Hs(s < 1) global well-posedness of the Cauchy problem for the cubic
convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the pre-
viously discussed conclusion, we obtain global solution for this non-scaling equation
with small initial data in Hs0×Hs0−1 where s0 = γ6 but not
−1, for this equation
that we consider is a subconformal equation in some sense. In doing so a number of
nonlinear prior estimates are already established by using Bony’s decomposition,
flexibility of Klein-Gordon admissible pairs which are slightly different from that of
wave equation and a commutator estimate. We establish this commutator estimate
by exploiting cancellation property and utilizing Coifman and Meyer multilinear
multiplier theorem. As far as we know, it seems that this is the first result on low
regularity for this Klein-Gordon-Hartree equation.
Key Words: Klein-Gordon-Hartree equation, Low regularity, Precise
Strichartz estimate, Bony’s para-product decomposition, Coifman and Meyer
multilinear multiplier theorem.
AMS Classification: 35Q40, 35Q55, 47J35.
1 Introduction
We study the following Cauchy problem for the Klein-Gordon-Hartree equation:
�φ+ φ+ (|x|−γ ∗ |φ|2)φ = 0 in R×R3
φ|t=0 = φ0, ∂tφ|t=0 = φ1.
(1.1)
Here φ(t, x) is a complex valued function defined in space time R1+3, and � = ∂tt −∆.
http://arxiv.org/abs/0704.1866v4
Recently the Cauchy problem (1.1) has been extensively studied in the case with
initial data (φ0, φ1) ∈ H1(Rn) × L2(Rn). The well-posedness and the asymptotic be-
havior of solution to the Cauchy problem (1.1) have been studied by G.P. Menzala and
W.Strauss [16, 17]. The scattering theory of solution to (1.1) has been established in
[23]. On the other hand, the time-dependent Schrödinger equation with interaction
term (|x|−γ ∗ |φ|2)φ has also been extensively studied. Ginibre and Velo [11] gave the
scattering theory of Hartree equation for the energy subcritical case. For the energy
critical case and mass critical, one can refer to [20, 21] with radial initial data.
Many authors [4, 9, 12, 18, 30] have studied the local well-posedness (as well as
global well-posedness) in fractional Sobolev spaces for the Cauchy problem of general
semilinear wave or Schrödinger equations under minimal regularity assumptions on
the initial data. For example, Tao [30] established the sharp local well-posedness of
nonlinear wave equation. Kenig, Ponce, and Vega [12] had established the global well-
posedness under the energy norm for the Cauchy problem of nonlinear wave equations
with rough initial data (in particular, in Ḣs(R3), 3
< s < 1 for cubic wave equation).
They used the Fourier truncation method discovered by Bourgain [4]. And also [18]
extended Kenig-Ponce-Vega’s result to the dimension n > 4. Recently, I. Gallagher
and F. Planchon [9] presented a different proof of the result in [12] for 3
< s < 1. H.
Bahouri and Jean-Yves Chemin [2] proved global well posedness for s = 3
by using a
nonlinear interpolation method and logarithmic estimates from S. Klainermann and D.
Tataru[14]. We also find Roy [26] obtains the global well-posedness for rough initial data
in Ḣs, 2
< s < 1 by following the I-method [5] and scaling transformation. However, if
one similarly deals with Klein-Gordon equation by using I-method, he or she may meet
a problem caused by the lack of the scaling property. More studies and discussions on
the low regularity of nonlinear wave or dispersive Schrödinger equations could be found
in [4, 31]. However, as far as we know, very few authors are engaged in studying the
global well-posedness of the Cauchy problems (1.1) with less regular initial data. It is
natural to ask whether a similar or better result holds for the problem (1.1).
This paper endeavors to find a global well-posedness solution to the Cauchy problem
(1.1) with initial data (φ0, φ1) ∈ Hs(R3) ×Hs−1(R3) for some s > γ4 with γ ∈ (2, 3).
Now we should remark some differences between (1.1) and cubic wave equation. If one
views (1.1) as a wave equation by dropping the massive term and then makes some
scaling analysis, we will find this nonlocal nonlinear term shares the scaling property of
the nonlinearity |u|
5−γ u. One can check that k := 4
+1 < 3 when 2 < γ < min{n, 4}
with n = 3 and this result shows that the equation which we consider is in subconformal
case. To obtain the global well-posedness theory, some previous literatures also show
the subconformal equations are slightly different from the superconformal ones. For
instance, Lindblad and Sogge [15] [27] have shown the global existence and scattering
theory for small data in a less regularity space for the superconformal case, while
not for the subconformal case. Inspired by [9], we also split the initial data into low
frequency part data in H1 and high frequency part data in Hs0 with a suitable s0.
Since the problem (1.1) is global well-posed for large data in H1 and small data in
Hs0 , one may be tempted to follow a general principle of nonlinear interpolation and
claim the problem (1.1) is global well-posed between them. Compared with the cubic
wave equation, speaking of the Strichartz estimate, we believe that the global solution
with high frequency data should exist in H
−1. It is well known that the Strichartz
estimate is associated with scaling transform and it is scaling invariant. Unfortunately,
the equation that we consider is a subconformal one, and its concentration effects take
over scaling. Since the Strichartz estimate is applied to our subconformal equation,
hence this brings about some loss to get a better result. In order to get a better result,
one should establish an estimate which is conformal invariant. Fortunately, we can take
0 6 θ 6 1 as a parameter for the flexible admissible pairs (see Definition 2.3)to make
the Strichartz estimate of Klein-Gordon more flexible than wave equation. This helps
us to get a global solution with the high frequency data, at the cost of 0 6 θ = 6
−2 6 1
which weakens the Strichartz estimate and causes 2 < γ < 3. One can refer the detail
in Section 3.
We point out that it is easy to have the result for
< s < 1 by rough Hölder’s
inequality. But how to get our low bound
< s < 1? A good way to think about this
is via precise Strichartz estimate to obtain index s as low as possible. The nonlinearity
including a formal negative derivative brings us some difficulties caused by the fact that
the negative derivative acts on the low frequency part. And this leads us to restricts
rather than s > max{1
}. At the end of this section, we also give some
intuitive analysis to show our result is reasonable. As a limited case, our result recovers
the result of [9, 12] when γ tends to 3.
During the process of proving our key estimate Lemma 5.1, the nonlocal nonlin-
earity brings about some essential difficulties when we try to make use of the precise
Strichartz estimate. Compared with the general semilinear nonlinearity, the convo-
lution nonlinearity not only essentially represents a negative derivation in it but also
has a difference construction of nonlinearity. These differences and difficulties prevent
us from obtaining directly our expected result s >
by restricting the range of the
parameter r. To overcome these difficulties, we firstly construct a commutator and
establish this commutator estimate by exploiting cancellation property and utilizing
Coifman and Meyer multilinear multiplier theorem and then go on our process through
using precise Strichartz estimate.
Now we state our main result:
Theorem 1.1 Let
< s < 1 with 2 < γ < 3. If (φ0, φ1) ∈ Hs(R3) ×Hs−1(R3), then
there exists a unique global solution φ of (1.1) such that φ ∈ C(R+;Hs(R3)).
We conclude this section by giving a sketch of the proof of Theorem 1.1 and one shall
read more detailed information in the rest of this paper. Without loss of generality,
we only consider φ as a real function for simplicity from now on. Since the problem
(1.1) is global well-posed for large data in H1 and small data in Hs0 with s0 =
, one
may be tempted to follow a general principle of nonlinear interpolation and believe the
problem (1.1) to be global well-posed between them, as well as the cubic defocusing
wave equation [9]. To make sense of this heuristic, we proceed it in the following steps.
Step 1. The purpose of this step is to show the global well-posedness for the high
frequency part. We split the initial data:
φi = (I− SJ)φi + SJφi
= vi + ui i = 0, 1
where I is identity operator and SJ is Littlewood-Paley operator, referring to Section
2. It is easy to see that
‖u0‖H1 . 2J(1−s)‖φ0‖Hs , ‖u1‖L2 . ‖φ1‖L2
‖v0‖Hβ . 2J(β−s)‖φ0‖Hs , ‖v1‖Hβ−1 . 2J(β−s)‖φ1‖Hs−1 for all β 6 s.
Thus it follows that
Eh,σ . 2J(σ−s)Es, for σ 6 s (1.2)
Eℓ,1 . 2J(1−s)Es, for s 6 1, (1.3)
where
== ‖φ0‖Hs + ‖φ1‖Hs−1 , (1.4)
== ‖v0‖Hσ + ‖v1‖Hσ−1 , (1.5)
== ‖u0‖Hσ + ‖u1‖Hσ−1 . (1.6)
Choosing J large enough, one can achieve Eh,s0 small enough, in other words, initial
data of the following problem
�v + v + (|x|−γ ∗ v2)v = 0 in R× R3,
v|t=0 = v0, ∂tv|t=0 = v1
(1.7)
is small enough inHs0(R3)×Hs0−1(R3) where s0 < s. Due to some technique difficulties
and this equation is subconformal one, we are restricted to choose s0 =
while not
− 1 proposed by scaling analysis or γ
proposed by conformal analysis. We will
get a global well-posed solution to the Cauchy problem (1.7), see Section 3 for details.
Step 2. In order to recover a solution to our problem (1.1), we solve a perturbed
equation with large initial data in H1 × L2,
�u+ u+ I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v = 0,
u|t=0 = u0 ∂tu|t=0 = u1,
(1.8)
where the operator I is the operator (−∆)
2 . We will prove there exists a unique
local solution to (1.8) in C([0, T ];H1).
Step 3. To complete the proof of Theorem 1.1, the key is how to extend the local
solution to a global solution. We should establish a priori bound on the energy of the
local solution u. In fact, the energy estimate yields
‖u(t)‖2H1 + ‖ut(t)‖
R3×R3
|x− y|−γu2(y, t)u2(x, t)dydx
‖u0‖2H1 + ‖u1‖
R3×R3
|x− y|−γu20(y)u20(x)dydx
I(v2)(x, τ)u(x, τ)∂τu(x, τ)dxdτ
I(uv)(x, τ)v(x, τ)∂τu(x, τ)dxdτ
I(u2)(x, τ)v(x, τ)∂τu(x, τ)dxdτ
I(uv)(x, τ)u(x, τ)∂τu(x, τ)dxdτ
∣∣∣∣ .
Let HT (u) := sup
H(u)(t) where
H(u)(t)
‖u(t)‖2H1 +
‖ut(t)‖2L2 +
R3×R3
|x− y|−γu2(y, t)u2(x, t)dydx
and then by making use of Hölder’s inequality and Sobolev embedding, it follows that
HT (u) .H(u)(0) +HT (u)
‖v(τ)‖2
dτ +H
T (u)
‖v(τ)‖
.H(u)(0) +HT (u)T
6 ‖v‖2
T (u)T
3 ‖v‖Xα
.22J(1−s) +HT (u)T
6 22J(β−s) +H
T (u)T
3 2J(α−s)
where α =
, β =
and the space Xα is defined in the coming section. What we
want to do is to control HT (u) for arbitrarily large T . As long as s > (α+1)/2 =
by choosing J large enough, bootstrap argument yields
HT (u) . 2
2J(1−s).
One can see that, if s >
, the argument is trivial, since the above mentioned
result can be deduced from some rough estimates such as the Hölder estimate. On
the other hand, since the scaling suggests us that X
−1 is the lowest regularity space
which v could belong to, it is tempting and reasonable to believe that the best result
obtained by this method is s > (
− 1+1)/2 = γ
instead of α by
− 1. To obtain this
optimal result s >
, we adopt some more sophisticated tools such as precise Strichartz
estimate, Bony’s paraproduct estimates and twice Bony’s decomposition. This result
is achieved under an assumption of a core estimate which will be shown through the
precise Strichartz estimate and a commutator estimate.
The paper is organized as follows: In the coming section, we recall some nota-
tions and recollect some well known results on Besov spaces in conjunction with the
Littlewood-Paley theory which will be used in the course of the proofs. Meanwhile,
we also introduce the precise Strichartz estimate. Section 3 provides the global well-
posedness of original equation evoking the high frequency part of initial data in Hs0 .
In Section 4, we prove prove a local well-posedness of perturbed equation with the low
frequency of the initial data in H1 by the standard fixed point theorem. In Section 5,
we give a energy estimate for the low frequency part provided an assumption the key
estimate in Lemma 5.1. We extend the local well-posedness of the perturbed equation
to globally well posed by the bootstrap argument in Section 6. In the final section,
we prove our essential and key lemma by the precise Strichartz estimate, commutator
estimate and Coifman and Meyer multiplier theorem.
2 Preliminaries
In this section, we shall present some well-known facts on the Littlewood-Paley theory
and introduce some notations, definitions and estimates which are needed in this paper.
Let S(R3) be the Schwarz class of rapidly decreasing functions. Given f ∈ S(R3), its
Fourier transform Ff = f̂ is defined by
f̂(ξ) = (2π)−
e−ix·ξf(x)dx, F−1f = f̂(−ξ).
Choose two nonnegative radial functions χ, ϕ ∈ S(R3) supported respectively in B =
{ξ ∈ R3, |ξ| 6 4
} and C = {ξ ∈ R3, 3
6 |ξ| 6 8
} such that
χ(ξ) +
ϕ(2−jξ) = 1, ξ ∈ R3,
ϕ(2−jξ) = 1, ξ ∈ R3\{0},
supp ϕ(2−j ·) ∩ supp ϕ(2−j′ ·) = ∅, |j − j′| > 2,
supp χ(·) ∩ supp ϕ(2−j ·) = ∅, j > 1.
Now we are in position to define the the Littlewood-Paley operators Sj, Ṡj, △j and
△̇j which are used to define Besov space.
0, j 6 −2,
χ(ξ)û(ξ)
, j = −1,
(F−1ϕ)(2jy)u(x− y)dy, j > 0,
j′6j−1
△j′u = 2jn
(F−1χ)(2jy)u(x− y)dy,
== 2jn
(F−1ϕ)(2jy)u(x− y)dy, j ∈ Z,
j′6j−1
△̇j′u.
One easily shows that △̇j = Ṡj+1 − Ṡj for j ∈ Z and
△−1 = S0, △̇j = △j , j > 0.
Now we give the Littlewood-Paley’s description of the Besov spaces.
Definition 2.1 Let s ∈ R, 1 6 p, q 6 ∞. The homogenous Besov space Ḃsp,q is defined
Ḃsp,q = {f ∈ Z ′(R3) : ‖f‖Ḃsp,q < ∞},
where
Ḃsp,q
2jsq‖△̇jf‖qp
, for q < ∞,
2js‖△̇jf‖p, for q = ∞,
and Z ′(R3) can be identified by the quotient space S ′/P with the space P of polynomials.
Definition 2.2 Let s ∈ R, 1 6 p, q 6 ∞. The inhomogeneous Besov space Bsp,q is
defined by
Bsp,q = {f ∈ S ′(R3) : ‖f‖Bsp,q < ∞},
where
‖f‖Bsp,q =
2jsq‖△jf‖qp
+ ‖S0(f)‖p, for q < ∞,
2js‖△jf‖p + ‖S0(f)‖p, for q = ∞.
If s > 0, then Bsp,q = L
p ∩ Ḃsp,q and ‖f‖Bsp,q ≈ ‖f‖p + ‖f‖Ḃsp,q . We refer the reader to
[1, 6, 22, 32] for details.
In order to investigate the low regularity solution of the Cauchy problem (1.1), we
require the use of the smoothing effect described by the Strichartz estimates and precise
Strichartz estimates. For the purpose of conveniently making use of the Strichartz
estimate, we introduce the admissible definition and the resolution space.
Definition 2.3 We shall say that a pair (q, r) is admissible, for 0 6 θ 6 1, if
q, r > 2, (q, r, θ) 6= (2,∞, 0) and 1
2 + θ
2 + θ
Remark 2.1 The above admissible pairs in Definition 2.3 is more flexible than wave
admissible pairs, since θ can vary from 0 to 1. Obviously, an admissible pair in Defini-
tion 2.3 will become a wave admissible pair when θ = 0. When we consider the global
existence for the high frequency part, we shall use θ = 6
− 2 since the equation that we
consider is a subconformal one.
The resolution space is defined in the following way based on the admissible defini-
tion.
Xµ(I) :=
06θ61
where
(I) :=
u : u ∈ (C ∩ L∞)(I;Hµ) ∩ Lq(I;Bσr,2)
(q,r) is admissible,
= (3 + θ)(
) + σ − µ
We go on this section by recalling the classical Strichartz estimate and the precise
Strichartz estimate. This kind of estimate goes back to Strichartz [29], and has been
proved in its generality by Ginibre and Velo [10], and Keel and Tao [13]. The Strichartz
estimates for the Klein-Gordon equation by using the above flexible admissible pairs
can be found in [19].
Proposition 2.1 Let u be a solution of
�u+ u = f in R× R3 with u|t=0 = u0, ∂tu|t=0 = u1.
Then, for any admissible pairs (q1, r1) and (q2, r2), we have that
‖△ju‖Lq1 (Lr1 ) + 2−j‖∂t△ju‖Lq1 (Lr1 )
j( 3+θ
− 3+θ
(‖△ju0‖L2 + 2−j‖△ju1‖L2)
j[(3+θ)(1− 1
−1]‖△jf‖
. (2.1)
We shall see that the classical Strichartz estimates are not enough to control some
nonlinearities, and this leads us to resort to the following precise Strichartz estimates
which were established by S.Klainerman and D.Tataru[14].
Proposition 2.2 Let u be a solution of
�u+ u = 0 with u|t=0 = u0, ∂tu|t=0 = u1.
Assume that the supports of the Fourier transform of u0 and u1 are included in a ball
B(ξj, h2
j) with |ξj | ∈ [2j−2, 2j+2] and h < 18 . Then we have that, for any admissible
couple (q, r),
‖u‖Lq(Lr) + 2−j‖∂tu‖Lq(Lr) 6 C2
r (‖u0‖L2 + 2−j‖u1‖L2). (2.2)
Let us recall the Hardy-Littlewood-Sobolev inequality [22, 28] and a proposition of
contraction which is generalization of Picard’s theorem [6]. We denote operator I by
Iu def== (−∆)
2 u = |x|−γ ∗ u,
‖Iu‖Lq(R3) 6 Cp,q‖u‖Lp(R3) (2.3)
0 < γ < 3, 1 < p < q < ∞, and 1
− 3− γ
Proposition 2.3 Let X be a Banach space and let B : X ×X × · · · ×X → X be a
m-linear continuous operator (m > 2) satisfying
‖B(u1, u2, · · · , um)‖X 6 M‖u1‖X‖u2‖X · · · ‖um‖X , ∀u1, u2, · · · , um ∈ X
for some constant M > 0. Let ε > 0 be such that m(2ε)m−1M < 1. Then for every
y ∈ X with ‖y‖X 6 ε the equation
u = y +B(u, u, · · · , u) (2.4)
has a unique solution u ∈ X satisfying that ‖u‖X 6 2ε. Moreover, the solution u
continuously depends on y in the sense that, if ‖y1‖X 6 ε and v = y1 +B(v, v, · · · , v),
‖v‖X 6 2ε then
‖u− v‖X 6
1−m(2ε)m−1M ‖y − y1‖X . (2.5)
For the sake of convenience, we conclude this section by giving some notations. The
solution φ to the Cauchy problem (1.1) is given by the following integral equation:
φ(t, x) = K̇(t)φ0 +K(t)φ1 +B(φ, φ, φ)
= T φ
where
K(t) :=
sin(t
I −∆)√
B(u1, u2, u3) := −
K(t− τ)(|x|−γ ∗ (u1u2))u3dτ.
Throughout this article we shall denote by the letter C all universal constant and ε > 0
is a arbitrary small data. We shall sometimes replace an inequality of the type f 6 Cg
by f . g. Also, we shall denote by (cj)j∈Z any sequence of norm less than 1 in ℓ
2(Z).
3 Global existence for the high frequency part
Let us consider the Cauchy problem with the high frequency data,
�v + v + (|x|−γ ∗ v2)v = 0, (t, x) ∈ R× R3
v|t=0 = v0, ∂tv|t=0 = v1, x ∈ R3.
(3.1)
and then its integral formation becomes
v(t, x) =K̇(t)v0(x) +K(t)v1(x)−
K(t− τ)(|x|−γ ∗ v2)vdτ
==K̇(t)v0(x) +K(t)v1(x) +B(v, v, v). (3.2)
Our goal in this section is to prove the global well-posedness of (3.1) or (3.2). More
precisely, we have the following proposition:
Proposition 3.1 Let s0 =
and suppose that (v0, v1) ∈ Hµ×Hµ−1 for any 0 6 µ 6 1.
There exists a constant ε0 > 0 such that if
‖v0‖Hs0 + ‖v1‖Hs0−1 6 ε0,
then there exists a unique global solution v to (3.1) or (3.2) in Xs0(R)∩Xµ(R). More-
over,
‖v‖Xµ 6 Cµ (‖v0‖Hµ + ‖v1‖Hµ−1) .
Remark 3.1 We focus on µ =
and µ =
in the coming section.
It is well known that the global existence theory for small initial data is a straightfor-
ward result of nonlinear estimate, thus how to obtain a suitable nonlinear estimate is
essential. Before proving this proposition, we make some analysis on nonlinear estimate.
As mentioned in the introduction, the nonlocal nonlinearity shares the scaling with a
subconformal nonlinearity when γ < 3 and this may bring some troubles when we make
a choice of a suitable resolution space Xs0 . Take 0 6 θ 6 1 as a parameter in the flexible
admissible pairs (see Definition 2.3), and we make analysis on the relationship between
θ and s0. The Strichartz estimate, Hölder inequality and Hardy-Littlewood-Sobolev
inequality imply that, for σ 6 0,
‖B(v, v, v)‖Xs0 6 ‖(|x|−γ ∗ |v|2)v‖
6 ‖v‖
Lq2 (B
‖v‖2Lq3 (Lr3 ),
with satisfying
= (3 + θ)(
) + σ + s0 − 1
= (3 + θ)(
)− σ − s0
= (3 + θ)(
)− s0
− 1 + γθ
We find the fact index s0 is increasing when the parameter θ increases. It is tempting to
choose θ = 0 to get the smallest s0 =
− 1 proposed by scaling. However, in addition
the admissible condition implies that
6 (2 + θ)(
6 (2 + θ)(
6 (2 + θ)(
then a direction computation gives that
) 6 (2 + θ)(2− 1
which yields that
6 1 +
If we choose θ = 0, then we are forced to γ > 3 which contradict with our requirement
γ < 3. But if we choose θ = 6
− 2 and then s0 = γ6 and we are allowed by 2 6 γ 6 3.
Proof of Proposition 3.1 Thanks to Strichartz estimate, we have
‖B(v, v, v)‖Xµ 6 ‖(|x|−γ ∗ |v|2)v‖
6 ‖v‖
Lq2 (B
‖v‖2Lq3 (Lr3 ),
where
3 + γ
(1− µ− σ), 1
3 + γ
(1− µ− σ)
3(µ+ σ)
3 + γ
− γ(µ+ σ)
3 + γ
2(3 + γ)
9 + 3γ − γ2
6(3 + γ)
When 0 6 µ 6 1
, we choose σ = 0; while 1
< µ 6 1, we choose σ = 1
Thus,
‖B(v, v, v)‖Xµ 6 ‖v‖Xµ‖v‖2Xs0 . (3.3)
Combining this nonlinear estimate, the Proposition 3.1 follows from a standard con-
traction argument and small initial data condition.
4 Local existence for the low frequency part
In this part, we shall study the following perturbed problem in R× R3:
�u+ u+ I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v = 0
u|t=0 = u0 ∂tu|t=0 = u1.
(4.1)
Proposition 4.1 Let α =
, β =
and assume that v be in Xα ∩ Xβ and
(u0, u1) ∈ H1 ×L2, then there exists a positive time T such that a unique solution u to
(4.1) satisfying
u ∈ C([0, T ];H1).
Proof of the Proposition 4.1 In practice, solving (4.1) on [0, T ] is equivalent to
solving the following integral equation
u =K̇(t)u0 +K(t)u1
K(t− τ)
I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v
,T̃ u.
Using the Strichartz estimate, we have
K(t− τ)I(u2)udτ
. ‖I(u2)u‖L1
(L2).
On one hand, we make use of Hölder’s inequality and Hardy-Littlewood-Sobolev in-
equality to deduce that
‖I(u2)u‖L1
(L2) 6 C‖I(u2)‖
6 C‖u‖3
6 CT‖u‖3L∞
H1 . (4.2)
For the rest of terms, arguing similarly as above, it can be obtained that
‖I(uv)u‖L1
(L2) 6 C‖u‖2L∞
L6‖v‖
3 ‖u‖2L∞
H1‖v‖Xα , (4.3)
‖I(v2)u‖L1
(L2) 6 C‖u‖L∞T L6‖v‖
3 ‖u‖L∞
H1‖v‖2Xβ , (4.4)
‖I(u2)v‖L1
(L2) 6 CT
3 ‖u‖2L∞
H1‖v‖Xα , (4.5)
‖I(uv)v‖L1
(L2) 6 CT
3 ‖u‖L∞
H1‖v‖2Xβ . (4.6)
A combination of (4.2), (4.3)-(4.6) and the Strichartz estimate in Proposition 2.1 lead
to the estimate
‖u‖L∞
(H1) . ‖u0‖H1 + ‖u1‖L2 + T‖u‖3L∞
3 ‖u‖2L∞
H1‖v‖Xα + T
3 ‖u‖L∞
H1‖v‖2Xβ .
As long as choosing T is small enough, T̃ is a contraction mapping in ball B(0, 2CEℓ,1).
By means of Picard’s fixed point argument we have an unique solution u to (4.1) in
L∞([0, T ];H1). Therefore, Proposition 4.1 is proved by the standard argument.
5 Energy estimate for the low frequency part
In order to extend the local solution to a global solution, we shall prove a prior estimate
for the Hamiltonian of u in this section. Let us recall the definition of Hamiltonian of
u defined by
H(u)(t)
‖u(t)‖2H1 +
‖ut(t)‖2L2 +
R3×R3
|x− y|−γu2(y, t)u2(x, t)dydx
Similarly we give another notation of the energy of u, which is denoted by
E(u)(t)
‖u(t)‖2H1 +
‖ut(t)‖2L2 .
HT (u)
== sup
H(u)(t), ET (u)
== sup
E(u)(t).
To extend the local existence to global existence, we have to do a number of nonlinear
a priori estimates provided that ET (u) 6 2CH(u)(0), see Proposition 5.1 and Lemma
5.1. As a direct consequence of the above assumption, we get an important relationship
between E(u) and Es defined in the introduction
ET (u) . 2
2J(1−s)(E2s + E4s ) . 22J(1−s). (5.1)
In fact, it follows from Hardy-Littlewood Sobolev inequality and the definition of u0
‖(|x|−γ ∗ u20)u20‖L1 . ‖u0‖412
6 ‖S0φ0‖412
06j6J
‖△jφ0‖412
And then the right hand of the above inequality can controlled that as soon as 1 > s >
by utilizing Bernstein inequality
‖S0φ0‖4L2 +
06j6J
−s)2j4s‖△jφ0‖4L2 . 2
2J(1−s)E4s .
From now on, we assume (5.1) to in our subsequence proof.
Proposition 5.1 Assume that (u0, u1) ∈ H1×L2, then the following estimate holds
for s0, α, β defined in Proposition 3.1 and Proposition 4.1,
HT (u) .H(u)(0) + T
3 2−2J(s−β)ET (u) + T
3 2−J(4s−α−2s0−1)ET (u)
−2J [s−(
−2J [s−(
+ T2−2J(s−
ET (u)
for max{2, 1
} < r1 < 23−γ and
6 r2 < ∞.
Proof. Multiplying (4.1) by ∂tu and integrating over x and t, we have
‖u(t)‖2H1 + ‖ut(t)‖
R3×R3
|x− y|−γu2(y, t)u2(x, t)dydx
‖u0‖2H1 + ‖u1‖
R3×R3
|x− y|−γu20(y)u20(x)dydx
I(v2)(x, τ)u(x, τ)∂τu(x, τ)dxdτ
I(uv)(x, τ)v(x, τ)∂τu(x, τ)dxdτ
I(u2)(x, τ)v(x, τ)∂τu(x, τ)dxdτ
I(uv)(x, τ)u(x, τ)∂τu(x, τ)dxdτ
∣∣∣∣ .
By taking the supermum over t 6 T , we have
HT (u) .H(u)(0) +
∥∥I(v2)u∂tu
+ ‖I(uv)v∂tu‖L1
I(u2)v∂tudxdt
I(uv)u∂tudxdt
=H(u)(0) + I + II + III + IV. (5.2)
The proof is broken down into the following several steps.
(i) Firstly, we estimate I and II. Making a similarly argument as (4.4) in the proof
of Proposition 4.1, it can be obtained that
I 6 ‖I(v2)u‖L1
L2‖ut‖L∞T L2 6 T
3 ET (u)‖v‖2Xβ ,
and then keeping in mind v has been estimated in Proposition 3.1, this together with
(1.2) yields that
I 6 T
3 ET (u)E2h,β 6 T
3 ET (u)2
−2J(s−β)E2s . (5.3)
Arguing similarly, we easily get
II 6 T
3 ET (u)2
−2J(s−β)E2s . (5.4)
(ii) Secondly, we estimate the terms III and IV . As mentioned in the introduction,
one can get the same type of estimate as above for the terms I and II, but that will
lead to s > α
, which is worse than the exponent given in the Theorem 1.1. To
improve the lower bound on s, we have to utilize more precise estimate on III and IV .
We first split III and IV into two different pieces, respectively. One can write
v = vF +B(v, v, v),
where vF is its free part and the other one comes from nonlinear term. For the nonlinear
part, it follows from (3.3) that
‖B(v, v, v)‖Xα 6 ‖v‖Xα‖v‖2Xs0 .
This along with (4.5), one can see that
‖I(u2)B(v, v, v)ut‖L1
L1 6 ‖I(u2)B(v, v, v)‖L1
L2‖ut‖L∞T L2
3 ‖u‖2L∞
H1‖B(v, v, v)‖Xα‖ut‖L∞T L2
3 ET (u)
2 ‖v‖Xα‖v‖2Xs0
Moreover, we get by (1.2),
‖I(u2)B(v, v, v)ut‖L1
L1 6 T
T (u)2
−J(3s−α−2s0)E3s . (5.5)
By the same way as leading to (5.5), we easily infer that
‖I(uB(v, v, v))uut‖L1
L1 6 T
T (u)2
−J(3s−α−2s0)E3s . (5.6)
Thus, it is sufficient to estimate these terms including free part vF since (5.5) and (5.6).
The following lemma gives estimates for the nonlinearity including free part vF .
Lemma 5.1 Let vF be a solution of the free Klein-Gordon equation, and u be such
that ET (u) . 2
2J(1−s). Then, for max{2, 1
} < r1 < 23−γ and
6 r2 < ∞
I(u2)vFutdxdt
∣∣∣ .
−2J [s−(
−2J [s−(
+ T2−2J(s−
ET (u), (5.7)
I(uvF )uutdxdt
∣∣∣ . T
−2J [s−(
ET (u). (5.8)
Hence these together with (5.5)-(5.6) yield that
III + IV . T
3 ET (u)2
−J [4s−α−2s0−1]E4s +
−2J [s−(
−2J [s−(
+ T2−2J(s−
ET (u). (5.9)
Therefore, we complete the proof of Proposition 5.1 provided that we had proved
Lemma 5.1, whose proof is postponed in the last section.
6 Proof of Theorem 1.1
Since the Cauchy problem (1.1) is split into equation (3.1) which is globally well-posed
by choosing J enough to make Eh,s0 < ε0 and equation (4.1) which is locally well-posed
(see Proposition 3.1 and Proposition 4.1), we have to show that the local solution to
equation (4.1) can be extended globally.
Let us denote T ∗J the maximum time of existence in Proposition 4.1. Theorem 1.1
will be proved if
T ∗J = +∞.
Let us consider TJ the supremum of the T < T
J such that
ET (u) 6 2CH(u)(0).
Thus, for any T < TJ , Proposition 5.1 gives us that
ET (u) 6 H(u)(0)
C + C1T
3 2−2J(s−β)E2s
+ C2T
3 2−J(4s−α−2s0−1)E4s + C3T
−2J [s−(
)]E2s
+ C4T2
−2J(s− 1
)E2s + C5T
−2J [s−(
)]E2s
By the assumption of Theorem 1.1 s >
, one easily verifies that
s > max
if choosing r1 sufficiently close to
and r2 large enough. We infer that TJ > T̃J if
we choose T̃J such that
= min
22J(s−β)
5C1E2s
24J(s−
5C2E4s
2J [s−(
5C3E2s
) 2r1
22J(s−
5C4E2s
2J [s−(
5C5E2s
) 2r2
By the definition of TJ , we get T
J > T̃J . Obviously, T̃J tends to infinity when J tend
to infinity. This completes the proof of Theorem 1.1.
7 Proof of Lemma 5.1
In order to make conveniently use of the precise Strichartz estimate on which mostly
the following proof relies, we begin this section by introducing a family of balls of center
ν )ν∈Λj,k of radius 2
k and a function χ ∈ C∞c (B(0, 1)) such that for j > 0
∀ξ ∈ 2jC,
ν∈Λj,k
χ(2−k(ξ − ξj,kν )) = 1 and C−10 6
ν∈Λj,k
χ2(2−k(ξ − ξj,kν )) 6 C0.
Let us define that, for some constant c
△νj,ka
== F−1
ϕ(2−jξ)χ(2−k(ξ − ξj,kν ))
â(ξ)
△̃νj,ka
== F−1
ϕ̃(2−jξ)χ(c2−k(ξ + ξj,kν ))
â(ξ)
As the support of the Fourier transform of a product belongs to the sum of the support
of each Fourier transform, we have
△ja =
ν∈Λj,k
△νj,ka, △jb =
ν′∈Λj,k
△ν′j,kb.
In view of this fact that if k 6 j − 2
ν,ν′∈Λj,k
△νj,ka△ν
is vanish when ξ
ν is close to ξ
, without loss of generality, we can write
△k(△ja△jb) ≈ △k
ν∈Λj,k
△νj,ka△̃νj,kb. (7.1)
For the sake of convenience, we also fix the notation in this section that, for 0 6=
f(t, x) ∈ L2TL2
ck = 2
‖△kv0‖L2 + 2−k‖△kv1‖L2
E−1h,σ, c̃k =
‖△kf‖L2
‖f‖L2
with σ = 1/2 + 1/r for 2 6 r < ∞.
Proof of Lemma 5.1. We first prove (5.7). In view of the fact that v̂F only has
high frequencies, Bony’s decomposition implies that there exists constant N0 such that
I(u2)vFut =
j>J−N0
Sj+2vF△jI(u2)ut +
j>J−N0
Sj−1I(u2)△jvFut. (7.2)
Since the negative derivative I acts on the high frequency for the former term while
on the low frequency for the latter one, the first term is much better than the second
one. We shall estimate the first term by using merely the Hölder inequality, Bernstein
inequality and classical Strichartz estimates. Firstly, we see that, for 2 6 r < ∞
j>J−N0
‖Sj+2vF△jI(u2)‖L2x .
j>J−N0
j′6j+1
‖△j′vF ‖L∞x ‖△jI(u
2)‖L2x
j>J−N0
j′6j+1
r ‖△j′vF ‖Lr2j(γ−
)‖u‖2L∞
Bernstein inequality and (2.1) in Proposition 2.1 with 1
for all 2 6 r < ∞
imply that
j>J−N0
‖Sj+2vF△jI(u2)‖L2x
p ‖u‖2L∞
j>J−N0
j′6j+1
r ‖△j′vF‖Lp
j(γ− 7
p ‖u‖2L∞
j>J−N0
2j(γ−
j′6j+1
j′( 3
‖△j′v0‖L2 + 2−j
′‖△j′v1‖L2
The right hand of the above inequality can be controlled by
p ‖u‖2L∞
j>J−N0
2j(γ−
j′6j+1
2 cj′Eh,σ
and moreover it follows from (1.2), the definition of Eh,σ and Sobolev embedding that
j>J−N0
‖Sj+2vF△jI(u2)ut‖L1x
pEh,σ‖u‖2L∞
j>J−N0
2j(γ−3)‖ut‖L∞
r 2−2J [s−(
)]E2s ‖u‖2L∞
(H1),
(7.3)
for 4
6 r < ∞.
Let us estimate the second term in (7.2) by the precise Strichartz estimates. Since
this term contains that the negative derivative acts on the low frequency part Sj−1(u
it leads to our new parameter r < 2
by some technique difficulties. Noting that
Fourier-Plancherel formula and Hölder’s inequality, we can see that
j>J−N0
Sj−1I(u2)△jvFutdxdt .
j>J−N0
−16k6j−2
△kI(u2)△jvF△jutdxdt
△kI(u2)△k
k6j−2,J−N06j
△jvF△jut
. ‖u2‖
2k(γ−
k6j−2,J−N06j
△jvF△jut
(7.4)
On one hand, we have
J−N06j
△jvF△jut
‖L2dt .
j>J−N0
‖△jvF△jut‖L1
j>J−N0
‖△jvF ‖L∞
L2‖△jut‖L2
j>J−N0
2−sjcj c̃jEh,s‖ut‖L2t,x .
If (7.4) is controlled by the term at k = −1, we can see that
j>J−N0
‖Sj−1I(u2)△jvF ‖L2
L2 . T
2 2−2J(s−
)E2s ‖u‖L∞T (H1). (7.5)
On the other hand, one denotes gk := △k
k6j−2
△jvF△jut
to estimate
2k(γ−
)‖gk‖
Let us write that
k6j−2
ν∈Λj,k
△νj,kvF△jut.
As the support of the Fourier transform of a product is included in the sum of the
support of each Fourier transform, we obtain
k6j−2
ν∈Λj,k
△νj,kvF △̃νj,kut,
as well as in (7.1). Using Hölder inequality, we get
k6j−2
ν∈Λj,k
‖△νj,kvF ‖Lr‖△̃νj,kut‖L2
and the Cauchy-Schwarz inequality and the L2 quasi-orthogonality properties yield that
k6j−2
ν∈Λj,k
‖△νj,kvF‖2Lr
ν∈Λj,k
ut‖2L2
k6j−2
ν∈Λj,k
‖△νj,kvF‖2Lr
2 ‖△jut‖L2 . (7.6)
Precise Strichartz estimate implies that, for 1
with 2 6 r < ∞,
r+2 )
06k6j−2
2(k−j)(
ν∈Λj,k
‖△νj,kv0‖2L2
2 + 2−j
ν∈Λj,k
‖△νj,kv1‖2L2
‖△jut‖L2t,x
and observe the quasi-orthogonality properties again, this can be dominated by
06k6j−2
2(k−j)(
‖△jv0‖L2 + 2−j‖△jv1‖L2
‖△jut‖L2t,x .
Keeping the definitions of Eh,σ and cj in mind, one can see that
r+2 )
p 2k(
k6j−2
r cj c̃jEh,σE
T (u) . T
p 2k(
)Eh,σE
T (u).
Therefore, we get that
2k(γ−
)‖gk‖
r+2 )
2k(γ−3)Eh,σE
T (u)
which implies nothing but
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T 1−
p Eh,σE
T (u). (7.7)
Finally, we get that, for 4
6 r < ∞
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T
r 2−2J [s−(
)]E2sET (u).
However, although the r ranges 4
6 r < ∞, the above estimate still needs s > 3
to continue our proof. If we only consider the high frequency k > J , the (7.7) can be
modified by
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T 1−
p 2J(γ−3)Eh,σE
T (u) (7.8)
and then we can obtain a better result
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T
r 2−2J [s−(
)]E2sET (u),
which implies the bad influence comes from the low frequency part and this is consist of
the effect of negative derivative acts on the low frequency. But if we choose σ̃ = γ− 5
instead of σ, we can improve (7.8), at cost of restricting r such that max{2, 1
} < r <
while not 2 6 r < ∞. Now we turn to details. It follows from similar argument
r+2 )
p 2k(
k6j−2
2−j(γ−3+
)cj c̃jEh,σ̃E
T (u)
where σ̃ = γ − 5
with 1
< 3− γ < 2
. We get
2k(γ−
)‖gk‖
r+2 )
k6j−2
2(k−j)(γ−3+
)cj c̃jEh,σ̃E
T (u)
which implies nothing but
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T 1−
pEh,σ̃E
T (u)
by Young’s inequality. Note that σ̃ 6 γ
< s when r sufficiently closes to 2
, therefore
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T
r 2−2J(s−
)E2sET (u).
Combining this with (7.3) and (7.5), we complete the proof of (5.7) by obtaining
j>J−N0
Sj−1I(u2)△jvFutdxdt
−2J [s−(
−2J [s−(
+ T2−2J(s−
E2sET (u)
with max{2, 1
} < r1 < 23−γ and
6 r2 < ∞.
We secondly prove (5.8) which is different from (5.7). To this end, we need to make
Bony’s decomposition more than once and establish a commutator estimate, which
helps us to complete our proof. In view of the fact that v̂F only has high frequencies
again, it follows from Bony’s decomposition that there exists N0 such that
I(uvF )uut =
j>J−N0
Sj+2vF△ju
uut +
j>J−N0
△jvFSj−1u
== I + II. (7.9)
In order to estimate the term I, we split it into two pieces with N1 ≫ N0 > 0
j>J−N0
uut△kI
Sj+2vF△ju
j>J−N0
k6J−N1
uut△kI
Sj+2vF△ju
j>J−N0
k>J−N1
uut△kI
Sj+2vF△ju
== I1 + I2.
The estimate of I1 is broken down into the following two cases.
Case 1. 2 < γ 6 5
In this case, to our purpose, we obtain the following coarse estimate by Hölder’s
inequality
‖I1‖L1x .
j>J−N0
k6J−N1
Sj+2vF△ju
‖L3‖u‖L∞
L6‖ut‖L∞
j>J−N0
k6J−N1
2k(γ−2)‖△k
Sj+2vF△ju
ET (u)
j>J−N0
k6J−N1
2k(γ−2)‖Sj+2vF‖L6‖△ju‖L2ET (u)
k6J−N1
2k(γ−2)
j>J−N0
2−j2j‖△ju‖L2
‖△j′vF ‖L6ET (u).
Choosing (p, r) such that 1
with 2 6 r 6 6, the Strichartz estimate yeilds
‖I1‖L1
k6J−N1
2k(γ−2)
j>J−N0
j′( 3
‖△j′v0‖L2 + 2−j
′‖△j′v1‖L2
T (u).
Arguing similarly as before it yields that
‖I1‖L1
k6J−N1
2k(γ−2)
j>J−N0
r cj′Eh,1/2E
T (u)
r 2−2J [s−(
)]E2sET (u)
with 2 6 r 6 6. If choose r = 6, one can easily check that γ
2 < γ 6 5
. Although this result is enough for us to prove the main theorem, we want
to improve the result for this term by loosen the upper bound of r from 6 to ∞ through
the precise Strichartz estimate. Arguing similarly as before, we have
‖I1‖L1x .
j>J−N0
k6J−N1
Sj+2vF△ju
‖L3‖u‖L∞
L6‖ut‖L∞
j>J−N0
k6J−N1
2k(γ−3)2k
2r ‖△k
Sj+2vF△ju
ET (u)
Since the Fourier transform of Sj−1vF△ju was supported in 2jC and k ≪ j, △k(Sj−1vF△ju)
vanishes which implies △k
Sj+2vF△ju
△̃jvF△ju
. As the support of the
Fourier transform of a product is included in the sum of the support of each Fourier
transform, we also have
△̃jvF△ju
ν,ν′∈Λj,k
△νj,kvF△ν
ν∈Λj,k
△νj,kvF △̃νj,ku
Choosing (p, r) such that 1
for 2 6 r < ∞, it follows from the Hölder inequality
and L2 quasi-orthogonality properties that
Sj+2vF△ju
r+2 )
ν∈Λj,k
‖△νj,kvF‖Lr‖△̃νj,ku‖L2
ν∈Λj,k
‖△νj,kvF ‖2LpLr
ν∈Λj,k
‖△νj,ku‖2L2
ν∈Λj,k
‖△νj,kvF ‖2LpLr
∥∥△ju
Then the precise Strichartz estimate yields that
‖I1‖L1
k6J−N1
2k(γ−3)2k
j>J−N0
2−j2j
∥∥△ju
2(k−j)(
ν∈Λj,k
‖△νj,kv0‖2L2
2 + 2−j
ν∈Λj,k
‖△νj,kv1‖2L2
ET (u).
By the L2quasi-orthogonality properties, it gives that
‖I1‖L1
k6J−N1
2k(γ−3)2k
j>J−N0
2−j2j
∥∥△ju
‖△jv0‖L2 + 2−j‖△jv1‖L2
ET (u).
Utilizing the technique as before yields that
‖I1‖L1
k6J−N1
2k(γ−2+
j>J−N0
2−j(1+
)cjEh,σE
T (u)
r 2J(γ−3)Eh,σE
T (u) . T
r 2−2J [s−(
)]E2sET (u),
with 4
6 r < ∞.
Case 2. 5
< γ < 3
In the this case, the fact γ− 5
> 0 helps us to obtain the the desirable result easily.
Arguing similarly as before, we have
‖I1‖L1x .
j>J−N0
k6J−N1
Sj+2vF△ju
‖L3‖u‖L∞
L6‖ut‖L∞
j>J−N0
k6J−N1
2k(γ−3)23k(
Sj+2vF△ju
‖L2ET (u)
j>J−N0
k6J−N1
2k(γ−
)‖Sj+2vF‖L∞‖△ju‖L2ET (u).
Choosing (p, r) such that 1
with 2 6 r < ∞, the Strichartz estimate yields
‖I1‖L1
k6J−N1
2k(γ−
j>J−N0
2 cj′Eh,σE
T (u)
p 2J(γ−
j>J−N0
2 Eh,σE
T (u)
r 2−2J [s−(
)]E2sE
T (u).
Combining these two cases, we have shown that
‖I1‖L1
r 2−2J [s−(
)]E2sET (u) (7.10)
with 4
6 r < ∞. To control ‖I‖L1
, it remains to estimate ‖I2‖L1
. Compared
with ‖I1‖L1
, since the negative derivative acts on the high frequency, the upper
bound of ‖I2‖L1
is much easier to get. Here is the details:
‖I2‖L1x .
j>J−N0
k>J−N1
Sj+2vF△ju
‖L3‖u‖L∞
L6‖ut‖L∞
j>J−N0
k>J−N1
2k(γ−3)‖Sj+2vF ‖L∞‖△ju‖L3ET (u).
Choosing (p, r) such that 1
with 2 6 r < ∞ again, the Strichartz estimate
yields
‖I2‖L1
k>J−N1
2k(γ−3)
j>J−N0
2 cj′Eh,σE
T (u)
r 2−2J [s−(
)]E2sET (u).
Combining this with (7.10), we obtain that
‖I‖L1
r 2−2J [s−(
)]E2sET (u) (7.11)
for 4
6 r < ∞.
To complete the proof the Lemma 5.1, it remains to estimate II. One can proceed
this as above by Hölder’s inequality to estimate
j>J−N0
2j(γ−3)‖△jvFSj−1u‖L3
ET (u). (7.12)
Resorting to the Hölder inequality and the classical Strichartz estimate, one can obtain
‖II‖L1
L1 . T
r 2−2J [s−(
)]E2sET (u).
with 2 6 r 6 6. One also can try to improve the result by using the precise Strichartz
estimate as before, but it fails and merely obtain that
‖II‖L1
L1 . T
r 2−2J [s−(
)]E2sET (u).
with 2 6 r 6 4.
One can easily check that the result is worse than the desirable result because of the
restriction of r. Compared with the second term in (7.2), the negative derivative acts
on the high frequency part so that it is tempting to obtain a better result than that of
(7.2). But △jvF is bound with Sj−1u by the operator I, and this structure prevents us
from using efficiently the precise Strichartz estimate. If one first resort to the Hölder
inequality, as shown in (7.12), he or she merely obtains a loss result because of the
range restriction of r. To go around this difficulty, we first establish a commutator
estimate through exploiting cancellation property. Now we turn to details. Our task is
to estimate
j>J−N0
I(△jvFSj−1u)uutdxdt
In order to drag the Sj−1u out of the operator I, we construct uI(△jvF )Sj−1u and
the triangle inequality yields that
j>J−N0
I(△jvFSj−1u)uutdxdt
∣∣∣ 6
j>J−N0
∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u
j>J−N0
I(△jvF )Sj−1uuutdxdt
We benefit from the cancellation when we deal with the first term. Since both the
Fourier transformation of I(△jvFSj−1u) and I(△jvF )Sj−1u are supported in a ring
sized 2j , the Hölder inequality and the Bernstein inequality lead to that
∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u
2 ‖I(△jvFSj−1u)− I(△jvF )Sj−1u‖L2x‖u‖L6 .
Before estimating its right hand, we recall the Coifman and Meyer multiplier theorem.
Consider an infinitely differentiable symbol m : Rnk 7→ C so that for all α ∈ Nnk and
all ξ = (ξ1, ξ2, · · · , ξk) ∈ Rnk, there is a constant c(α) such that
|∂αξ m(ξ)| 6 c(α)(1 + |ξ|)−|α|. (7.13)
Define the multilinear operator T by
[T (f1, · · · , fk)](x) =
eix·(ξ1+···+ξk)m(ξ1, · · · , ξk)f̂1(ξ1), · · · , f̂k(ξk)dξ1 · · · dξk,
(7.14)
F [T (f1, · · · , fk)](ξ) =
ξ=ξ1+···+ξk
m(ξ1, · · · , ξk)f̂1(ξ1), · · · , f̂k(ξk)dξ1 · · · dξk−1. (7.15)
Proposition 7.1 ([8],Page 179.) Suppose pj ∈ (1,∞), j = 1, · · · k, are such that 1p =
+ · · ·+ 1
6 1. Assume m(ξ1, · · · , ξk) a smooth symbol as in (7.13). Then there
is a constant C = C(pi, n, k, c(α)) so that for all Schwarz class functions f1, · · · , fk,
‖[T (f1, · · · , fk)](x)‖Lp(Rn) 6 C‖f1‖Lp1 (Rn) · · · ‖fk‖Lpk (Rn). (7.16)
Since the operator I is a convolution operator with kernel |x|−γ in R3, we can write
F [I(△jvFSj−1u)−I(△jvF )Sj−1u](ξ) =
ξ=ξ1+ξ2
|ξ1+ξ2|γ−3−|ξ1|γ−3
△̂jvF (ξ1)Ŝj−1u(ξ2)dξ2.
By the mean value theorem, the right hand of the above formula becomes that
ξ=ξ1+ξ2
|ξ1 + λξ2|γ−4
(ξ1 + λξ2) · ξ2
|ξ1 + λξ2|
△̂jvF (ξ1)Ŝj−1u(ξ2)dξ2,
for a certain λ ∈ [0, 1]. Moreover, we rewrite it as follow:
ξ=ξ1+ξ2
m(ξ1, ξ2)f̂1(ξ1)f̂2(ξ2)dξ2,
m(ξ1, ξ2) = (ξ1 + λξ2)|ξ1 + λξ2|γ−5|ξ1|4−γ , f1 = |∇|γ−4△jvF , f2 = ∇Sj−1u.
Observe that |ξ1| > 2j−1 and 2j−2 > |ξ2|, we have that |ξ1+λξ2| ∼ |ξ1| > 2J−N0 . Hence,
we can check that the symbol m(ξ1, ξ2) satisfies the estimate (7.13). Finally, it follows
from Proposition 7.1 that
‖I(△jvFSj−1u)− I(△jvF )Sj−1u‖L2x . ‖f1‖Lrx‖f2‖
with 2 < r < ∞. After making use of the Bernstein inequality, the right hand can be
controlled by
2j(γ−4+
)‖△jvF ‖Lrx‖∇u‖L2x .
Keeping in mind j > J−N0 and recalling the definition of Eh,σ, the Strichartz estimate
and a direct calculation of summing in j show that
j>J−N0
2 2j(γ−4+
)‖△jvF ‖Lp
j>J−N0
2j(γ−3+
−s)Eh,s.
with 1
and 2 < r < ∞. Choosing r such that max{2, 1
} 6 r < ∞, we have
j>J−N0
∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u
r 2−2J [s−(
)]E2sET (u). (7.17)
Now the rest of the paper devotes to estimate this term
j>J−N0
I(△jvF )Sj−1uuutdxdt
In order to use precise Strichartz estimate, we need to decompose this term by Bony’s
para-product decomposition again,
I(△jvF )Sj−1uuut =
Sk−1(uSj−1u)△kI(△jvF )ut +△k(uSj−1u)Sk+2I(△jvF )
= II1 + II2.
After decomposing this, the term II1 is similar to the second term in the (7.2) and the
negative derivative acts on the high frequency △jvF leading to a better result than the
second term in the (7.2). Thanks to Fourier-Plancherel formula and Hölder inequality,
we obtain
j>J−N0
II1dxdt ≈
j>J−N0
Sk−1(uSj−1u)△kI(△jvF )△kutdxdt
j>J−N0
k′6k−2
△k′(uSj−1u)△kI(△jvF )△kutdxdt
j>J−N0
△k′(uSj−1u)△k′
k′6k−2
(△kI(△jvF )△kut)dxdt
j>J−N0
‖uSj−1u‖
2 ‖△k′
k′6k−2
(△kI(△jvF )△kut)‖L2
j>J−N0
‖u‖2L∞H1
2 ‖△k′
k′6k−2
(△kI(△jvF )△kut)‖L2
On the other hand, one denotes
gk′,j = △k′
k′6k−2
△kI(△jvF )△kut
to estimate ∑
′(− 1
)‖gk′,j‖
Let us write that
gk′,j =
k′6k−2
ν∈Λk,k′
△νk,k′I(△jvF )△kut
As the support of the Fourier transform of a product is included in the sum of the
support of each Fourier transform, we obtain
gk′,j =
k′6k−2
ν∈Λk,k′
△νk,k′I(△jvF )△̃νk,k′ut
Using Hölder inequality, we get
‖gk′,j‖
k′6k−2
ν∈Λk,k′
‖△νk,k′I(△jvF )‖Lr‖△̃νk,k′ut‖L2
6 2j(γ−3)
k′6k−2
ν∈Λk,k′
‖△νk,k′vF ‖2Lr
ν∈Λk,k′
‖△νk,k′ut‖2L2
6 2j(γ−3)
k′6k−2
ν∈Λk,k′
‖△νk,k′vF ‖2Lr
2‖△kut‖L2
the use of quasi-orthogonality properties is made in the last inequality.
Precise Strichartz estimate and the quasi-orthogonality properties imply that
‖gk′,j‖
r+2 )
p 2j(γ−3)
k′6k−2
′−k)( 1
ν∈Λk,k′
‖△νk,k′v0‖2L2
2 + 2−k
ν∈Λk,k′
‖△νk,k′v1‖2L2
‖△kut‖L2
p 2j(γ−3)
k′6k−2
′−k)( 1
‖△kv0‖L2 + 2−k‖△kv1‖L2
‖△kut‖L2
with 1
for 2 6 r < ∞. Therefore
′(− 1
)‖gk′,j‖
r+2 )
p 2j(γ−3)
k′6k−2
′−k) 2
r ck c̃kEh,σE
T (u).
A direct computation shows that
2 ‖gk′,j‖L1
′(− 1
)‖gk′,j‖
r+2 )
p 2j(γ−3)Eh,σE
T (u).
Hence, we have that
j>J−N0
II1dxdt
∣∣∣ . 2−2J [s−(
r E2sET (u) (7.18)
with 4
6 r < ∞. Finally, we conclude this section by giving the estimate of II2.
j>J−N0
II2dxdt
∣∣∣ . T
j>J−N0
‖△k(uSj−1u)Sk+1I(△jvF )‖L2
L2‖ut‖L∞T L2
j>J−N0
2j(γ−3)
‖△k(uSj−1u)‖L∞
r ‖△k′△jvF‖Lp
T (u)
j>J−N0
2j(γ−3)
‖△k(uSj−1u)‖L∞
2 ck′Eh,σE
T (u)
j>J−N0
2j(γ−3)
2 ‖△k(uSj−1u)‖L∞
(k′−k) 1
2Eh,σE
T (u)
j>J−N0
2j(γ−3)‖ck′‖ℓ2(Z)‖2
2 ‖△k(uSj−1u)‖L∞
L2‖ℓ2(Z)‖2−
2 ‖ℓ2(N)Eh,σE
T (u)
p 2J(γ−3)E
T (u)Eh,σ . 2
−2J [s−(
r E2sET (u).
(7.19)
Collecting (7.18) and (7.19), we have been proved that
j>J−N0
I(△jvF )Sj−1uuutdxdt
∣∣∣ . T
r 2−2J [s−(
)]E2sET (u), (7.20)
with 4
6 r < ∞. Finally, we complete the proof of (5.8) by (7.11) and (7.20), hence
it ends the proof of Lemma 5.1.
Acknowledgements: The authors are grateful to Prof. J.Chemin for sending his
lecture to us. The authors were partly supported by the NSF of China, No.10725102.
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http://arxiv.org/abs/math/0612028
	Introduction
	Preliminaries
	Global existence for the high frequency part
	Local existence for the low frequency part
	Energy estimate for the low frequency part
	Proof of Theorem ??
	Proof of Lemma ??
	References

  In this paper, we consider the Cauchy problem for Klein-Gordon equation with
a cubic convolution nonlinearity in $\R^3$. By making use of Bourgain's method
in conjunction with a precise Strichartz estimate of S.Klainerman and D.Tataru,
we establish the $H^s (s<1)$ global well-posedness of the Cauchy problem for
the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving
at the previously discussed conclusion, we obtain global solution for this
non-scaling equation with small initial data in $H^{s_0}\times H^{s_0-1}$ where
$s_0=\frac\gamma 6$ but not $\frac\gamma2-1$, for this equation that we
consider is a subconformal equation in some sense. In doing so a number of
nonlinear prior estimates are already established by using Bony's
decomposition, flexibility of Klein-Gordon admissible pairs which are slightly
different from that of wave equation and a commutator estimate. We establish
this commutator estimate by exploiting cancellation property and utilizing
Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems
that this is the first result on low regularity for this Klein-Gordon-Hartree
equation.

Introduction
We study the following Cauchy problem for the Klein-Gordon-Hartree equation:
�φ+ φ+ (|x|−γ ∗ |φ|2)φ = 0 in R×R3
φ|t=0 = φ0, ∂tφ|t=0 = φ1.
(1.1)
Here φ(t, x) is a complex valued function defined in space time R1+3, and � = ∂tt −∆.
http://arxiv.org/abs/0704.1866v4
Recently the Cauchy problem (1.1) has been extensively studied in the case with
initial data (φ0, φ1) ∈ H1(Rn) × L2(Rn). The well-posedness and the asymptotic be-
havior of solution to the Cauchy problem (1.1) have been studied by G.P. Menzala and
W.Strauss [16, 17]. The scattering theory of solution to (1.1) has been established in
[23]. On the other hand, the time-dependent Schrödinger equation with interaction
term (|x|−γ ∗ |φ|2)φ has also been extensively studied. Ginibre and Velo [11] gave the
scattering theory of Hartree equation for the energy subcritical case. For the energy
critical case and mass critical, one can refer to [20, 21] with radial initial data.
Many authors [4, 9, 12, 18, 30] have studied the local well-posedness (as well as
global well-posedness) in fractional Sobolev spaces for the Cauchy problem of general
semilinear wave or Schrödinger equations under minimal regularity assumptions on
the initial data. For example, Tao [30] established the sharp local well-posedness of
nonlinear wave equation. Kenig, Ponce, and Vega [12] had established the global well-
posedness under the energy norm for the Cauchy problem of nonlinear wave equations
with rough initial data (in particular, in Ḣs(R3), 3
< s < 1 for cubic wave equation).
They used the Fourier truncation method discovered by Bourgain [4]. And also [18]
extended Kenig-Ponce-Vega’s result to the dimension n > 4. Recently, I. Gallagher
and F. Planchon [9] presented a different proof of the result in [12] for 3
< s < 1. H.
Bahouri and Jean-Yves Chemin [2] proved global well posedness for s = 3
by using a
nonlinear interpolation method and logarithmic estimates from S. Klainermann and D.
Tataru[14]. We also find Roy [26] obtains the global well-posedness for rough initial data
in Ḣs, 2
< s < 1 by following the I-method [5] and scaling transformation. However, if
one similarly deals with Klein-Gordon equation by using I-method, he or she may meet
a problem caused by the lack of the scaling property. More studies and discussions on
the low regularity of nonlinear wave or dispersive Schrödinger equations could be found
in [4, 31]. However, as far as we know, very few authors are engaged in studying the
global well-posedness of the Cauchy problems (1.1) with less regular initial data. It is
natural to ask whether a similar or better result holds for the problem (1.1).
This paper endeavors to find a global well-posedness solution to the Cauchy problem
(1.1) with initial data (φ0, φ1) ∈ Hs(R3) ×Hs−1(R3) for some s > γ4 with γ ∈ (2, 3).
Now we should remark some differences between (1.1) and cubic wave equation. If one
views (1.1) as a wave equation by dropping the massive term and then makes some
scaling analysis, we will find this nonlocal nonlinear term shares the scaling property of
the nonlinearity |u|
5−γ u. One can check that k := 4
+1 < 3 when 2 < γ < min{n, 4}
with n = 3 and this result shows that the equation which we consider is in subconformal
case. To obtain the global well-posedness theory, some previous literatures also show
the subconformal equations are slightly different from the superconformal ones. For
instance, Lindblad and Sogge [15] [27] have shown the global existence and scattering
theory for small data in a less regularity space for the superconformal case, while
not for the subconformal case. Inspired by [9], we also split the initial data into low
frequency part data in H1 and high frequency part data in Hs0 with a suitable s0.
Since the problem (1.1) is global well-posed for large data in H1 and small data in
Hs0 , one may be tempted to follow a general principle of nonlinear interpolation and
claim the problem (1.1) is global well-posed between them. Compared with the cubic
wave equation, speaking of the Strichartz estimate, we believe that the global solution
with high frequency data should exist in H
−1. It is well known that the Strichartz
estimate is associated with scaling transform and it is scaling invariant. Unfortunately,
the equation that we consider is a subconformal one, and its concentration effects take
over scaling. Since the Strichartz estimate is applied to our subconformal equation,
hence this brings about some loss to get a better result. In order to get a better result,
one should establish an estimate which is conformal invariant. Fortunately, we can take
0 6 θ 6 1 as a parameter for the flexible admissible pairs (see Definition 2.3)to make
the Strichartz estimate of Klein-Gordon more flexible than wave equation. This helps
us to get a global solution with the high frequency data, at the cost of 0 6 θ = 6
−2 6 1
which weakens the Strichartz estimate and causes 2 < γ < 3. One can refer the detail
in Section 3.
We point out that it is easy to have the result for
< s < 1 by rough Hölder’s
inequality. But how to get our low bound
< s < 1? A good way to think about this
is via precise Strichartz estimate to obtain index s as low as possible. The nonlinearity
including a formal negative derivative brings us some difficulties caused by the fact that
the negative derivative acts on the low frequency part. And this leads us to restricts
rather than s > max{1
}. At the end of this section, we also give some
intuitive analysis to show our result is reasonable. As a limited case, our result recovers
the result of [9, 12] when γ tends to 3.
During the process of proving our key estimate Lemma 5.1, the nonlocal nonlin-
earity brings about some essential difficulties when we try to make use of the precise
Strichartz estimate. Compared with the general semilinear nonlinearity, the convo-
lution nonlinearity not only essentially represents a negative derivation in it but also
has a difference construction of nonlinearity. These differences and difficulties prevent
us from obtaining directly our expected result s >
by restricting the range of the
parameter r. To overcome these difficulties, we firstly construct a commutator and
establish this commutator estimate by exploiting cancellation property and utilizing
Coifman and Meyer multilinear multiplier theorem and then go on our process through
using precise Strichartz estimate.
Now we state our main result:
Theorem 1.1 Let
< s < 1 with 2 < γ < 3. If (φ0, φ1) ∈ Hs(R3) ×Hs−1(R3), then
there exists a unique global solution φ of (1.1) such that φ ∈ C(R+;Hs(R3)).
We conclude this section by giving a sketch of the proof of Theorem 1.1 and one shall
read more detailed information in the rest of this paper. Without loss of generality,
we only consider φ as a real function for simplicity from now on. Since the problem
(1.1) is global well-posed for large data in H1 and small data in Hs0 with s0 =
, one
may be tempted to follow a general principle of nonlinear interpolation and believe the
problem (1.1) to be global well-posed between them, as well as the cubic defocusing
wave equation [9]. To make sense of this heuristic, we proceed it in the following steps.
Step 1. The purpose of this step is to show the global well-posedness for the high
frequency part. We split the initial data:
φi = (I− SJ)φi + SJφi
= vi + ui i = 0, 1
where I is identity operator and SJ is Littlewood-Paley operator, referring to Section
2. It is easy to see that
‖u0‖H1 . 2J(1−s)‖φ0‖Hs , ‖u1‖L2 . ‖φ1‖L2
‖v0‖Hβ . 2J(β−s)‖φ0‖Hs , ‖v1‖Hβ−1 . 2J(β−s)‖φ1‖Hs−1 for all β 6 s.
Thus it follows that
Eh,σ . 2J(σ−s)Es, for σ 6 s (1.2)
Eℓ,1 . 2J(1−s)Es, for s 6 1, (1.3)
where
== ‖φ0‖Hs + ‖φ1‖Hs−1 , (1.4)
== ‖v0‖Hσ + ‖v1‖Hσ−1 , (1.5)
== ‖u0‖Hσ + ‖u1‖Hσ−1 . (1.6)
Choosing J large enough, one can achieve Eh,s0 small enough, in other words, initial
data of the following problem
�v + v + (|x|−γ ∗ v2)v = 0 in R× R3,
v|t=0 = v0, ∂tv|t=0 = v1
(1.7)
is small enough inHs0(R3)×Hs0−1(R3) where s0 < s. Due to some technique difficulties
and this equation is subconformal one, we are restricted to choose s0 =
while not
− 1 proposed by scaling analysis or γ
proposed by conformal analysis. We will
get a global well-posed solution to the Cauchy problem (1.7), see Section 3 for details.
Step 2. In order to recover a solution to our problem (1.1), we solve a perturbed
equation with large initial data in H1 × L2,
�u+ u+ I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v = 0,
u|t=0 = u0 ∂tu|t=0 = u1,
(1.8)
where the operator I is the operator (−∆)
2 . We will prove there exists a unique
local solution to (1.8) in C([0, T ];H1).
Step 3. To complete the proof of Theorem 1.1, the key is how to extend the local
solution to a global solution. We should establish a priori bound on the energy of the
local solution u. In fact, the energy estimate yields
‖u(t)‖2H1 + ‖ut(t)‖
R3×R3
|x− y|−γu2(y, t)u2(x, t)dydx
‖u0‖2H1 + ‖u1‖
R3×R3
|x− y|−γu20(y)u20(x)dydx
I(v2)(x, τ)u(x, τ)∂τu(x, τ)dxdτ
I(uv)(x, τ)v(x, τ)∂τu(x, τ)dxdτ
I(u2)(x, τ)v(x, τ)∂τu(x, τ)dxdτ
I(uv)(x, τ)u(x, τ)∂τu(x, τ)dxdτ
∣∣∣∣ .
Let HT (u) := sup
H(u)(t) where
H(u)(t)
‖u(t)‖2H1 +
‖ut(t)‖2L2 +
R3×R3
|x− y|−γu2(y, t)u2(x, t)dydx
and then by making use of Hölder’s inequality and Sobolev embedding, it follows that
HT (u) .H(u)(0) +HT (u)
‖v(τ)‖2
dτ +H
T (u)
‖v(τ)‖
.H(u)(0) +HT (u)T
6 ‖v‖2
T (u)T
3 ‖v‖Xα
.22J(1−s) +HT (u)T
6 22J(β−s) +H
T (u)T
3 2J(α−s)
where α =
, β =
and the space Xα is defined in the coming section. What we
want to do is to control HT (u) for arbitrarily large T . As long as s > (α+1)/2 =
by choosing J large enough, bootstrap argument yields
HT (u) . 2
2J(1−s).
One can see that, if s >
, the argument is trivial, since the above mentioned
result can be deduced from some rough estimates such as the Hölder estimate. On
the other hand, since the scaling suggests us that X
−1 is the lowest regularity space
which v could belong to, it is tempting and reasonable to believe that the best result
obtained by this method is s > (
− 1+1)/2 = γ
instead of α by
− 1. To obtain this
optimal result s >
, we adopt some more sophisticated tools such as precise Strichartz
estimate, Bony’s paraproduct estimates and twice Bony’s decomposition. This result
is achieved under an assumption of a core estimate which will be shown through the
precise Strichartz estimate and a commutator estimate.
The paper is organized as follows: In the coming section, we recall some nota-
tions and recollect some well known results on Besov spaces in conjunction with the
Littlewood-Paley theory which will be used in the course of the proofs. Meanwhile,
we also introduce the precise Strichartz estimate. Section 3 provides the global well-
posedness of original equation evoking the high frequency part of initial data in Hs0 .
In Section 4, we prove prove a local well-posedness of perturbed equation with the low
frequency of the initial data in H1 by the standard fixed point theorem. In Section 5,
we give a energy estimate for the low frequency part provided an assumption the key
estimate in Lemma 5.1. We extend the local well-posedness of the perturbed equation
to globally well posed by the bootstrap argument in Section 6. In the final section,
we prove our essential and key lemma by the precise Strichartz estimate, commutator
estimate and Coifman and Meyer multiplier theorem.
2 Preliminaries
In this section, we shall present some well-known facts on the Littlewood-Paley theory
and introduce some notations, definitions and estimates which are needed in this paper.
Let S(R3) be the Schwarz class of rapidly decreasing functions. Given f ∈ S(R3), its
Fourier transform Ff = f̂ is defined by
f̂(ξ) = (2π)−
e−ix·ξf(x)dx, F−1f = f̂(−ξ).
Choose two nonnegative radial functions χ, ϕ ∈ S(R3) supported respectively in B =
{ξ ∈ R3, |ξ| 6 4
} and C = {ξ ∈ R3, 3
6 |ξ| 6 8
} such that
χ(ξ) +
ϕ(2−jξ) = 1, ξ ∈ R3,
ϕ(2−jξ) = 1, ξ ∈ R3\{0},
supp ϕ(2−j ·) ∩ supp ϕ(2−j′ ·) = ∅, |j − j′| > 2,
supp χ(·) ∩ supp ϕ(2−j ·) = ∅, j > 1.
Now we are in position to define the the Littlewood-Paley operators Sj, Ṡj, △j and
△̇j which are used to define Besov space.
0, j 6 −2,
χ(ξ)û(ξ)
, j = −1,
(F−1ϕ)(2jy)u(x− y)dy, j > 0,
j′6j−1
△j′u = 2jn
(F−1χ)(2jy)u(x− y)dy,
== 2jn
(F−1ϕ)(2jy)u(x− y)dy, j ∈ Z,
j′6j−1
△̇j′u.
One easily shows that △̇j = Ṡj+1 − Ṡj for j ∈ Z and
△−1 = S0, △̇j = △j , j > 0.
Now we give the Littlewood-Paley’s description of the Besov spaces.
Definition 2.1 Let s ∈ R, 1 6 p, q 6 ∞. The homogenous Besov space Ḃsp,q is defined
Ḃsp,q = {f ∈ Z ′(R3) : ‖f‖Ḃsp,q < ∞},
where
Ḃsp,q
2jsq‖△̇jf‖qp
, for q < ∞,
2js‖△̇jf‖p, for q = ∞,
and Z ′(R3) can be identified by the quotient space S ′/P with the space P of polynomials.
Definition 2.2 Let s ∈ R, 1 6 p, q 6 ∞. The inhomogeneous Besov space Bsp,q is
defined by
Bsp,q = {f ∈ S ′(R3) : ‖f‖Bsp,q < ∞},
where
‖f‖Bsp,q =
2jsq‖△jf‖qp
+ ‖S0(f)‖p, for q < ∞,
2js‖△jf‖p + ‖S0(f)‖p, for q = ∞.
If s > 0, then Bsp,q = L
p ∩ Ḃsp,q and ‖f‖Bsp,q ≈ ‖f‖p + ‖f‖Ḃsp,q . We refer the reader to
[1, 6, 22, 32] for details.
In order to investigate the low regularity solution of the Cauchy problem (1.1), we
require the use of the smoothing effect described by the Strichartz estimates and precise
Strichartz estimates. For the purpose of conveniently making use of the Strichartz
estimate, we introduce the admissible definition and the resolution space.
Definition 2.3 We shall say that a pair (q, r) is admissible, for 0 6 θ 6 1, if
q, r > 2, (q, r, θ) 6= (2,∞, 0) and 1
2 + θ
2 + θ
Remark 2.1 The above admissible pairs in Definition 2.3 is more flexible than wave
admissible pairs, since θ can vary from 0 to 1. Obviously, an admissible pair in Defini-
tion 2.3 will become a wave admissible pair when θ = 0. When we consider the global
existence for the high frequency part, we shall use θ = 6
− 2 since the equation that we
consider is a subconformal one.
The resolution space is defined in the following way based on the admissible defini-
tion.
Xµ(I) :=
06θ61
where
(I) :=
u : u ∈ (C ∩ L∞)(I;Hµ) ∩ Lq(I;Bσr,2)
(q,r) is admissible,
= (3 + θ)(
) + σ − µ
We go on this section by recalling the classical Strichartz estimate and the precise
Strichartz estimate. This kind of estimate goes back to Strichartz [29], and has been
proved in its generality by Ginibre and Velo [10], and Keel and Tao [13]. The Strichartz
estimates for the Klein-Gordon equation by using the above flexible admissible pairs
can be found in [19].
Proposition 2.1 Let u be a solution of
�u+ u = f in R× R3 with u|t=0 = u0, ∂tu|t=0 = u1.
Then, for any admissible pairs (q1, r1) and (q2, r2), we have that
‖△ju‖Lq1 (Lr1 ) + 2−j‖∂t△ju‖Lq1 (Lr1 )
j( 3+θ
− 3+θ
(‖△ju0‖L2 + 2−j‖△ju1‖L2)
j[(3+θ)(1− 1
−1]‖△jf‖
. (2.1)
We shall see that the classical Strichartz estimates are not enough to control some
nonlinearities, and this leads us to resort to the following precise Strichartz estimates
which were established by S.Klainerman and D.Tataru[14].
Proposition 2.2 Let u be a solution of
�u+ u = 0 with u|t=0 = u0, ∂tu|t=0 = u1.
Assume that the supports of the Fourier transform of u0 and u1 are included in a ball
B(ξj, h2
j) with |ξj | ∈ [2j−2, 2j+2] and h < 18 . Then we have that, for any admissible
couple (q, r),
‖u‖Lq(Lr) + 2−j‖∂tu‖Lq(Lr) 6 C2
r (‖u0‖L2 + 2−j‖u1‖L2). (2.2)
Let us recall the Hardy-Littlewood-Sobolev inequality [22, 28] and a proposition of
contraction which is generalization of Picard’s theorem [6]. We denote operator I by
Iu def== (−∆)
2 u = |x|−γ ∗ u,
‖Iu‖Lq(R3) 6 Cp,q‖u‖Lp(R3) (2.3)
0 < γ < 3, 1 < p < q < ∞, and 1
− 3− γ
Proposition 2.3 Let X be a Banach space and let B : X ×X × · · · ×X → X be a
m-linear continuous operator (m > 2) satisfying
‖B(u1, u2, · · · , um)‖X 6 M‖u1‖X‖u2‖X · · · ‖um‖X , ∀u1, u2, · · · , um ∈ X
for some constant M > 0. Let ε > 0 be such that m(2ε)m−1M < 1. Then for every
y ∈ X with ‖y‖X 6 ε the equation
u = y +B(u, u, · · · , u) (2.4)
has a unique solution u ∈ X satisfying that ‖u‖X 6 2ε. Moreover, the solution u
continuously depends on y in the sense that, if ‖y1‖X 6 ε and v = y1 +B(v, v, · · · , v),
‖v‖X 6 2ε then
‖u− v‖X 6
1−m(2ε)m−1M ‖y − y1‖X . (2.5)
For the sake of convenience, we conclude this section by giving some notations. The
solution φ to the Cauchy problem (1.1) is given by the following integral equation:
φ(t, x) = K̇(t)φ0 +K(t)φ1 +B(φ, φ, φ)
= T φ
where
K(t) :=
sin(t
I −∆)√
B(u1, u2, u3) := −
K(t− τ)(|x|−γ ∗ (u1u2))u3dτ.
Throughout this article we shall denote by the letter C all universal constant and ε > 0
is a arbitrary small data. We shall sometimes replace an inequality of the type f 6 Cg
by f . g. Also, we shall denote by (cj)j∈Z any sequence of norm less than 1 in ℓ
2(Z).
3 Global existence for the high frequency part
Let us consider the Cauchy problem with the high frequency data,
�v + v + (|x|−γ ∗ v2)v = 0, (t, x) ∈ R× R3
v|t=0 = v0, ∂tv|t=0 = v1, x ∈ R3.
(3.1)
and then its integral formation becomes
v(t, x) =K̇(t)v0(x) +K(t)v1(x)−
K(t− τ)(|x|−γ ∗ v2)vdτ
==K̇(t)v0(x) +K(t)v1(x) +B(v, v, v). (3.2)
Our goal in this section is to prove the global well-posedness of (3.1) or (3.2). More
precisely, we have the following proposition:
Proposition 3.1 Let s0 =
and suppose that (v0, v1) ∈ Hµ×Hµ−1 for any 0 6 µ 6 1.
There exists a constant ε0 > 0 such that if
‖v0‖Hs0 + ‖v1‖Hs0−1 6 ε0,
then there exists a unique global solution v to (3.1) or (3.2) in Xs0(R)∩Xµ(R). More-
over,
‖v‖Xµ 6 Cµ (‖v0‖Hµ + ‖v1‖Hµ−1) .
Remark 3.1 We focus on µ =
and µ =
in the coming section.
It is well known that the global existence theory for small initial data is a straightfor-
ward result of nonlinear estimate, thus how to obtain a suitable nonlinear estimate is
essential. Before proving this proposition, we make some analysis on nonlinear estimate.
As mentioned in the introduction, the nonlocal nonlinearity shares the scaling with a
subconformal nonlinearity when γ < 3 and this may bring some troubles when we make
a choice of a suitable resolution space Xs0 . Take 0 6 θ 6 1 as a parameter in the flexible
admissible pairs (see Definition 2.3), and we make analysis on the relationship between
θ and s0. The Strichartz estimate, Hölder inequality and Hardy-Littlewood-Sobolev
inequality imply that, for σ 6 0,
‖B(v, v, v)‖Xs0 6 ‖(|x|−γ ∗ |v|2)v‖
6 ‖v‖
Lq2 (B
‖v‖2Lq3 (Lr3 ),
with satisfying
= (3 + θ)(
) + σ + s0 − 1
= (3 + θ)(
)− σ − s0
= (3 + θ)(
)− s0
− 1 + γθ
We find the fact index s0 is increasing when the parameter θ increases. It is tempting to
choose θ = 0 to get the smallest s0 =
− 1 proposed by scaling. However, in addition
the admissible condition implies that
6 (2 + θ)(
6 (2 + θ)(
6 (2 + θ)(
then a direction computation gives that
) 6 (2 + θ)(2− 1
which yields that
6 1 +
If we choose θ = 0, then we are forced to γ > 3 which contradict with our requirement
γ < 3. But if we choose θ = 6
− 2 and then s0 = γ6 and we are allowed by 2 6 γ 6 3.
Proof of Proposition 3.1 Thanks to Strichartz estimate, we have
‖B(v, v, v)‖Xµ 6 ‖(|x|−γ ∗ |v|2)v‖
6 ‖v‖
Lq2 (B
‖v‖2Lq3 (Lr3 ),
where
3 + γ
(1− µ− σ), 1
3 + γ
(1− µ− σ)
3(µ+ σ)
3 + γ
− γ(µ+ σ)
3 + γ
2(3 + γ)
9 + 3γ − γ2
6(3 + γ)
When 0 6 µ 6 1
, we choose σ = 0; while 1
< µ 6 1, we choose σ = 1
Thus,
‖B(v, v, v)‖Xµ 6 ‖v‖Xµ‖v‖2Xs0 . (3.3)
Combining this nonlinear estimate, the Proposition 3.1 follows from a standard con-
traction argument and small initial data condition.
4 Local existence for the low frequency part
In this part, we shall study the following perturbed problem in R× R3:
�u+ u+ I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v = 0
u|t=0 = u0 ∂tu|t=0 = u1.
(4.1)
Proposition 4.1 Let α =
, β =
and assume that v be in Xα ∩ Xβ and
(u0, u1) ∈ H1 ×L2, then there exists a positive time T such that a unique solution u to
(4.1) satisfying
u ∈ C([0, T ];H1).
Proof of the Proposition 4.1 In practice, solving (4.1) on [0, T ] is equivalent to
solving the following integral equation
u =K̇(t)u0 +K(t)u1
K(t− τ)
I(u2)u+ 2I(uv)u+ I(v2)u+ I(u2)v + 2I(uv)v
,T̃ u.
Using the Strichartz estimate, we have
K(t− τ)I(u2)udτ
. ‖I(u2)u‖L1
(L2).
On one hand, we make use of Hölder’s inequality and Hardy-Littlewood-Sobolev in-
equality to deduce that
‖I(u2)u‖L1
(L2) 6 C‖I(u2)‖
6 C‖u‖3
6 CT‖u‖3L∞
H1 . (4.2)
For the rest of terms, arguing similarly as above, it can be obtained that
‖I(uv)u‖L1
(L2) 6 C‖u‖2L∞
L6‖v‖
3 ‖u‖2L∞
H1‖v‖Xα , (4.3)
‖I(v2)u‖L1
(L2) 6 C‖u‖L∞T L6‖v‖
3 ‖u‖L∞
H1‖v‖2Xβ , (4.4)
‖I(u2)v‖L1
(L2) 6 CT
3 ‖u‖2L∞
H1‖v‖Xα , (4.5)
‖I(uv)v‖L1
(L2) 6 CT
3 ‖u‖L∞
H1‖v‖2Xβ . (4.6)
A combination of (4.2), (4.3)-(4.6) and the Strichartz estimate in Proposition 2.1 lead
to the estimate
‖u‖L∞
(H1) . ‖u0‖H1 + ‖u1‖L2 + T‖u‖3L∞
3 ‖u‖2L∞
H1‖v‖Xα + T
3 ‖u‖L∞
H1‖v‖2Xβ .
As long as choosing T is small enough, T̃ is a contraction mapping in ball B(0, 2CEℓ,1).
By means of Picard’s fixed point argument we have an unique solution u to (4.1) in
L∞([0, T ];H1). Therefore, Proposition 4.1 is proved by the standard argument.
5 Energy estimate for the low frequency part
In order to extend the local solution to a global solution, we shall prove a prior estimate
for the Hamiltonian of u in this section. Let us recall the definition of Hamiltonian of
u defined by
H(u)(t)
‖u(t)‖2H1 +
‖ut(t)‖2L2 +
R3×R3
|x− y|−γu2(y, t)u2(x, t)dydx
Similarly we give another notation of the energy of u, which is denoted by
E(u)(t)
‖u(t)‖2H1 +
‖ut(t)‖2L2 .
HT (u)
== sup
H(u)(t), ET (u)
== sup
E(u)(t).
To extend the local existence to global existence, we have to do a number of nonlinear
a priori estimates provided that ET (u) 6 2CH(u)(0), see Proposition 5.1 and Lemma
5.1. As a direct consequence of the above assumption, we get an important relationship
between E(u) and Es defined in the introduction
ET (u) . 2
2J(1−s)(E2s + E4s ) . 22J(1−s). (5.1)
In fact, it follows from Hardy-Littlewood Sobolev inequality and the definition of u0
‖(|x|−γ ∗ u20)u20‖L1 . ‖u0‖412
6 ‖S0φ0‖412
06j6J
‖△jφ0‖412
And then the right hand of the above inequality can controlled that as soon as 1 > s >
by utilizing Bernstein inequality
‖S0φ0‖4L2 +
06j6J
−s)2j4s‖△jφ0‖4L2 . 2
2J(1−s)E4s .
From now on, we assume (5.1) to in our subsequence proof.
Proposition 5.1 Assume that (u0, u1) ∈ H1×L2, then the following estimate holds
for s0, α, β defined in Proposition 3.1 and Proposition 4.1,
HT (u) .H(u)(0) + T
3 2−2J(s−β)ET (u) + T
3 2−J(4s−α−2s0−1)ET (u)
−2J [s−(
−2J [s−(
+ T2−2J(s−
ET (u)
for max{2, 1
} < r1 < 23−γ and
6 r2 < ∞.
Proof. Multiplying (4.1) by ∂tu and integrating over x and t, we have
‖u(t)‖2H1 + ‖ut(t)‖
R3×R3
|x− y|−γu2(y, t)u2(x, t)dydx
‖u0‖2H1 + ‖u1‖
R3×R3
|x− y|−γu20(y)u20(x)dydx
I(v2)(x, τ)u(x, τ)∂τu(x, τ)dxdτ
I(uv)(x, τ)v(x, τ)∂τu(x, τ)dxdτ
I(u2)(x, τ)v(x, τ)∂τu(x, τ)dxdτ
I(uv)(x, τ)u(x, τ)∂τu(x, τ)dxdτ
∣∣∣∣ .
By taking the supermum over t 6 T , we have
HT (u) .H(u)(0) +
∥∥I(v2)u∂tu
+ ‖I(uv)v∂tu‖L1
I(u2)v∂tudxdt
I(uv)u∂tudxdt
=H(u)(0) + I + II + III + IV. (5.2)
The proof is broken down into the following several steps.
(i) Firstly, we estimate I and II. Making a similarly argument as (4.4) in the proof
of Proposition 4.1, it can be obtained that
I 6 ‖I(v2)u‖L1
L2‖ut‖L∞T L2 6 T
3 ET (u)‖v‖2Xβ ,
and then keeping in mind v has been estimated in Proposition 3.1, this together with
(1.2) yields that
I 6 T
3 ET (u)E2h,β 6 T
3 ET (u)2
−2J(s−β)E2s . (5.3)
Arguing similarly, we easily get
II 6 T
3 ET (u)2
−2J(s−β)E2s . (5.4)
(ii) Secondly, we estimate the terms III and IV . As mentioned in the introduction,
one can get the same type of estimate as above for the terms I and II, but that will
lead to s > α
, which is worse than the exponent given in the Theorem 1.1. To
improve the lower bound on s, we have to utilize more precise estimate on III and IV .
We first split III and IV into two different pieces, respectively. One can write
v = vF +B(v, v, v),
where vF is its free part and the other one comes from nonlinear term. For the nonlinear
part, it follows from (3.3) that
‖B(v, v, v)‖Xα 6 ‖v‖Xα‖v‖2Xs0 .
This along with (4.5), one can see that
‖I(u2)B(v, v, v)ut‖L1
L1 6 ‖I(u2)B(v, v, v)‖L1
L2‖ut‖L∞T L2
3 ‖u‖2L∞
H1‖B(v, v, v)‖Xα‖ut‖L∞T L2
3 ET (u)
2 ‖v‖Xα‖v‖2Xs0
Moreover, we get by (1.2),
‖I(u2)B(v, v, v)ut‖L1
L1 6 T
T (u)2
−J(3s−α−2s0)E3s . (5.5)
By the same way as leading to (5.5), we easily infer that
‖I(uB(v, v, v))uut‖L1
L1 6 T
T (u)2
−J(3s−α−2s0)E3s . (5.6)
Thus, it is sufficient to estimate these terms including free part vF since (5.5) and (5.6).
The following lemma gives estimates for the nonlinearity including free part vF .
Lemma 5.1 Let vF be a solution of the free Klein-Gordon equation, and u be such
that ET (u) . 2
2J(1−s). Then, for max{2, 1
} < r1 < 23−γ and
6 r2 < ∞
I(u2)vFutdxdt
∣∣∣ .
−2J [s−(
−2J [s−(
+ T2−2J(s−
ET (u), (5.7)
I(uvF )uutdxdt
∣∣∣ . T
−2J [s−(
ET (u). (5.8)
Hence these together with (5.5)-(5.6) yield that
III + IV . T
3 ET (u)2
−J [4s−α−2s0−1]E4s +
−2J [s−(
−2J [s−(
+ T2−2J(s−
ET (u). (5.9)
Therefore, we complete the proof of Proposition 5.1 provided that we had proved
Lemma 5.1, whose proof is postponed in the last section.
6 Proof of Theorem 1.1
Since the Cauchy problem (1.1) is split into equation (3.1) which is globally well-posed
by choosing J enough to make Eh,s0 < ε0 and equation (4.1) which is locally well-posed
(see Proposition 3.1 and Proposition 4.1), we have to show that the local solution to
equation (4.1) can be extended globally.
Let us denote T ∗J the maximum time of existence in Proposition 4.1. Theorem 1.1
will be proved if
T ∗J = +∞.
Let us consider TJ the supremum of the T < T
J such that
ET (u) 6 2CH(u)(0).
Thus, for any T < TJ , Proposition 5.1 gives us that
ET (u) 6 H(u)(0)
C + C1T
3 2−2J(s−β)E2s
+ C2T
3 2−J(4s−α−2s0−1)E4s + C3T
−2J [s−(
)]E2s
+ C4T2
−2J(s− 1
)E2s + C5T
−2J [s−(
)]E2s
By the assumption of Theorem 1.1 s >
, one easily verifies that
s > max
if choosing r1 sufficiently close to
and r2 large enough. We infer that TJ > T̃J if
we choose T̃J such that
= min
22J(s−β)
5C1E2s
24J(s−
5C2E4s
2J [s−(
5C3E2s
) 2r1
22J(s−
5C4E2s
2J [s−(
5C5E2s
) 2r2
By the definition of TJ , we get T
J > T̃J . Obviously, T̃J tends to infinity when J tend
to infinity. This completes the proof of Theorem 1.1.
7 Proof of Lemma 5.1
In order to make conveniently use of the precise Strichartz estimate on which mostly
the following proof relies, we begin this section by introducing a family of balls of center
ν )ν∈Λj,k of radius 2
k and a function χ ∈ C∞c (B(0, 1)) such that for j > 0
∀ξ ∈ 2jC,
ν∈Λj,k
χ(2−k(ξ − ξj,kν )) = 1 and C−10 6
ν∈Λj,k
χ2(2−k(ξ − ξj,kν )) 6 C0.
Let us define that, for some constant c
△νj,ka
== F−1
ϕ(2−jξ)χ(2−k(ξ − ξj,kν ))
â(ξ)
△̃νj,ka
== F−1
ϕ̃(2−jξ)χ(c2−k(ξ + ξj,kν ))
â(ξ)
As the support of the Fourier transform of a product belongs to the sum of the support
of each Fourier transform, we have
△ja =
ν∈Λj,k
△νj,ka, △jb =
ν′∈Λj,k
△ν′j,kb.
In view of this fact that if k 6 j − 2
ν,ν′∈Λj,k
△νj,ka△ν
is vanish when ξ
ν is close to ξ
, without loss of generality, we can write
△k(△ja△jb) ≈ △k
ν∈Λj,k
△νj,ka△̃νj,kb. (7.1)
For the sake of convenience, we also fix the notation in this section that, for 0 6=
f(t, x) ∈ L2TL2
ck = 2
‖△kv0‖L2 + 2−k‖△kv1‖L2
E−1h,σ, c̃k =
‖△kf‖L2
‖f‖L2
with σ = 1/2 + 1/r for 2 6 r < ∞.
Proof of Lemma 5.1. We first prove (5.7). In view of the fact that v̂F only has
high frequencies, Bony’s decomposition implies that there exists constant N0 such that
I(u2)vFut =
j>J−N0
Sj+2vF△jI(u2)ut +
j>J−N0
Sj−1I(u2)△jvFut. (7.2)
Since the negative derivative I acts on the high frequency for the former term while
on the low frequency for the latter one, the first term is much better than the second
one. We shall estimate the first term by using merely the Hölder inequality, Bernstein
inequality and classical Strichartz estimates. Firstly, we see that, for 2 6 r < ∞
j>J−N0
‖Sj+2vF△jI(u2)‖L2x .
j>J−N0
j′6j+1
‖△j′vF ‖L∞x ‖△jI(u
2)‖L2x
j>J−N0
j′6j+1
r ‖△j′vF ‖Lr2j(γ−
)‖u‖2L∞
Bernstein inequality and (2.1) in Proposition 2.1 with 1
for all 2 6 r < ∞
imply that
j>J−N0
‖Sj+2vF△jI(u2)‖L2x
p ‖u‖2L∞
j>J−N0
j′6j+1
r ‖△j′vF‖Lp
j(γ− 7
p ‖u‖2L∞
j>J−N0
2j(γ−
j′6j+1
j′( 3
‖△j′v0‖L2 + 2−j
′‖△j′v1‖L2
The right hand of the above inequality can be controlled by
p ‖u‖2L∞
j>J−N0
2j(γ−
j′6j+1
2 cj′Eh,σ
and moreover it follows from (1.2), the definition of Eh,σ and Sobolev embedding that
j>J−N0
‖Sj+2vF△jI(u2)ut‖L1x
pEh,σ‖u‖2L∞
j>J−N0
2j(γ−3)‖ut‖L∞
r 2−2J [s−(
)]E2s ‖u‖2L∞
(H1),
(7.3)
for 4
6 r < ∞.
Let us estimate the second term in (7.2) by the precise Strichartz estimates. Since
this term contains that the negative derivative acts on the low frequency part Sj−1(u
it leads to our new parameter r < 2
by some technique difficulties. Noting that
Fourier-Plancherel formula and Hölder’s inequality, we can see that
j>J−N0
Sj−1I(u2)△jvFutdxdt .
j>J−N0
−16k6j−2
△kI(u2)△jvF△jutdxdt
△kI(u2)△k
k6j−2,J−N06j
△jvF△jut
. ‖u2‖
2k(γ−
k6j−2,J−N06j
△jvF△jut
(7.4)
On one hand, we have
J−N06j
△jvF△jut
‖L2dt .
j>J−N0
‖△jvF△jut‖L1
j>J−N0
‖△jvF ‖L∞
L2‖△jut‖L2
j>J−N0
2−sjcj c̃jEh,s‖ut‖L2t,x .
If (7.4) is controlled by the term at k = −1, we can see that
j>J−N0
‖Sj−1I(u2)△jvF ‖L2
L2 . T
2 2−2J(s−
)E2s ‖u‖L∞T (H1). (7.5)
On the other hand, one denotes gk := △k
k6j−2
△jvF△jut
to estimate
2k(γ−
)‖gk‖
Let us write that
k6j−2
ν∈Λj,k
△νj,kvF△jut.
As the support of the Fourier transform of a product is included in the sum of the
support of each Fourier transform, we obtain
k6j−2
ν∈Λj,k
△νj,kvF △̃νj,kut,
as well as in (7.1). Using Hölder inequality, we get
k6j−2
ν∈Λj,k
‖△νj,kvF ‖Lr‖△̃νj,kut‖L2
and the Cauchy-Schwarz inequality and the L2 quasi-orthogonality properties yield that
k6j−2
ν∈Λj,k
‖△νj,kvF‖2Lr
ν∈Λj,k
ut‖2L2
k6j−2
ν∈Λj,k
‖△νj,kvF‖2Lr
2 ‖△jut‖L2 . (7.6)
Precise Strichartz estimate implies that, for 1
with 2 6 r < ∞,
r+2 )
06k6j−2
2(k−j)(
ν∈Λj,k
‖△νj,kv0‖2L2
2 + 2−j
ν∈Λj,k
‖△νj,kv1‖2L2
‖△jut‖L2t,x
and observe the quasi-orthogonality properties again, this can be dominated by
06k6j−2
2(k−j)(
‖△jv0‖L2 + 2−j‖△jv1‖L2
‖△jut‖L2t,x .
Keeping the definitions of Eh,σ and cj in mind, one can see that
r+2 )
p 2k(
k6j−2
r cj c̃jEh,σE
T (u) . T
p 2k(
)Eh,σE
T (u).
Therefore, we get that
2k(γ−
)‖gk‖
r+2 )
2k(γ−3)Eh,σE
T (u)
which implies nothing but
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T 1−
p Eh,σE
T (u). (7.7)
Finally, we get that, for 4
6 r < ∞
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T
r 2−2J [s−(
)]E2sET (u).
However, although the r ranges 4
6 r < ∞, the above estimate still needs s > 3
to continue our proof. If we only consider the high frequency k > J , the (7.7) can be
modified by
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T 1−
p 2J(γ−3)Eh,σE
T (u) (7.8)
and then we can obtain a better result
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T
r 2−2J [s−(
)]E2sET (u),
which implies the bad influence comes from the low frequency part and this is consist of
the effect of negative derivative acts on the low frequency. But if we choose σ̃ = γ− 5
instead of σ, we can improve (7.8), at cost of restricting r such that max{2, 1
} < r <
while not 2 6 r < ∞. Now we turn to details. It follows from similar argument
r+2 )
p 2k(
k6j−2
2−j(γ−3+
)cj c̃jEh,σ̃E
T (u)
where σ̃ = γ − 5
with 1
< 3− γ < 2
. We get
2k(γ−
)‖gk‖
r+2 )
k6j−2
2(k−j)(γ−3+
)cj c̃jEh,σ̃E
T (u)
which implies nothing but
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T 1−
pEh,σ̃E
T (u)
by Young’s inequality. Note that σ̃ 6 γ
< s when r sufficiently closes to 2
, therefore
j>J−N0
Sj−1I(u2)△jvFutdxdt
∣∣∣ . T
r 2−2J(s−
)E2sET (u).
Combining this with (7.3) and (7.5), we complete the proof of (5.7) by obtaining
j>J−N0
Sj−1I(u2)△jvFutdxdt
−2J [s−(
−2J [s−(
+ T2−2J(s−
E2sET (u)
with max{2, 1
} < r1 < 23−γ and
6 r2 < ∞.
We secondly prove (5.8) which is different from (5.7). To this end, we need to make
Bony’s decomposition more than once and establish a commutator estimate, which
helps us to complete our proof. In view of the fact that v̂F only has high frequencies
again, it follows from Bony’s decomposition that there exists N0 such that
I(uvF )uut =
j>J−N0
Sj+2vF△ju
uut +
j>J−N0
△jvFSj−1u
== I + II. (7.9)
In order to estimate the term I, we split it into two pieces with N1 ≫ N0 > 0
j>J−N0
uut△kI
Sj+2vF△ju
j>J−N0
k6J−N1
uut△kI
Sj+2vF△ju
j>J−N0
k>J−N1
uut△kI
Sj+2vF△ju
== I1 + I2.
The estimate of I1 is broken down into the following two cases.
Case 1. 2 < γ 6 5
In this case, to our purpose, we obtain the following coarse estimate by Hölder’s
inequality
‖I1‖L1x .
j>J−N0
k6J−N1
Sj+2vF△ju
‖L3‖u‖L∞
L6‖ut‖L∞
j>J−N0
k6J−N1
2k(γ−2)‖△k
Sj+2vF△ju
ET (u)
j>J−N0
k6J−N1
2k(γ−2)‖Sj+2vF‖L6‖△ju‖L2ET (u)
k6J−N1
2k(γ−2)
j>J−N0
2−j2j‖△ju‖L2
‖△j′vF ‖L6ET (u).
Choosing (p, r) such that 1
with 2 6 r 6 6, the Strichartz estimate yeilds
‖I1‖L1
k6J−N1
2k(γ−2)
j>J−N0
j′( 3
‖△j′v0‖L2 + 2−j
′‖△j′v1‖L2
T (u).
Arguing similarly as before it yields that
‖I1‖L1
k6J−N1
2k(γ−2)
j>J−N0
r cj′Eh,1/2E
T (u)
r 2−2J [s−(
)]E2sET (u)
with 2 6 r 6 6. If choose r = 6, one can easily check that γ
2 < γ 6 5
. Although this result is enough for us to prove the main theorem, we want
to improve the result for this term by loosen the upper bound of r from 6 to ∞ through
the precise Strichartz estimate. Arguing similarly as before, we have
‖I1‖L1x .
j>J−N0
k6J−N1
Sj+2vF△ju
‖L3‖u‖L∞
L6‖ut‖L∞
j>J−N0
k6J−N1
2k(γ−3)2k
2r ‖△k
Sj+2vF△ju
ET (u)
Since the Fourier transform of Sj−1vF△ju was supported in 2jC and k ≪ j, △k(Sj−1vF△ju)
vanishes which implies △k
Sj+2vF△ju
△̃jvF△ju
. As the support of the
Fourier transform of a product is included in the sum of the support of each Fourier
transform, we also have
△̃jvF△ju
ν,ν′∈Λj,k
△νj,kvF△ν
ν∈Λj,k
△νj,kvF △̃νj,ku
Choosing (p, r) such that 1
for 2 6 r < ∞, it follows from the Hölder inequality
and L2 quasi-orthogonality properties that
Sj+2vF△ju
r+2 )
ν∈Λj,k
‖△νj,kvF‖Lr‖△̃νj,ku‖L2
ν∈Λj,k
‖△νj,kvF ‖2LpLr
ν∈Λj,k
‖△νj,ku‖2L2
ν∈Λj,k
‖△νj,kvF ‖2LpLr
∥∥△ju
Then the precise Strichartz estimate yields that
‖I1‖L1
k6J−N1
2k(γ−3)2k
j>J−N0
2−j2j
∥∥△ju
2(k−j)(
ν∈Λj,k
‖△νj,kv0‖2L2
2 + 2−j
ν∈Λj,k
‖△νj,kv1‖2L2
ET (u).
By the L2quasi-orthogonality properties, it gives that
‖I1‖L1
k6J−N1
2k(γ−3)2k
j>J−N0
2−j2j
∥∥△ju
‖△jv0‖L2 + 2−j‖△jv1‖L2
ET (u).
Utilizing the technique as before yields that
‖I1‖L1
k6J−N1
2k(γ−2+
j>J−N0
2−j(1+
)cjEh,σE
T (u)
r 2J(γ−3)Eh,σE
T (u) . T
r 2−2J [s−(
)]E2sET (u),
with 4
6 r < ∞.
Case 2. 5
< γ < 3
In the this case, the fact γ− 5
> 0 helps us to obtain the the desirable result easily.
Arguing similarly as before, we have
‖I1‖L1x .
j>J−N0
k6J−N1
Sj+2vF△ju
‖L3‖u‖L∞
L6‖ut‖L∞
j>J−N0
k6J−N1
2k(γ−3)23k(
Sj+2vF△ju
‖L2ET (u)
j>J−N0
k6J−N1
2k(γ−
)‖Sj+2vF‖L∞‖△ju‖L2ET (u).
Choosing (p, r) such that 1
with 2 6 r < ∞, the Strichartz estimate yields
‖I1‖L1
k6J−N1
2k(γ−
j>J−N0
2 cj′Eh,σE
T (u)
p 2J(γ−
j>J−N0
2 Eh,σE
T (u)
r 2−2J [s−(
)]E2sE
T (u).
Combining these two cases, we have shown that
‖I1‖L1
r 2−2J [s−(
)]E2sET (u) (7.10)
with 4
6 r < ∞. To control ‖I‖L1
, it remains to estimate ‖I2‖L1
. Compared
with ‖I1‖L1
, since the negative derivative acts on the high frequency, the upper
bound of ‖I2‖L1
is much easier to get. Here is the details:
‖I2‖L1x .
j>J−N0
k>J−N1
Sj+2vF△ju
‖L3‖u‖L∞
L6‖ut‖L∞
j>J−N0
k>J−N1
2k(γ−3)‖Sj+2vF ‖L∞‖△ju‖L3ET (u).
Choosing (p, r) such that 1
with 2 6 r < ∞ again, the Strichartz estimate
yields
‖I2‖L1
k>J−N1
2k(γ−3)
j>J−N0
2 cj′Eh,σE
T (u)
r 2−2J [s−(
)]E2sET (u).
Combining this with (7.10), we obtain that
‖I‖L1
r 2−2J [s−(
)]E2sET (u) (7.11)
for 4
6 r < ∞.
To complete the proof the Lemma 5.1, it remains to estimate II. One can proceed
this as above by Hölder’s inequality to estimate
j>J−N0
2j(γ−3)‖△jvFSj−1u‖L3
ET (u). (7.12)
Resorting to the Hölder inequality and the classical Strichartz estimate, one can obtain
‖II‖L1
L1 . T
r 2−2J [s−(
)]E2sET (u).
with 2 6 r 6 6. One also can try to improve the result by using the precise Strichartz
estimate as before, but it fails and merely obtain that
‖II‖L1
L1 . T
r 2−2J [s−(
)]E2sET (u).
with 2 6 r 6 4.
One can easily check that the result is worse than the desirable result because of the
restriction of r. Compared with the second term in (7.2), the negative derivative acts
on the high frequency part so that it is tempting to obtain a better result than that of
(7.2). But △jvF is bound with Sj−1u by the operator I, and this structure prevents us
from using efficiently the precise Strichartz estimate. If one first resort to the Hölder
inequality, as shown in (7.12), he or she merely obtains a loss result because of the
range restriction of r. To go around this difficulty, we first establish a commutator
estimate through exploiting cancellation property. Now we turn to details. Our task is
to estimate
j>J−N0
I(△jvFSj−1u)uutdxdt
In order to drag the Sj−1u out of the operator I, we construct uI(△jvF )Sj−1u and
the triangle inequality yields that
j>J−N0
I(△jvFSj−1u)uutdxdt
∣∣∣ 6
j>J−N0
∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u
j>J−N0
I(△jvF )Sj−1uuutdxdt
We benefit from the cancellation when we deal with the first term. Since both the
Fourier transformation of I(△jvFSj−1u) and I(△jvF )Sj−1u are supported in a ring
sized 2j , the Hölder inequality and the Bernstein inequality lead to that
∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u
2 ‖I(△jvFSj−1u)− I(△jvF )Sj−1u‖L2x‖u‖L6 .
Before estimating its right hand, we recall the Coifman and Meyer multiplier theorem.
Consider an infinitely differentiable symbol m : Rnk 7→ C so that for all α ∈ Nnk and
all ξ = (ξ1, ξ2, · · · , ξk) ∈ Rnk, there is a constant c(α) such that
|∂αξ m(ξ)| 6 c(α)(1 + |ξ|)−|α|. (7.13)
Define the multilinear operator T by
[T (f1, · · · , fk)](x) =
eix·(ξ1+···+ξk)m(ξ1, · · · , ξk)f̂1(ξ1), · · · , f̂k(ξk)dξ1 · · · dξk,
(7.14)
F [T (f1, · · · , fk)](ξ) =
ξ=ξ1+···+ξk
m(ξ1, · · · , ξk)f̂1(ξ1), · · · , f̂k(ξk)dξ1 · · · dξk−1. (7.15)
Proposition 7.1 ([8],Page 179.) Suppose pj ∈ (1,∞), j = 1, · · · k, are such that 1p =
+ · · ·+ 1
6 1. Assume m(ξ1, · · · , ξk) a smooth symbol as in (7.13). Then there
is a constant C = C(pi, n, k, c(α)) so that for all Schwarz class functions f1, · · · , fk,
‖[T (f1, · · · , fk)](x)‖Lp(Rn) 6 C‖f1‖Lp1 (Rn) · · · ‖fk‖Lpk (Rn). (7.16)
Since the operator I is a convolution operator with kernel |x|−γ in R3, we can write
F [I(△jvFSj−1u)−I(△jvF )Sj−1u](ξ) =
ξ=ξ1+ξ2
|ξ1+ξ2|γ−3−|ξ1|γ−3
△̂jvF (ξ1)Ŝj−1u(ξ2)dξ2.
By the mean value theorem, the right hand of the above formula becomes that
ξ=ξ1+ξ2
|ξ1 + λξ2|γ−4
(ξ1 + λξ2) · ξ2
|ξ1 + λξ2|
△̂jvF (ξ1)Ŝj−1u(ξ2)dξ2,
for a certain λ ∈ [0, 1]. Moreover, we rewrite it as follow:
ξ=ξ1+ξ2
m(ξ1, ξ2)f̂1(ξ1)f̂2(ξ2)dξ2,
m(ξ1, ξ2) = (ξ1 + λξ2)|ξ1 + λξ2|γ−5|ξ1|4−γ , f1 = |∇|γ−4△jvF , f2 = ∇Sj−1u.
Observe that |ξ1| > 2j−1 and 2j−2 > |ξ2|, we have that |ξ1+λξ2| ∼ |ξ1| > 2J−N0 . Hence,
we can check that the symbol m(ξ1, ξ2) satisfies the estimate (7.13). Finally, it follows
from Proposition 7.1 that
‖I(△jvFSj−1u)− I(△jvF )Sj−1u‖L2x . ‖f1‖Lrx‖f2‖
with 2 < r < ∞. After making use of the Bernstein inequality, the right hand can be
controlled by
2j(γ−4+
)‖△jvF ‖Lrx‖∇u‖L2x .
Keeping in mind j > J−N0 and recalling the definition of Eh,σ, the Strichartz estimate
and a direct calculation of summing in j show that
j>J−N0
2 2j(γ−4+
)‖△jvF ‖Lp
j>J−N0
2j(γ−3+
−s)Eh,s.
with 1
and 2 < r < ∞. Choosing r such that max{2, 1
} 6 r < ∞, we have
j>J−N0
∥∥(I(△jvFSj−1u)− I(△jvF )Sj−1u
r 2−2J [s−(
)]E2sET (u). (7.17)
Now the rest of the paper devotes to estimate this term
j>J−N0
I(△jvF )Sj−1uuutdxdt
In order to use precise Strichartz estimate, we need to decompose this term by Bony’s
para-product decomposition again,
I(△jvF )Sj−1uuut =
Sk−1(uSj−1u)△kI(△jvF )ut +△k(uSj−1u)Sk+2I(△jvF )
= II1 + II2.
After decomposing this, the term II1 is similar to the second term in the (7.2) and the
negative derivative acts on the high frequency △jvF leading to a better result than the
second term in the (7.2). Thanks to Fourier-Plancherel formula and Hölder inequality,
we obtain
j>J−N0
II1dxdt ≈
j>J−N0
Sk−1(uSj−1u)△kI(△jvF )△kutdxdt
j>J−N0
k′6k−2
△k′(uSj−1u)△kI(△jvF )△kutdxdt
j>J−N0
△k′(uSj−1u)△k′
k′6k−2
(△kI(△jvF )△kut)dxdt
j>J−N0
‖uSj−1u‖
2 ‖△k′
k′6k−2
(△kI(△jvF )△kut)‖L2
j>J−N0
‖u‖2L∞H1
2 ‖△k′
k′6k−2
(△kI(△jvF )△kut)‖L2
On the other hand, one denotes
gk′,j = △k′
k′6k−2
△kI(△jvF )△kut
to estimate ∑
′(− 1
)‖gk′,j‖
Let us write that
gk′,j =
k′6k−2
ν∈Λk,k′
△νk,k′I(△jvF )△kut
As the support of the Fourier transform of a product is included in the sum of the
support of each Fourier transform, we obtain
gk′,j =
k′6k−2
ν∈Λk,k′
△νk,k′I(△jvF )△̃νk,k′ut
Using Hölder inequality, we get
‖gk′,j‖
k′6k−2
ν∈Λk,k′
‖△νk,k′I(△jvF )‖Lr‖△̃νk,k′ut‖L2
6 2j(γ−3)
k′6k−2
ν∈Λk,k′
‖△νk,k′vF ‖2Lr
ν∈Λk,k′
‖△νk,k′ut‖2L2
6 2j(γ−3)
k′6k−2
ν∈Λk,k′
‖△νk,k′vF ‖2Lr
2‖△kut‖L2
the use of quasi-orthogonality properties is made in the last inequality.
Precise Strichartz estimate and the quasi-orthogonality properties imply that
‖gk′,j‖
r+2 )
p 2j(γ−3)
k′6k−2
′−k)( 1
ν∈Λk,k′
‖△νk,k′v0‖2L2
2 + 2−k
ν∈Λk,k′
‖△νk,k′v1‖2L2
‖△kut‖L2
p 2j(γ−3)
k′6k−2
′−k)( 1
‖△kv0‖L2 + 2−k‖△kv1‖L2
‖△kut‖L2
with 1
for 2 6 r < ∞. Therefore
′(− 1
)‖gk′,j‖
r+2 )
p 2j(γ−3)
k′6k−2
′−k) 2
r ck c̃kEh,σE
T (u).
A direct computation shows that
2 ‖gk′,j‖L1
′(− 1
)‖gk′,j‖
r+2 )
p 2j(γ−3)Eh,σE
T (u).
Hence, we have that
j>J−N0
II1dxdt
∣∣∣ . 2−2J [s−(
r E2sET (u) (7.18)
with 4
6 r < ∞. Finally, we conclude this section by giving the estimate of II2.
j>J−N0
II2dxdt
∣∣∣ . T
j>J−N0
‖△k(uSj−1u)Sk+1I(△jvF )‖L2
L2‖ut‖L∞T L2
j>J−N0
2j(γ−3)
‖△k(uSj−1u)‖L∞
r ‖△k′△jvF‖Lp
T (u)
j>J−N0
2j(γ−3)
‖△k(uSj−1u)‖L∞
2 ck′Eh,σE
T (u)
j>J−N0
2j(γ−3)
2 ‖△k(uSj−1u)‖L∞
(k′−k) 1
2Eh,σE
T (u)
j>J−N0
2j(γ−3)‖ck′‖ℓ2(Z)‖2
2 ‖△k(uSj−1u)‖L∞
L2‖ℓ2(Z)‖2−
2 ‖ℓ2(N)Eh,σE
T (u)
p 2J(γ−3)E
T (u)Eh,σ . 2
−2J [s−(
r E2sET (u).
(7.19)
Collecting (7.18) and (7.19), we have been proved that
j>J−N0
I(△jvF )Sj−1uuutdxdt
∣∣∣ . T
r 2−2J [s−(
)]E2sET (u), (7.20)
with 4
6 r < ∞. Finally, we complete the proof of (5.8) by (7.11) and (7.20), hence
it ends the proof of Lemma 5.1.
Acknowledgements: The authors are grateful to Prof. J.Chemin for sending his
lecture to us. The authors were partly supported by the NSF of China, No.10725102.
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http://arxiv.org/abs/math/0612028
	Introduction
	Preliminaries
	Global existence for the high frequency part
	Local existence for the low frequency part
	Energy estimate for the low frequency part
	Proof of Theorem ??
	Proof of Lemma ??
	References
