Global well-posedness and scattering for the energy-critical, defocusing
  Hartree equation for radial data
Global well-posedness and scattering for the energy-critical,
defocusing Hartree equation for radial data
Changxing Miao, Guixiang Xu, and Lifeng Zhao
Institute of Applied Physics and Computational Mathematics
P. O. Box 8009, Beijing, China, 100088
(miao changxing@iapcm.ac.cn, xu guixiang@iapcm.ac.cn, zhao lifeng@iapcm.ac.cn )
Abstract
We consider the defocusing, Ḣ1-critical Hartree equation for the radial data in
all dimensions (n ≥ 5). We show the global well-posedness and scattering results in
the energy space. The new ingredient in this paper is that we first take advantage
of the term −
|x|≤A|I|1/2
|u|2∆
dxdt in the localized Morawetz identity to
rule out the possibility of energy concentration, instead of the classical Morawetz
estimate dependent of the nonlinearity.
Key Words: Hartree equation, Global well-posedness, Scattering, Morawetz
estimate.
AMS Classification: 35Q40, 35Q55, 47J35.
1 Introduction
In this paper, we study the Cauchy problem for the Hartree equation
iut +∆u = f(u), in R
n × R, n ≥ 5,
u(0) = ϕ(x), in Rn.
(1.1)
Here f(u) =
V ∗ |u|2
u is a nonlinear function of Hartree type for V (x) = |x|−γ , 0 <
γ < n, where ∗ denotes the convolution in Rn. In practice, we use the integral formula
of (1.1)
u(t) = U(t)ϕ− i
U(t− s)f(u(s))ds, (1.2)
where U(t) = eit∆.
http://arxiv.org/abs/0704.0665v2
If the solution u of (1.1) has sufficient smoothness and decay at infinity, it satisfies
two conservation laws :
M(u(t)) =
∥u(t)
E(u(t)) =
∥∇u(t)
|x− y|γ
|u(t, x)|2|u(t, y)|2 dxdy = E(ϕ).
(1.3)
As explained in [6], the energy is also conserved for the energy solutions u ∈ C0t (R,H
From the viewpoint of the fractional integral, we rewrite the equation (1.1) as
iut +∆u =
(−∆)−
2 |u|2
For dimension n ≥ 5, the exponent γ = 4 is the unique exponent which is energy critical
in the sense that the natural scale transformation
uλ(t, x) = λ
2 u(λ2t, λx),
leaves the energy invariant, in other words, the energy E(u) is a dimensionless quantity.
The Cauchy problem of the Hartree equation has been intensively studied ([4-10],
[15, 16, 18, 19]. With regard to the global well-posedness and scattering results, they
all dealt with the Ḣ1-subcritical case
2 < γ < min(4, n)
in the energy space or some
weighted spaces. In [16], we obtained the small data scattering result for the Ḣ1-critical
case in the energy space. For the large initial data for the Ḣ1-critical case
γ = 4, n ≥ 5
in the energy space , the argument in [16] can not yield the global well-posedness, even
with the conservation of the energy (1.3), because the time of existence given by the
local theory depends on the profile of the data as well as on the energy.
Concerning the Ḣ1-subcritical case
2 < γ < min(4, n)
, using the method of
Morawetz and Strauss [17], J. Ginibre and G. Velo [6] developed the scattering the-
ory in the energy space, where they exploited the properties of ∆ and obtain the usual
Morawetz estimate
∣u(t, x)
|x− y|γ
∇|u(t, y)|2dydxdt . CE(u).
Later, K. Nakanishi [18] exploited the properties of i∂t + ∆ and used a certain related
Sobolev-type inequality to obtain a new Morawetz estimate
|t|1+ν |u(t, x)|
(|t|+ |x|)2+ν
dxdt ≤ C(E, ν), for any ν > 0,
which was independent of the nonlinearity.
In this paper, we deal with the Cauchy problem of the Hartree equation with the large
data for the Ḣ1-critical case
γ = 4, n ≥ 5
. Inspired by the approach of Bourgain [1]
and Tao [22] in the case of the Ḣ1-critical Schrödinger equation with the local nonlinear
term, we obtain the global well-posedness and scattering results for the Hartree equation
for the large radial data in Ḣ1. The new ingredient is that we take advantage of the
following localized estimate for the first time
|x|≤A|I|1/2
|u(t, x)|2∆
dxdt = (n− 3)
|x|≤A|I|1/2
|u(t, x)|2
dxdt ≤ A|I|1/2C(E)
to rule out the possibility of energy concentration, instead of the classical Morawetz
estimate
∇V (x− y)|u(x)|2|u(y)|2dydxdt . C(E)
due to the nonlinear term .
Our main result is the following global well-posedness result in the energy space.
Theorem 1.1. Let n ≥ 5, and ϕ ∈ Ḣ1 be radial. then there exists a unique global
solution u ∈ C0t (Ḣ
(iut +∆u)(t, x) =
uV ∗ |u|2
(t, x), in Rn × R,
u(0) = ϕ(x), in Rn.
(1.4)
where V (x) = |x|−4 and on each compact time interval [t−, t+], we have
x ([t−,t+]×Rn)
). (1.5)
As the right hand side of (1.5) is independent of t−, t+, we can obtain the global
spacetime estimate. As a direct consequence of the global L6tL
x estimate, we have
scattering, asymptotic completeness, and uniform regularity.
Corollary 1.1. Let ϕ be radial and have finite energy. Then there exists finite energy
solutions u±(t, x) to the free Schrödinger equation iut +∆u = 0 such that
∥u±(t)− u(t)
→ 0 as t → ±∞.
Furthermore, the maps ϕ 7→ u±(0) are homeomorphisms from Ḣ
1(Rn) to Ḣ1(Rn). Fi-
nally, if ϕ ∈ Hs for some s > 1, then u(t) ∈ Hs for all time t, and one has the uniform
bounds
∥u(t)
≤ C(E(ϕ), s)
The paper is organized as follows.
In Section 2, we introduce notations and the basic estimates; In Section 3, we derive
the local mass conservation and Morawetz inequality; In Section 4, we discuss the local
theory for (1.4); In Section 5, we obtain the perturbation theory; Finally, we prove the
main theorem in Section 6.
2 Notations and basic estimates
We will often use the notations a . b and a = O(b) to denote the estimate a ≤ Cb for
some C. The derivative operator ∇ refers to the space variable only. We also occasionally
use subscripts to denote the spatial derivatives and use the summation convention over
repeated indices.
We define 〈a, b〉 = Re(ab), ∂ = (∂t,∇), D = (−
,∇); For 1 ≤ p ≤ ∞, we denote by
p′ the dual exponent, that is, 1
For any time interval I, we use L
x(I×R
n) to denote the mixed spacetime Lebesgue
x(I×R
Lr(Rn)
with the usual modifications when q = ∞. When q = r, we abbreviate L
x by L
We use U(t) = eit∆ to denote the free group generated by the free Schrödinger
equation iut +∆u = 0. It can commute with derivatives, and obeys the inequality
∥eit∆f
Lp(Rn)
. |t|
−n( 1
(2.1)
for t 6= 0, 2 ≤ p ≤ ∞.
We say that a pair (q, r) is admissible if
2 ≤ r
≤ ∞, n = 1;
< ∞, n = 2;
, n ≥ 3.
For a spacetime slab I ×Rn, we define the Strichartz norm Ṡ0(I) by
Ṡ0(I)
:= sup
(q,r) admissible
x(I×R
and define Ṡ1(I) by
Ṡ1(I)
Ṡ0(I)
When n ≥ 3, the spaces
Ṡ0(I), ‖ · ‖
Ṡ0(I)
Ṡ1(I), ‖ · ‖
Ṡ1(I)
are Banach spaces,
respectively.
Based on the above notations, we have the following Strichartz inequalities
Lemma 2.1. [11], [21] Let u be an Ṡ0 solution to the Schrödinger equation (1.1). Then
∥u(t0)
∥f(u)
x (I×R
for any t0 ∈ I and any admissible pairs (q, r). The implicit constant is independent of
the choice of interval I.
From Sobolev embedding, we have
Lemma 2.2. For any function u on I × Rn, we have
L∞t L
L∞t L
where all spacetime norms are on I × Rn.
For convenience, we introduce two abbreviated notations. For a time interval I, we
denote
x (I×R
W (I)
x (I×R
Lemma 2.3. Let f(u)(t, x) =
uV ∗ |u|2
(t, x), where V (x) = |x|−4. For any time
interval I and t0 ∈ I, we have
ei(t−s)∆f(u)(s, x)ds
Ṡ1(I)
W (I)
Proof: By Strichartz estimates, Hardy-Littlewood-Sobolev inequality and Hölder
inequality, we have
ei(t−s)∆f(u)(s, x)ds
Ṡ1(I)
. ‖∇f(u)(t, x)‖
x (I×R
. ‖∇uV ∗ |u|2‖
x (I×Rn)
+ ‖uV ∗ (u∇u)‖
x (I×Rn)
. ‖∇u‖
x (I×Rn)
‖V ∗ |u|2‖
x (I×Rn)
+ ‖u‖
x (I×Rn)
‖V ∗ (u∇u)‖
x (I×R
W (I)
3 Local mass conservation and Morawetz inequality
In this section, we will prove two useful estimates. One is a local mass conservation
estimate and the other is a Morawetz inequality, which appears in Morawetz identity.
The local mass conservation estimate is used to control the flow of mass through a region
of space, and the Morawetz inequality is used to prevent concentration.
3.1 Local mass conservation
We recall a local mass conservation law that has appeared in [1], [13] and [22]. For
completeness, we give the sketch of the proof. Let χ be a bump function supported on
the ball B(0, 1) that equals 1 on the ball B(0, 1/2). Observe that if u is a finite energy
solution of (1.4), then
∣u(t, x)
= −2∇ · Im(u∇u(t, x)).
We define
Mass(u(t), B(x0, R)) :=
(x− x0
u(t, x)
Differentiating the above quantity with respect to time, we obtain by the integration by
parts
∂tMass(u(t), B(x0, R)) =
(x− x0
∣u(t, x)
(x− x0
∇ · Im(u∇u)dx
(x− x0
(x− x0
Im(u∇u)dx
∥∇u(t)
Mass(u(t), B(x0, R))
hence, we have
Mass(u(t1), B(x0, R))
1/2 −Mass(u(t2), B(x0, R))
∣t1 − t2
∣. (3.1)
This implies that if the local mass Mass(u(t), B(x0, R)) is large for some time t, then it
can also be shown to be similarly large for nearly time t, by increasing the radius R if
necessary to reduce the rate of change of the mass.
On the other hand, from Sobolev and Hölder inequalities, we have
Mass(u(t), B(x0, R)) ≤
(x− x0
. (3.2)
This gives the control of mass in small volumes.
3.2 A Morawetz inequality
To prevent the concentration of the energy, we need a Morawetz estimate. The Morawetz
estimate is based on some integral identity derived by variation of the lagrangian.
We define ℓ(u) by
2ℓ(u) = 〈iut, u〉+ |∇u|
|u|2(V ∗ |u|2)
ℓ(u) is the lagrangian density associated to the equation (1.1).
From the definition of the variation of the functional ℓ, we have
δvℓ(u) : = lim
ℓ(u+ ǫv)− ℓ(u)
= 〈iut +∆u− u(V ∗ |u|
2), v〉 + ∂ · 〈Du, v〉.
Using this identity together with h =
, q =
Re(D·h) =
and Mu = h ·Du+qu,
we obtain the following formula:
〈iut +∆u− u(V ∗ |u|
2),Mu〉 =− ∂ · 〈Du,Mu〉 +D ·
hℓ(u) +
〈Du, ∂hαDαu〉 −
D · ∂q − (V ∗ ∇|u|2) ·
|u|2.
As a consequence of the above dilation identity, we have the following Morawetz
estimate, which plays an important role in our proof.
Proposition 3.1 (Morawetz estimate). Let u be a solution to (1.4) on a spacetime slab
I × Rn. Then for any A ≥ 1, we have
|x|≤A|I|1/2
dxdt−
∇V (x− y)|u(x)|2|u(y)|2dydxdt
. A|I|1/2E,
where Ω =
(x, y) ∈ Rn ×Rn; |x| ≤ A|I|1/2; |y| ≤ A|I|1/2
Remark 3.1. Since
∇V (x− y) = 4
|x||y| − x · y
|x− y|6
we have
∇V (x− y)|u(x)|2|u(y)|2dydxdt ≥ 0.
Proof: We define V a0 (t) =
a(x)|u(t, x)|2dx, then
Ma0 (t) =: ∂tV
0 (t) = 2Im
ajujudx.
0 (t) = −2Im
ajjutudx− 4Im
ajujutdx
△△a|u|2dx+ 4Re
ajkujukdx
− 2Re
∇a(x)∇V (x− y)|u(y)|2|u(x)|2dxdy
△△a|u|2dx+ 4Re
ajkujukdx
∇a(x)−∇a(y)
∇V (x− y)|u(y)|2|u(x)|2dxdy
where we use the symmetry of a(x) and V (x). Let R > 0 and let η be a bump function
adapted to the ball |x| ≤ R which equals 1 on the ball |x| ≤ R/2. We set a(x) := |x|η(x).
For |x| ≤ R/2, we have
; ajk =
; △a =
; −△△a =
(n− 1)(n − 3)
and for R/2 ≤ |x| ≤ R, we have bounds
aj = O(1); ajk = O(R
−1); △△a = O(R−3).
Thus we have
0 (t) = (n− 1)(n − 3)
|x|≤R/2
dx+ 4
|x|≤R/2
|∇u|2 − |∂ru|
∇V (x− y)|u(x)|2|u(y)|2dydx
|x|∼R
( |u|2
|∇u|2
aj(x)− aj(y)
) xj − yj
|x− y|γ+2
|u(x)|2|u(y)|2dydx
where γ = 4,
(x, y) ∈ Rn × Rn; |x| ≤ R/2, |y| ≤ R/2
(x, y) ∈ Rn × Rn; |x| ∼ R
(x, y) ∈ Rn × Rn; |y| ∼ R
Meanwhile
|x|∼R
( |u|2
|∇u|2
dx . R−1E,
aj(x)− aj(y)
) xj − yj
|x− y|γ+2
|u(x)|2|u(y)|2dydx
Ω2: |x−y|≤R/4
aj(x)− aj(y)
) xj − yj
|x− y|γ+2
|u(x)|2|u(y)|2dydx
Ω2: |x−y|≥R/4
aj(x)− aj(y)
) xj − yj
|x− y|γ+2
|u(x)|2|u(y)|2dydx
. R−1
|x− y|γ
|u(x)|2|u(y)|2dydx
. R−1E.
Moreover, from Sobolev and Hölder inequalities, we have
Ma0 (t) .
|x|.R
|u||∇u| . ‖u‖
‖∇u‖L2x
|x|.R
. RE.
So if we integrate by parts on a time interval I and take R = 2A|I|1/2, we obtain
|x|≤A|I|1/2
dxdt−
∇V (x− y)|u(x)|2|u(y)|2dydxdt
. A|I|1/2E
for n ≥ 4. The proof is completed.
4 Local theory
In this section, we develop a local well-posedness and blow-up criterion for the Ḣ1-critical
Hartree equation. First, we have
Proposition 4.1 (Local well-posedness). Let u(t0) ∈ Ḣ
1, and I be a compact time
interval that contains t0 such that
∥U(t− t0)u(t0)
for a sufficiently small absolute constant η > 0. Then there exists a unique strong solution
to (1.4) on I × Rn such that
∥u(t0)
Proof: The proof of this proposition is standard and based on the contraction
mapping arguments. We define the solution map to be
Φ(u)(t) := U(t− t0)u(t0)− i
U(t− s)f(u(s))ds,
then Φ is a map from
B = {u : ‖u‖X(I) ≤ 2η, ‖u‖W (I) ≤ 2C‖u(t0)‖Ḣ1}
with the metric
‖u‖B = ‖u‖X(I) + ‖u‖W (I)
onto itself because
‖Φ(u)‖X(I) ≤ ‖U(t− t0)u(t0)‖X(I) + C‖u‖
X(I)‖u‖W (I) ≤ η + 8Cη
2‖u(t0)‖Ḣ1 ≤ 2η;
‖Φ(u)‖W (I) ≤ C‖u(t0)‖Ḣ1+C‖u‖
X(I)‖u‖W (I) ≤ C‖u(t0)‖Ḣ1+8Cη
2‖u(t0)‖Ḣ1 ≤ 2C‖u(t0)‖Ḣ1 .
It suffices to prove Φ is a contraction map. Let u, v ∈ B, then
‖Φ(u)− Φ(v)‖W (I) ≤
U(t− s)(V ∗ (ū− v̄)u)u(s, x)ds
W (I)
U(t− s)(V ∗ v̄(u− v))u(s, x)ds
W (I)
U(t− s)(V ∗ v̄v)(u− v)(s, x)ds
W (I)
By Lemma 2.3, we have
‖Φ(u)− Φ(v)‖W (I) ≤ ‖u− v‖X(I)
‖u‖W (I)‖u‖X(I) + ‖u‖W (I)‖v‖X(I) + ‖v‖W (I)‖v‖X(I)
+ ‖u− v‖W (I)
‖u‖X(I)‖u‖X(I) + ‖u‖X(I)‖v‖X(I) + ‖v‖X(I)‖v‖X(I)
≤ 12Cη‖u(t0)‖Ḣ1‖u− v‖X(I) + 12η
2‖u− v‖W (I)
(‖u− v‖X(I) + ‖u− v‖W (I))
In the same way, we have
‖Φ(u)− Φ(v)‖X(I) ≤ 12Cη‖u(t0)‖Ḣ1‖u− v‖X(I) + 12η
2‖u− v‖W (I)
(‖u− v‖X(I) + ‖u− v‖W (I))
as long as η is chosen sufficiently small. Then the contraction mapping theorem implies
the existence of the unique solution to (1.4) on I.
Next, we give the blow-up criterion of the solutions for (1.4). The usual form is
similar to those in [2], [12], which is in the form of a maximal interval of existence. For
convenience, we obtain
Proposition 4.2 (Blow-up criterion). Let ϕ ∈ Ḣ1, and let u be a strong solution to
(1.4) on the slab [0, T ) × Rn such that
X([0,T ))
Then there exists δ > 0 such that the solution u extends to a strong solution to (1.4) on
the slab [0, T + δ] × Rn.
Proof: By the absolute continuity of integrals, there exists a t0 ∈ [0, T ), such that
‖u‖X([t0,T )) ≤ η/4,
then by Lemma 2.3, we have
‖u‖W ([t0,T )) . ‖u(t0)‖Ḣ1 + ‖u‖
X([t0,T ))
‖u‖W ([t0,T )),
therefore
‖u‖W ([t0,T )) . ‖u(t0)‖Ḣ1 .
Now we write
U(t− t0)u(t0) = u(t) + i
U(t− s)(V ∗ |u|2)u(s, x)ds,
‖U(t−t0)u(t0)‖X([t0,T )) ≤ ‖u‖X([t0,T ))+C‖u‖
X([t0,T ))
‖u‖W ([t0,T )) ≤
+Cη2‖u(t0)‖Ḣ1 ≤
By the absolute continuity of integrals again, there exists a δ, such that
‖U(t− t0)u(t0)‖X([t0,T+δ)) ≤ η.
Thus we may apply Proposition 4.1 on the interval [t0, T + δ] to complete the proof.
In other words, this lemma asserts that if [t0, T
∗) is the maximal interval of existence
and T ∗ < ∞, then
‖u‖X([t0,T ∗)) = ∞.
5 Perturbation result
In this section, we obtain the perturbation for Hartree equation, which shows that the
solution can not be large if the linear part of the solution is not large. This is an
analogue of Lemma 3.2 in [22], and later, Killip, Visan and Zhang [13] gave the similar
perturbation result for the Schrödinger equation with the quadric potentials.
Lemma 5.1 (Perturbation lemma). Let u be a solution to (1.4) on I = [t1, t2] such that
where η is sufficiently small constant depending on the norm of the initial data, then
Ṡ1(I)
where uk(t) = U(t− tk)u(tk) for k = 1, 2.
Proof: From Strichartz estimate and Lemma 2.3, we obtain
Ṡ1(I)
∥u(t1)
W (I)
∥u(t1)
Ṡ1(I)
∥u(t1)
Ṡ1(I)
If η is sufficiently small, we have the first claim
Ṡ1(I)
As for the second claim, we give the proof for k = 1, the case k = 2 is similar. Using
Strichartz estimate and Lemma 2.3 again, we have
∥u− u1
Ṡ1(I)
. η2,
therefore, the second claim follows by the triangle inequality and choosing η sufficiently
small.
6 Global well-posedness
In this section, we give the proof of Theorem 1.1. The new ingredient is that we first
take advantage of the the estimate of the term
|x|≤A|I|1/2
dxdt in the localized
Morawetz identity to rule out the possibility of energy concentration, which is indepen-
dent of the nonlinear term. For the Schrödinger equation, Tao [22] used the classical
Morawetz estimate, which depends on the nonlinearity, to prevent the concentration.
For readability, we first take some constants
C1 = 6n; C2 = 3; C3 = 18n. (6.1)
which come from several constraints in the rest of this section. All implicit constants in
this section are permitted to depend on the dimension n and the energy.
Fix E, [t−, t+], u. We may assume that the energy is large, E > c > 0, otherwise the
claim follows from the small energy theory [16]. From the boundedness of energy and
Sobolev embedding, we can obtain
∥u(t)
∥u(t)
. 1 (6.2)
for all t ∈ [t−, t+].
Assume that the solution u already exists on [t−, t+]. By Lemma 4.2, it suffices to
obtain a priori estimate
X([t−,t+])
≤ O(1), (6.3)
where O(1) is independent of t−, t+.
We may assume that
X([t−,t+])
≥ 2η,
otherwise it is trivial. We divide [t−, t+] into J subintervals Ij = [tj , tj+1] for some J ≥ 2
such that
X(Ij)
≤ η, (6.4)
where η is a small constant depending on the dimension n and the energy. As a conse-
quence, it suffices to estimate the number J .
Now let u± = U(t − t±)u(t±). By Sobolev embedding and Strichartz estimates, we
X([t−,t+])
. 1. (6.5)
We adapt the following definition of Tao [22].
Definition 6.1. We call Ij exceptional if
X(Ij)
> ηC3
for at least one sign ±. Otherwise, we call Ij unexceptional.
From (6.5), we obtain the upper bound on the number of exceptional intervals,
O(η−6C3). We may assume that there exist unexceptional intervals, otherwise the claim
would follow from this bound and (6.4). Therefore, it suffices to compute the number of
unexceptional intervals.
We first prove the existence of a bubble of mass concentration in each unexceptional
interval.
Proposition 6.1 (Existence of a bubble). Let Ij be an unexceptional interval. Then
there exists xj ∈ R
n such that
Mass(u(t), B(xj , η
−C1 |Ij |
1/2)) & ηC1 |Ij|
for all t ∈ Ij .
Proof: By time translation invariance and scale invariance, we may assume that
Ij = [0, 1]. We subdivide Ij further into [0,
] and [1
, 1]. By (6.4) and the pigeonhole
principle and time reflection symmetry if necessary, we may assume that
X([ 1
Thus by Lemma 5.1, we have
X([ 1
. (6.6)
By Duhamel formula, we have
) = U(t− t−)u(t−)− i
U(t− s)f(u(s))ds
U(t− s)f(u(s))ds.
(6.7)
Since [0, 1] is unexceptional interval, we have
∥U(t− t−)u(t−)
X([ 1
∥u−(t)
X([ 1
≤ ηC3 .
On the other hand, by (6.4), Lemma 2.2 , Lemma 2.3 and Lemma 5.1, we have
U(t− s)f(u(s))ds
X([ 1
X([ 1
W ([ 1
Ṡ1([ 1
. η2.
Thus the triangle inequality implies that
U(t− s)f(u(s))ds
X([ 1
provided η is chosen sufficiently small. Hence, if we define
v(t) :=
U(t− s)f(u(s))ds,
then we have
X([ 1
η. (6.8)
Next, we estimate the upper bound on v. We have by (6.7) and the triangle inequality
Ṡ1([ 1
Ṡ1([ 1
∥U(t− t−)u(t−)
Ṡ1([ 1
U(t− s)f(u(s))ds
Ṡ1([ 1
X([0, 1
W ([0, 1
X([0, 1
Ṡ1([0, 1
(6.9)
where we use Strichartz estimate, (6.4) and Lemma 5.1.
We shall need some additional regularity control on v. For any h ∈ Rn, let u(h)
denote the translation of u by h, i.e. u(h)(t, x) = u(t, x− h).
Lemma 6.1. Let χ be a bump function supported on the ball B(0, 1) of total mass one,
and define
vav(t, x) =
χ(y)v(t, x+ ηC2y)dy,
then we have
∥v − vav
X([ 1
. ηC2 .
Proof: By the chain rule, Hölder inequality and Sobolev embedding, we have
∥∇f(u)(s)
∥(V ∗ |u|2)∇u
∥u(V ∗ ∇|u|2)
∥V ∗ |u|2
∥V ∗ ∇|u|2
∥|u|2
∥∇|u|2
it follows by (2.1)
L∞t L
,1]×Rn)
≤ sup
t∈[ 1
|t− s|2
∥∇f(u)(s)
ds . 1.
From (6.9) and interpolation, we have
L∞t L
,1]×Rn)
L∞t L
,1]×Rn)
L∞t L
,1]×Rn)
From the fundamental theorem of calculus, we have
∥v − v(h)
L∞t L
,1]×Rn)
. |h|.
This implies
∥v − vav
L∞t L
,1]×Rn)
∥v(t, x+ ηC2y)− v(x)
L∞t L
,1]×Rn)
χ(y)|ηC2y|dy
. ηC2 .
Hence from Hölder inequality, we obtain
∥v − vav
X([ 1
∥v − vav
L∞t L
,1]×Rn)
. ηC2 .
This completes the proof of Lemma.
Now we return to the proof of Proposition 6.1. By Lemma 6.1 and (6.8), we have
X([ 1
& η. (6.10)
On the other hand, by Hölder inequality, Young inequalities and (6.9), we have
2(3n−8)
,1]×Rn)
L∞t L
,1]×Rn)
L∞t L
,1]×Rn)
Interpolating with (6.10) gives
L∞t,x([
,1]×Rn))
X([ 1
− 3n−8
2(3n−8)
,1]×Rn)
Thus there exists (sj, xj) ∈ [
, 1] × Rn such that
∣vav(sj, xj)
∣ & η
Hence, by Cauchy-Schwarz inequality, we have
∣vav(sj , xj)
χ(y)v(sj , xj + η
C2y)dy
= η−nC2
x− xj
)v(sj , x)dx
. η−nC2η
C2Mass(v(sj), B(xj , η
C2))1/2,
that is
Mass(v(sj), B(xj , η
C2)) & η3n−6+nC2 & ηC1 . (6.11)
Observe that (3.1) also holds for v. If we take R = η−C1 and choose η sufficiently
small, we have
Mass(v(t), B(xj , η
−C1)) &
Mass(v(sj), B(xj , η
−C1))1/2 −
& (Mass(v(sj), B(xj , η
C2))1/2 − ηC1)2
& ηC1
(6.12)
for all t ∈ [0, 1].
The last step is to show that this mass concentration holds for u. We first show mass
concentration for u at time 0.
Since [0, 1] is unexceptional interval, by the pigeonhole principle, there is a τj ∈ [0, 1]
such that
∥u−(τj)
. ηC3 ,
and so by Hölder inequality,
Mass(u−(τj), B(xj , η
−C1)) .
(x− xj
∥u−(τj)
C1+2C3 . η2C1 .
From (3.1), we have
Mass(u−(0), B(xj , η
−C1)) . η2C1 . (6.13)
Recall that u(0) = u−(0) − iv(0). Combing (6.12) and (6.13) with the triangle
inequality, we obtain
Mass(u(0), B(xj , η
−C1)) & ηC1 . (6.14)
Using (3.1) again, we obtain the result.
Next, we use the radial assumption to show that the bubble of mass concentration
must occur at the spatial origin. In the forthcoming paper, we shall use the interaction
Morawetz estimate with the frequency localized L2 almost-conservation law to rule out
the possibility of the energy concentration at any place and deal with the non-radial
data. The corresponding results for the Schrödinger equation with local nonlinearity,
please see [3], [20] and [23].
Corollary 6.1 (Bubble at the origin). Let Ij be an unexceptional interval. Then
Mass(u(t), B(0, η−3C1 |Ij|
1/2)) & ηC1 |Ij |
for all t ∈ Ij .
Proof: If xj in Proposition 6.1 is within
η−3C1 |Ij |
1/2 of the origin, then the result
follows immediately. Otherwise by the radial assumption, there would be at least
((η−3C1 |Ij |
1/2)n−1
(η−C1 |Ij|1/2)n−1
η−2(n−1)C1
many distinct balls each containing at least ηC1 |Ij | amount of mass. By Hölder inequality,
this implies
η−2(n−1)C1 × ηC1 |Ij | .
(η−3C1−η−C1 )|Ij |
1/2≤|x|≤(η−3C1+η−C1 )|Ij |
|u(t, x)|2dx
(η−3C1−η−C1 )|Ij |
1/2≤|x|≤(η−3C1+η−C1 )|Ij |
η−3C1 |Ij |
× η−C1 |Ij |
that is
2n2−9n+4
Because 2n2 − 9n+ 4 > 0 for n ≥ 5, this contradicts the boundedness on the energy
of (6.2). This completes the proof.
Next, we use Proposition 3.1 to show that if there are many unexceptional intervals,
they must form a cascade and must concentrate at some time t∗.
Corollary 6.2. Assume that the solution u is spherically symmetric. For any interval
I ⊆ [t−, t+] and I be a union of consecutive unexceptional intervals Ij. Then
. η−13C1
and moreover, there exists a j such that
∣ & η26C1
Proof: For any unexceptional interval Ij , from Hölder inequality and Corollary 6.1,
we have
∣ . Mass
u(t), B(0, η−3C1 |Ij |
η−3C1 |Ij |1/2
2η−3C1 |Ij |1/2
2 |u(t, x)|2
η−3C1 |Ij|
|x|≤2η−3C1 |Ij |
|u(t, x)|2
therefore
|x|≤2η−3C1 |Ij|
|u(t, x)|2
dx & η10C1
We integrate this over each unexceptional interval Ij and sum over j,
η10C1
|x|≤2η−3C1 |Ij|
|u(t, x)|2
|x|≤2η−3C1 |I|1/2
|u(t, x)|2
|x|≤2η−3C1 |I|1/2
|u(t, x)|2
. η−3C1 |I|1/2.
The second claim follows from the first and the fact that
)−1/2
This completes the proof.
Proposition 6.2 (Interval cascade). Let I be an interval tiled by finitely many intervals
I1, · · · , IN . Suppose that for any continuous family
Ij : j ∈ J
of the unexceptional
intervals, there exists j∗ ∈ J such that
∣ ≥ a
∣ (6.15)
for some small a > 0. Then there exist K ≥ log(N)/ log(2a−1) distinct indices j1, · · · , jK
such that
∣ ≥ 2
∣ ≥ · · · ≥ 2K−1
and for any t∗ ∈ IjK ,
dist(Ijk , t∗) .
hold for 1 ≤ k ≤ K.
I(1) 
I(2) 
I(k) 
I(K−1) 
I(K) 
(t) t+ 
exceptional interval 
at most O(η−6C3)
exceptional interval 
at most 2a−1−1 
Figure 1: Iteration process in Proposition 6.2.
Proof: Here we use an algorithm in [1] and [22] to assign a generation to each Ij .
By hypothesis, I contains at least one interval of length a|I|. All intervals with length
larger than a|I|/2 belong to the first generation. By the total measure, we see that there
are at most 2a−1 − 1 intervals in the first generation. Removing there intervals from I
leaves at most 2a−1 gaps, which are tiled by intervals Ij .
By (6.15) and the contradiction argument, we know that there is not gap with length
larger than |I|/2.
We now apply this argument recursively to all gaps generated by the previous itera-
tion until every Ij has been labeled with a generation number.
Each iteration of the algorithm removes at most 2a−1−1 many intervals and produces
at most 2a−1 gaps. Suppose that there areN consecutive unexceptional intervals initially,
and we perform at most K times iterations. Then the number K obeys
N ≤ (2a−1 − 1) + (2a−1 − 1)2a−1 + · · ·+ (2a−1 − 1)(2a−1)K−1
≤ (2a−1)K ,
which leads to the claim K ≥ log(N)/ log(2a−1).
Let I(K) be the interval obtained after K − 1 iterations and IjK be any interval in
I(K). For 1 ≤ i ≤ K − 1, let I(i) be the (i − 1)-generation gap which contains the IjK ,
and assign the Iji be any ith-generation interval which is contained in I
(i) (see Figure
1). By the construction, for any t∗ ∈ IjK , we have
dist(t∗, Ijk) ≤ |I
(k)| ≤ 2a−1
for all 1 ≤ k ≤ K.
Proposition 6.3 (Energy non-evacuation). Let Ij1 , · · · , IjK be a disjoint family of un-
exceptional intervals obeying
∣ ≥ 2
∣ ≥ · · · ≥ 2K−1
∣ (6.16)
and for any t∗ ∈ IjK ,
dist(Ijk , t∗) . η
−26C1
hold for 1 ≤ k ≤ K. Then
K ≤ η−100C1 .
Proof: By Corollary 6.1,
Mass(u(t), B(0, η−3C1 |Ijk |
1/2)) & ηC1 |Ijk |
for all t ∈ Ijk . By (3.1), we have
Mass(u(t∗), B(0, η
−27C1 |Ijk |
1/2)) &
ηC1 |Ijk |
dist(t∗, Ijk)
η−27C1 |Ijk |
& ηC1 |Ijk |.
On the other hand, from (3.2), we have
Mass(u(t∗), B(0, 2η
C1 |Ijk |
1/2)) . η2C1 |Ijk |.
Define
A(k) =
x : ηC1 |Ijk |
1/2 ≤ |x| ≤ η−27C1 |Ijk |
then we have
∣u(t∗, x)
dx & Mass(u(t), B(0, η−27C1 |Ijk |
1/2))−Mass(u(t), B(0, 2ηC1 |Ijk |
1/2))
& ηC1 |Ijk |.
By Hölder inequality, we have
∣u(t∗, x)
n−2 dx &
ηC1 |Ijk |
η−27C1 |Ijk |
)− 2n
& η95C1
Choosing M = −56C1 log η, then we obtain by (6.16)
η−27C1 |IjM+1 |
1/2 ≤ ηC1 |Ij1 |
η−27C1 |Ij2M+1 |
1/2 ≤ ηC1 |IjM+1 |
· · ·
Hence the annuli A(k) associated to k = 1,M +1, 2M +1, · · · , are disjoint. The number
of such annuli is O(K/M).
Therefore from (6.2), we obtain
η95C1 .
∣u(t∗, x)
n−2 dx . 1.
That is
K . Mη−95C1 . η−100C1 .
We now return to the proof of Theorem 1.1. As explained at the beginning of this
section, it suffices to bound the number of the unexceptional intervals.
Note that the number of exceptional interval is at most O(η−6C3). We first bound
the number N of unexceptional intervals that can occur consecutively.
Let us denote the union of these consecutive unexceptional intervals by I. By Corol-
lary 6.2, the hypotheses of Proposition 6.2 are satisfied with a = η26C1 and so we can
find a cascade of K intervals and they satisfied the hypotheses of Proposition 6.3. The
bound on K implies the bound on N , namely,
N . (2a−26C1)K ≈ (2η−26C1)η
−100C1
At last, since there are at most O(η−6C3) exceptional intervals, the total number of
intervals is
J . η−6C3 + η−6C3N . eη
−200C1
This completes the proof of Theorem 1.1.
Acknowledgements: The authors were partly supported by the NNSF of China.
G. Xu wish to thank Xiaoyi Zhang for providing the paper [13] and some discussions.
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	Introduction
	Notations and basic estimates
	Local mass conservation and Morawetz inequality
	Local mass conservation
	A Morawetz inequality
	Local theory
	Perturbation result
	Global well-posedness
	References
