Profiles and Quantization of the Blow Up Mass for critical
nonlinear Schrödinger Equation
Frank Merle∗,∗∗ , Pierre Raphael∗
∗
Université de Cergy–Pontoise and Institute for Advanced Study
∗∗
CNRS
Abstract
We consider finite time blow up solutions to the critical nonlinear Schrödinger
4
equation iut = −∆u − |u| N u for which limt↑T <+∞ |∇u(t)|L2 = +∞. For a suitable
class of initial data in the energy space H 1 , we prove that the solution splits in two
parts: the first part corresponds to the singular part and accumulates a quantized
amount of L2 mass at the blow up point, the second part corresponds to the regular
part and has a strong L2 limit at blow up time.
1
Introduction
1.1
Setting of the problem
We consider in this paper the critical nonlinear Schrödinger equation
(
(N LS)
4
iut = −∆u − |u| N u, (t, x) ∈ [0, T ) × RN
u(0, x) = u0 (x), u0 : RN → C
(1)
with u0 ∈ H 1 = H 1 (RN ) in dimension N ≥ 1. From a result of Ginibre Velo [4], (1)
is locally well-posed in H 1 and thus, for u0 ∈ H 1 , there exists 0 < T ≤ +∞ such that
u(t) ∈ C([0, T ), H 1 ) and either T = +∞, we say the solution is global, or T < +∞ and
then lim supt↑T |∇u(t)|L2 = +∞, we say the solution blows up in finite time. Moreover,
the Cauchy problem is locally well-posed in L2 from Cazenave, Weissler, [3].
(1) admits the following conservation laws in energy space H 1 :
L2 norm :
|u(t, x)|2 = |u0 (x)|2 ;
R
Energy : E(u(t, x)) = 21 |∇u(t, x)|2 −
R
R
R
R
1
4
2+ N
R
Momentum : Im ( ∇uu(t, x)) = Im ( ∇u0 u0 (x)) .
1
4
|u(t, x)|2+ N = E(u0 );
For notational purpose, we shall introduce the following invariant:
1
E (u) = E(u) −
2
G
R
|Im( ∇uu)|
|u|L2
2
.
(2)
It is classical from the conservation of the energy and the L2 norm that the power non
linearity in (1) is the smallest one for which blow up may occur, and existence of blow up
solutions is known from the virial identity: let an initial data u0 ∈ Σ = H 1 ∩ {xu ∈ L2 },
then the corresponding solution u(t) to (1) satisfies:
d2
u(t) ∈ Σ and
dt2
Z
|x|2 |u(t, x)|2 = 16E(u0 ).
Thus if u0 ∈ Σ with E(u0 ) < 0, the positive quantity
times and u blows up in finite time.
R
(3)
|x|2 |u(t, x)|2 cannot exist for whole
Equation (1) admits a number of symmetries in energy space H 1 : if u(t, x) is a solution
N
N
to (1) then ∀(λ0 , t0 , x0 , β0 , γ0 ) ∈ R+
∗ × R × R × R × R, so is
N
v(t, x) = λ02 u(t + t0 , λ0 x + x0 − β0 t)ei
β
β0
·(x− 20 t)
2
eiγ0 .
A last symmetry is not in energy space H 1 but in the virial space Σ, the pseudo conformal
transformation: if u(t, x) solves (1), then so does
v(t, x) =
1 x i |x|2
u(
, )e 4t .
N
t t
|t| 2
1
Special solutions play a fundamental role for the description of the dynamics of (1). They
are the so called solitary waves of the form u(t, x) = eiωt Wω (x), ω > 0, where Wω solves
4
∆Wω + Wω |Wω | N = ωWω .
(4)
Equation (4) is a standard nonlinear elliptic equation, and from [1] and [6], there is a
unique positive solution up to translation Qω (x). Qω is in addition radially symmetric.
N
1
Letting Q = Qω=1 , then Qω (x) = ω 4 Q(ω 2 x) from scaling property, and from direct
computation and Pohozaev identity:
E(Qω ) = ωE(Q) = 0 and |Qω |L2 = |Q|L2 .
Recall also that in dimension N ≥ 2, (4) for ω = 1 admits a family of non zero radial
solutions Qi ∈ H 1 which is unbounded in L2 .
For |u0 |L2 < |Q|L2 , the solution is global in H 1 from the conservation of the energy,
the L2 norm and Gagliardo-Nirenberg inequality as exhibited by Weinstein in [20]:
∀u ∈ H 1
1
, E(u) ≥
2
Z

|∇u|2 1 −
2
!2
R
|u|2 N 
R
.
Q2
(5)
In addition, this condition is sharp: for |u0 |L2 ≥ |Q|L2 , blow up may occur. Indeed,
the pseudo-conformal transformation applied to the stationary solution eit Q(x) yields an
explicit solution
|x|2
i
1
x
S(t, x) = N Q( )e−i 4t + t
(6)
t
|t| 2
which blows up at T = 0 with |S(t)|L2 = |Q|L2 . Note that blow up speed for S(t) is:
|∇S(t)|L2 ∼
1
.
|t|
Moreover, from [10], S(t) is the unique minimal mass blow up solution up to the symmetries.
Most results concerning blow up dynamics of (1) now concern the perturbative situation when
Z
Z
Z
u0 ∈ Bα∗ = {u0 ∈ H 1 with
Q2 ≤ |u0 |2 < Q2 + α∗ }
for some small constant α∗ > 0. At least two different blow up behaviors are known to
possibly occur:
• There exist in dimension N = 1, 2 a family of solutions of type S(t) by a result of
Bourgain, Wang, [2], that is solutions with |∇u(t)|L2 ∼ T 1−t near blow up time.
• On the other hand, numerical simulations, [7], and formal arguments, [19], suggest
1
−t)| 2
the existence of solutions blowing up like |∇u(t)|L2 ∼ log|log(T
in dimension
T −t
N = 2. Perelman proves in [17] in dimension N = 1 the existence of an even solution
of this type and its stability in some space E ⊂ H 1 .
The situation has been clarified in sequel of papers [11], [12], [13], [14], [18]. More precisely, let us consider the following property:
Spectral Property Let N ≥ 2. Consider the two real Schrödinger operators
2
L1 = −∆ +
N
4
4
2 4
+ 1 Q N −1 y · ∇Q , L2 = −∆ + Q N −1 y · ∇Q,
N
N
(7)
and the real valued quadratic form for ε = ε1 + iε2 ∈ H 1 :
H(ε, ε) = (L1 ε1 , ε1 ) + (L2 ε2 , ε2 ).
(8)
Then there exists a universal constant δ̃1 > 0 such that ∀ε ∈ H 1 , if (ε1 , Q) = (ε1 , Q1 ) =
(ε1 , yQ) = (ε2 , Q1 ) = (ε2 , Q2R) = (ε2 , ∇Q)
= 0, then:
R
−
(i) for N = 2, H(ε, ε) ≥ δ̃1 ( R |∇ε|2 + |ε|2 e−2 |y| ) for some universal constant 2− < 2;
(ii) for N ≥ 3, H(ε, ε) ≥ δ̃1 |∇ε|2 ;
3
where Q1 =
N
2Q
+ y · ∇Q and Q2 =
N
2 Q1
+ y · ∇Q1 .
This property has been proved in [11] for dimension N = 1 and constant 2− = 59 , and
will always be implicitly assumed in higher dimension N ≥ 2.
We first have the following theorem which exhibits two different blow up behaviors in
H 1:
Theorem 1 (Dynamics of (1), [11], [12], [13], [14], [18]) Let N = 1 or N ≥ 2 assuming Spectral Property holds true. There exist universal constants α∗ > 0, C ∗ > 0 such
that the following holds true. For u0 ∈ H 1 , let u(t) the corresponding solution to (1) with
[0, T ) its maximum time interval existence on the right in H 1 . Let the set
O = {u0 ∈ Bα∗
|∇u(t)|L2
with T < +∞ and lim
t→T |∇Q|L2
T −t
log|log(T − t)|
1
2
1
= √ },
2π
(9)
then:
Log-log regime:
(i) Dynamic of non positive energy solutions:
{u0 ∈ Bα∗ with E0G ≤ 0 and
Z
|u0 |2 >
Z
Q2 } ⊂ O.
(10)
(ii) Stability of the log-log regime: O is open in H 1 .
(iii) Universality of blow up profile: if u0 ∈ O, then there exist parameters λ0 (t) =
|∇Q|L2
N and γ (t) ∈ R such that
0
|∇u(t)| 2 , x0 (t) ∈ R
L
N
eiγ0 (t) λ02 (t)u(t, λ0 (t)x + x0 (t)) → Q in Ḣ 1 as t → T.
(11)
S(t) type of regime:
(iv) If 0 < T < +∞ and u0 ∈ Bα∗ does not belong to O, then E0G > 0 and the following
lower bound holds:
C∗
q
|∇u(t)|L2 ≥
.
(12)
(T − t) E0G
Moreover, asymptotic stability (11) holds on a sequence tn → T .
1.2
Statement of the results
Our aim in this paper is to precise the nature of the singularity formation. This question
was first investigated in [13] but in the rescaled variable, what corresponds to the proof of
asymptotic stability of the soliton in the blow up regime (11). Here, we further investigate
this question and exhibit the structure of the singularity formation in the original space
4
variable.
Since the 60’s, a question of fundamental physical importance is the one of the amount
of mass which is focused by the blow up dynamic. Recall that (1) is in dimension N = 2
a model for the self focusing of laser beams, and in this frame, the amount of mass which
goes into the blow up point is related to the energy of the laser beam. It was conjectured
-at least in the stable regime- that this focused mass is quantized.
Another question regards the localization in space of the singularity formation. More
precisely, considering a blow up solution, one asks the question of the size of the singular set
of blow up points and of the behavior of the solution outside this set. One can conjecture
from the criticality of the problem that the number of blow up points is finite and outside
these points, |u|2 converges in the distributional sense to a L1 function. In the setting of
small excess of mass u0 ∈ Bα∗ , the conjecture writes: there exist x(T ) ∈ RN and f ∈ L1
such that:
Z
2
2
|u(t)| *
Q δx=x(T ) + f as t → T.
(13)
We in fact claim a stronger result which is more natural in the context of solving the
Cauchy problem locally in time in L2 : up to a singular part which structure is universal,
the solution remains smooth at blow up time in L2 .
Theorem 2 (Existence of a L2 profile at blow up time) Let N = 1 or N ≥ 2 assuming Spectral Property holds true. There exists a universal constant α∗ > 0 such that the
following holds true. Let u0 ∈ Bα∗ and assume the corresponding solution to (1) blows up
in finite time 0 < T < +∞, then there exist parameters (λ(t), x(t), γ(t)) ∈ R∗+ × RN × R
and an asymptotic profile u∗ ∈ L2 such that
1
u(t) −
N
Q
λ(t) 2
x − x(t) iγ(t)
e
→ u∗ in L2 as t → T.
λ(t)
Moreover, blow up point is finite in the sense that
x(t) → x(T ) ∈ RN as t → T.
Comments on the result
1. Quantization of the blow up mass: Observe that (14) implies:
2
|u(t)| *
with
R
|u0 |2 =
R
Z
2
Q
δx=x(T ) + |u∗ |2 as t → T
Q2 + |u∗ |2 , and thus the quantization of mass (13).
R
5
(14)
2. About concentration points: In the general large data case, the only facts known
about L2 concentration have been obtained in [15]: there is a function x(t) ∈ RN without
any a priori limit such that:
∀R > 0, lim inf
t→T
Z
|u(t)|2 ≥
Z
Q2 .
|x−x(t)|<R
In the radial case, x(t) = 0.
In the small super critical mass case u0 ∈ Bα∗ , open problems following this work were:
- prove that x(t) does not oscillate in time and has a finite limit as t → T ;
- prove that the amount of mass focused at x(T ) has a limit;
- avoid small concentration of L2 norm outside the blow up point x(T ), and then prove
existence of the strong L2 limit.
Note that existence and finiteness of blow up point x(T ) for data u0 ∈ O was already
proved in [13].
R
Of course, the main feature of the result is to say that the focused mass is exactly Q2
which is the mass of the minimal mass blow up solution.
3. Asymptotic stability and mass quantization: On the one hand, it is elementary to
check that strong convergence (14) implies the asymptotic stability of blow up profile (11).
In particular, a first step in the proof will be to prove asymptotic stability of Q as the
blow up profile on the whole sequence in time in the S(t) regime. On the other hand, the
proof of (11) in [13] is the starting point of our analysis. Indeed, it gives the exact shape
of the solution in the rescaled variables -in both blow up regimes, log-log and S(t)-. The
proof of Theorem 2 follows by propagating in original variables this information outside
blow up point, and corresponding estimates are based on those used in [13], [14], but of a
different type. We conjecture in a general context that the universality of blow up profile
and the mass quantization are equivalent formulations of the same property. For example,
for the generalized critical KdV equation, the blow up picture is similar in some sense to
the one of (1), and indeed universality of Q as a blow up profile has been proved in [8].
Nevertheless, the quantization of the blow up mass is still an open problem.
4. Comparison with the Zakharov model: In dimension N = 2, if we consider the next
term in the physical approximation leading to (NLS), we get Zakharov equation:
(
iut = −∆u + nu
1
n = ∆n + ∆|u|2
c2 tt
(15)
0
for some large constant c0 . Now as exhibited in [5], this system admits a one parameter
family of radially symmetric finite
time blow up solutions which each concentrate at x = 0
R
in L2 an amount of mass m0 ∈ ( Q2 , +∞).
5. Regularity outside blow up point: From Theorem 2, the formation of the singularity
is localized in space. Indeed, outside the blow up point x(T ), the solution has a strong L2
6
limit, whereas the Cauchy problem for (1) is well posed in L2 . It means in particular that
the phase of the solution is not oscillatory outside blow up point, whereas the phase γ(t)
of the singularity is known to satisfy γ(t) → +∞ as t → T . This strong regularity of the
solution outside the blow up point is a surprise to us.
Following this Theorem, we conjecture the following for the large initial data case in
H 1:
Conjecture: Let u(t) ∈ H 1 a solution to (1)R which blows up in finite time 0 < T <
2
|u0 |
+∞. Then there exist (xi )1≤i≤L ∈ RN with L ≤ R Q2 , and u∗ ∈ L2 such that: ∀R > 0,
u(t) → u∗ in L2 (RN −
[
B(xi , R))
1≤i≤L
Z
and |u(t)|2 * Σ1≤i≤L mi δx=xi + |u∗ |2 with mi ∈ [
Q2 , +∞).
The set M of admissible focused mass mi for N ≥ 2 is known to contain the unbounded
set of the L2 masses of excited bound states Qi solutions to (4), see for example [9]. These
are the only known examples.
In the case of the log-log regime, the singular part of the solution is up to a phase shift
completely universal and independent of the Cauchy data -this is an open problem for the
S(t) regime.
Proposition 1 (Universality of the singular structure in the log-log regime) Under
assumptions of Theorem 2, let u0 ∈ O, x(T ) its blow up point and u∗ its L2 profile. Set
λ0 (t) =
√
s
2π
T −t
,
log|log(T − t)|
(16)
then there exists a phase parameter γ0 (t) ∈ R such that:
x − x(T ) iγ0 (t)
u(t) −
e
→ u∗ in L2 as t → T.
N Q
λ
(t)
0
λ0 (t) 2
1
(17)
Remark 1 Regarding the universal behavior of the phase shift, we know:
γ0 (t) ∼
1
|log(T − t)|log|log(T − t)| as t → T.
2π
Nevertheless, we do not know whether γ0 (t) minus its equivalent has a finite limit as t → T .
7
Remark 2 Continuity of u∗ in the log-log regime: From [18], the set O of log-log blow up
is open. Moreover, from [13], blow up time T and concentration point x(T ) are continuous
functions of the data. From the proof of existence of u∗ , same kind of arguments apply
and u∗ is in O a continuous function of the data.
We claim that the two different blow up dynamics can be characterized by regularity
properties at blow up point of the profile u∗ :
Theorem 3 (Asymptotic behavior of u∗ at blow up point) Let N = 1 or N ≥ 2
assuming Spectral Property holds true. There exist universal constants α∗ > 0, C ∗ > 0
such that the following holds true. Let u0 ∈ Bα∗ and assume the corresponding solution
u(t) to (1) blows up in finite time 0 < T < +∞. Let x(T ) its blow up point and u∗ ∈ L2
its profile given by Theorem 2, then for R > 0 small enough, we have:
(i) Log-log case: if u0 ∈ O, then
1
≤
∗
C (log|log(R)|)2
Z
|u∗ (x)|2 dx ≤
|x−x(T )|≤R
C∗
,
(log|log(R)|)2
(18)
and in particular:
u∗ ∈
/ H 1 and u∗ ∈
/ Lp f or p > 2.
(19)
(ii) S(t) case: if u(t) satisfies (12), then
Z
|x−x(T )|≤R
|u∗ |2 ≤ C ∗ E0 R2 ,
(20)
and
u∗ ∈ H 1 .
Comments on the result
1. The two blow up scenarios: The fact that one can separate within the two blow up
dynamics and see the different blow up speeds on asymptotic profile u∗ is a completely
new feature for (NLS), and was not even expected at the formal level. Moreover, this results strengthens our belief that S(t) type of solutions are in some sense on the boundary
of the set of finite time blow up solutions. Indeed, the stable log-log blow up scenario
is based on the ejection of a radiative mass which strongly couples the singular and the
regular parts of the solution and induces singular behavior (18) of the profile at blow up
point; on the contrary, the S(t) regime corresponds to formation of a minimal mass blow
up bubble very decoupled from the regular part which indeed remains in the Cauchy space.
2. Degeneracy of u∗ in the S(t) regime: For N = 1, (20) implies u∗ (0) = 0. In [2],
di ∗
Bourgain and Wang construct for a given radial profile u∗ smooth with dr
i u (r)|r=0 = 0,
1 ≤ i ≤ A, a solution to (1) with blow up point x = 0 and asymptotic profile u∗ . In their
8
proof, A is very large, what is used to decouple the regular and the singular part of the
solutions. In this sense, estimate (20) proves in general a decoupling of this kind for the
S(t) dynamic. It is an open problem to estimate the exact degeneracy of u∗ .
The proofs of Theorems 2 and 3 rely on deep estimates on the solution in some rescaled
variables which have been established in our previous works on this problem. Our analysis
here collects these results altogether to derive information in the original variables. Nevertheless, the proof of Theorem 3, and in particular of asymptotic (18), follows a different
scheme and involves a new type of estimates.
Part of this work has been supported by grant DMS-0111298.
2
Dispersive estimates on the solution
We now recall tools and dispersive estimates needed to describe the blow up dynamics.
These estimates depend on the considered blow up regime. We first recall them from [12],
[14] in the log-log regime, and then from [18] in the S(t) regime. We derive the asymptotic stability of Q as a blow up profile in this last case what generalizes a corollary in [13].
Let u0 ∈ Bα∗ and assume that u(t) blows up in finite time 0 < T < +∞. As first
observed in [11], we may always assume up to a fixed Galilean transform:
Z
Im
∇u0 u0 = 0,
(21)
estimate the solution in this context, and then conclude using Galilean invariance, see
subsection 3.1.
2.1
Dynamical controls in the log-log regime
In this subsection, we assume u0 ∈ O, ie (9) holds.
step 1: Localized self similar profiles
We recall from [12] the existence of a one parameter family of localized self similar
profiles in the vicinity of ground state solution Q. Let a parameter 0 < η << 1 small
√
2 √
enough to be fixed later. For b 6= 0, set Rb = |b|
1 − η and Rb− = 1 − ηRb . Denote
BRb = {y ∈ RN , |y| ≤ Rb } and ∂BRb = {y ∈ RN , |y| = Rb }. We introduce a regular
radially symmetric cut-off function φb (x) = 0 for |x| ≥ Rb and φb (x) = 1 for |x| ≤ Rb− ,
0 ≤ φb (x) ≤ 1, such that |φ0b |L∞ + |∆φb |L∞ → 0 as |b| → 0. We claim:
Proposition 2 (Localized self similar profiles) See Propositions 8 and 9 of [13]. There
exist universal constants C > 0, η ∗ > 0 such that the following holds true. For all
9
0 < η < η ∗ , there exist constants ε∗ (η) > 0, b∗ (η) > 0 going to zero as η → 0 such that
for all |b| < b∗ (η), there exists a unique radial solution Q0b to

4
o − Qo + ib N Qo + y · ∇Qo + Qo |Qo | N

= 0,
∆Q

b
b
b
b
b
b
2

b|y|2
P o = Qob ei 4 > 0 in BRb ,


 bo
Qb (0) ∈ (Q(0) − ε∗ (η), Q(0) + ε∗ (η)), Qob (Rb ) = 0.
Moreover, let
Q̃b (r) = Qob (r)φb (r),
then:
keCr (Q̃b − Q)kC 3 → 0 as b → 0.
From now on, we note: Q̃b = Σ + iΘ in terms of real and imaginary parts.
Profiles Q̃b are not exact self similar solutions and we define error term Ψb by:
4
∆Q̃b − Q̃b + ib(Q̃b )1 + Q̃b |Q̃b | N = −Ψb .
(22)
We next introduce outgoing radiation escaping the soliton core according to the following
Lemma:
Lemma 1 (Linear outgoing radiation) See Lemma 15 in [13]. There exist universal
constants C > 0 and η ∗ > 0 such that ∀0 < η < η ∗ , there exists b∗ (η) > 0 such that
∀0 < b < b∗ (η), the following holds true: let Ψb given by (22), there exists a unique radial
solution ζb to
(
∆ζ
− ζb + ib(ζb )1 = Ψb
R b
|∇ζb |2 < +∞.
Moreover, let
Γb =
then there holds:
lim |y|N |ζb (y)|2 ,
|y|→+∞
π
π
e−(1+Cη) b ≤ Γb ≤ e−(1−Cη) b .
(23)
step 2: Geometrical decomposition close to Q̃b
The solution u(t) admits for t close enough to T the geometrical decomposition close
to the four dimensional manifold
N
M = {eiγ λ 2 Q̃b (λy + x)} :
10
Lemma 2 (Nonlinear modulation of the solution with respect to M) See Lemma
2 in [12]. There exist some time t(u0 ) ∈ [0, T ) and some C 1 functions (λ, γ, x, b) :
[t(u0 ), T ) → (0, +∞) × R × RN × R such that
N
∀t ∈ [t(u0 ), T ), ε(t, y) = eiγ(t) λ 2 (t)u(t, λ(t)y + x(t)) − Q̃b(t) (y)
satisfies the following:
(i)
ε1 (t), |y|2 Σb(t) + ε2 (t), |y|2 Θb(t) = ε1 (t), yΣb(t) + ε2 (t), yΘb(t) = 0,
− ε1 (t), (Θb(t) )2 + ε2 (t), (Σb(t) )2 = − ε1 (t), (Θb(t) )1 + ε2 (t), (Σb(t) )1 = 0,
where ε = ε1 + iε2 in terms of real and imaginary part;
(ii) |1 − λ(t)
|∇u(t)|L2
| + |ε(t)|H 1 + |b(t)| ≤ δ(α∗ ) where δ(α∗ ) → 0 as α∗ → 0.
|∇Q|L2
(iii) Let the rescaled time
Z t
dτ
s(t) =
t(u0 )
λ2 (τ )
,
then:
|
xs λs
+ b| + |bs | + |γ̃s | + ≤ C
λ
λ
Z
|∇ε|2 +
Z
− |y|
|ε|2 e−2
1
2
+ Γb1−Cη .
(24)
step 3: Control of the parameters in the log-log dynamic
We proved in [14] the following sharp controls of the parameters for the log-log dynamic:
Proposition 3 (Sharp controls of the log-log regime) (i) Equivalent of the geometrical parameters as t → T :
log|log(T − t)|
λ(t)
T −t
1
2
→
√
2π,
(25)
b(t)log|log(T − t)| → π,
(26)
s(t)
1
→
.
|log(T − t)| log|log(T − t)|
2π
(27)
(ii) Pointwise dispersive control of ε: There holds for t close enough to T ,
Z
|∇ε(t)|2 +
Z
|ε(t)|2 e−2
for some universal constant C > 0.
11
− |y|
1−Cη
≤ Γb(t)
,
(28)
Remark 3 From (23) and (26), we have:
e−(1+Cη)log|log(T −t)| ≤ e
and thus:
π
−(1+Cη) b(t)
≤ Γb(t) ≤ e
π
−(1−Cη) b(t)
≤ e−(1−Cη)log|log(T −t)|
1
1
≤ Γb (t) ≤
,
1+Cη
|log(T − t)|
|log(T − t)|1−Cη
and
λ(t) ≤ e
−
1
ΓC
b
(29)
,
(30)
for some universal constant C > 0.
Estimates of Proposition 3 are pointwise in time. Yet, the analysis in [14] provides us
with a slight time averaging improvement of (28) and (29) which will be crucial in our
further analysis. In what follows, given a parameter A > 0, we note χA (r) = χ Ar a
radial cut-off function with χ(r) = 1 for 0 ≤ r ≤ 1 and χ(r) = 0 for r ≥ 2.
Proposition 4 (Dispersive control of ε) There exist universal constants η ∗ , C ∗ > 0
such that the following holds true. ∀0 < η < η ∗ , let
a=
and
π
a b(t)
A = A(t) = e
√
η
(31)
−a(1−Cη)
so that Γb
−a(1+Cη)
≤ A ≤ Γb
,
(32)
and consider approximate radiation
ζ̃ = χA ζb
(33)
where ζb is the linear outgoing radiation of Lemma 1. Consider the new variable
ε̃ = ε − ζ̃,
(34)
we then have for s large enough:
(i) Full dispersive control:
Z +∞ Z
|∇ε̃(s)|2 +
Z
− |y|
|ε(s)|2 e−2
s
+ Γb(s) ds ≤
C∗
.
log(s)
(35)
(ii) Space localization of the L2 mass: ∀K 0 ≥ K > 0,
C2 (K, K 0 )
≤
log(s)
Z +∞
s
Z K0A
KA
12
!
2
|ε(s)|
ds ≤
C1 (K, K 0 )
.
log(s)
(36)
2.2
Dynamical controls in the S(t) regime
Let u0 ∈ Bα∗ satisfying (21) such that the corresponding solution u(t) to (1) blows up in
finite time 0 < T < +∞ with lower bound (12). First observe from Theorem 1 and (21)
that:
E0 > 0.
step 1: Geometrical decomposition of the solution
From blow up assumption on u(t) and variational characterization of Q, the solution
admits for t close enough to T a geometrical decomposition close to the four dimensional
manifold
N
M̂ = {eiγ λ 2 Q̂b (λy + x)}
where profiles Q̂b are adapted to the S(t) blow up behavior:
Q̂b = Qe−i
b|y|2
4
= Σ̂ + iΘ̂
(37)
in terms of real and imaginary parts.
Remark 4 Introduction of profiles (37) was needed in [18] to obtain some optimal monotonicity results. This is not needed here and we could work with profiles Q̃b . Nevertheless,
as all key estimates in [18] have been written under this decomposition, we stick to it.
We then have:
Lemma 3 (Nonlinear modulation of the solution with respect to M̂) See Lemma
2 in [18]. There exist some time t(u0 ) ∈ [0, T ) and some continuous functions (λ, γ, x, b) :
[t(u0 ), T ) → (0, +∞) × R × RN × R such that
N
∀t ∈ [t(u0 ), T ), ε(t, y) = eiγ(t) λ 2 (t)u(t, λ(t)y + x(t)) − Q̂b(t) (y)
satisfies the following:
(i)
ε1 (t), |y|2 Σ̂ + ε2 (t), |y|2 Θ̂ = ε1 (t), y Σ̂ + ε2 (t), y Θ̂ = 0,
− ε1 (t), Θ̂2 + ε2 (t), Σ̂2 = − ε1 (t), Θ̂1 + ε2 (t), Σ̂1 = 0,
where ε = ε1 + iε2 in terms of real and imaginary part;
(ii) |1 − λ(t)
|∇u(t)|L2
| + |ε(t)|H 1 + |b(t)| ≤ δ(α∗ ) where δ(α∗ ) → 0 as α∗ → 0.
|∇Q|L2
(iii) Let the rescaled time be s(t) =
dt0
t(u0 ) λ2 (t0 ) ,
Rt
then
xs λs
| + b| + |bs + b2 | + |γ̃s | + ≤ C
λ
λ
13
Z
2
|∇ε| +
Z
2 −2− |y|
|ε| e
1
2
,
(38)
and
Z
Z
2
2 −|y|
|∇ε| ≤ C
|ε| e
1
2
+ C(b2 + λ2 E0 ).
(39)
step 2: Virial estimate and the sharp monotonicity property
In this regime, we do not know the analogue of Proposition 3 and it is indeed an open
problem to get the exact laws for the parameters -if any in general. Nevertheless, we have
the following:
Proposition 5 (Control of the parameters in the S(t) regime) See Proposition 1
in [18]. There exist universal constants δ0 , C ∗ > 0 such that for t close enough to T :
(i) Pointwise control of b by λ:
|b(t)| ≤ λ(t)C ∗ E0 ,
(40)
λ(t) ≤ C ∗ E0 (T − t),
(41)
p
(ii) Pointwise control of the speed:
p
(iii) Dispersive virial relation: for all s large enough,
Z
bs ≥ δ0 (
|∇ε|2 +
Z
− |y|
|ε|2 e−2
)−
1 2
λ E0 .
δ0
(42)
step 3: Asymptotic stability in the S(t) regime
We now prove as a consequence of local virial estimate (42) coupled with sharp control
(40) time averaged dispersive estimates on ε which in particular imply the asymptotic
stability of Q as a blow up profile in the S(t) regime.
Proposition 6 (Asymptotic stability of the blow up profile in the S(t) regime)
There exists some universal constant C > 0 such that the following holds true:
(i) Asymptotic stability of Q as blow up profile:
Z
b(t) +
2
|∇ε(t)| +
Z
|ε(t)|2 e−|y| → 0 as t → T.
(43)
(ii) Estimate of the speed of dispersion: we have for t close enough to T ,
Z T
t
dt
2
λ (t)
Z
2
|∇ε(t)| +
Z
2 −|y|
|ε(t)| e
≤ CE0 (T − t).
(44)
In particular, there exists a sequence tn → T such that: ∀n,
Z
2
|∇ε(tn )| +
Z
|ε(tn )|2 e−|y| ≤ Cλ2 (tn )E0 .
14
(45)
Remark 5 Asymptotic stability (43) has been proved for the log-log regime in [13], [14].
The proof there is the most difficult in our analysis. This fact is related to the proximity
of the log-log regime to a possible scaling regime, |∇u(t)|L2 ∼ √T1−t , which existence would
contradict asymptotic stability. In general, results in [13] ensure (43) on a sequence in
time, or on all the sequence under monotonicity assumptions on λ -this last fact is still
open-. Observe now that in the S(t) regime, lower bound (12) is far above the scaling
estimate, and the proof of (43) will then be direct.
Proof of Proposition 6
First observe from
dt
ds
= λ2 and finite time blow up assumption on u(t) that:
Z +∞
λ2 (s)ds = T − t.
s
Integrating (42) in time s and using b(s) → 0 as s → +∞ from (40), we get:
Z T
t
≤ C
dt
2
λ (t)
Z +∞
Z
|∇ε(t)|2 +
Z
|ε(t)|2 e−|y| =
Z +∞
Z
ds
|∇ε(s)|2 +
Z
|ε(s)|2 e−|y|
s
λ2 (s)E0 ds + C|b(s)| ≤ CE0 (T − t).
s
This proves (44). In particular,
Z +∞
Z
2 −|y|
|ε(s)| e
ds
< +∞.
(46)
s
Moreover, from
the equation
satisfied by ε and control on the parameters (38), we have
R
2
−|y|
-see [13]-: |ε(s)| e
< +∞, and thus (46) implies:
s
Z
|ε(s)|2 e−|y| → 0 as s → +∞.
(39) with (40) now yields (43).
(45) directly follows from (44). This concludes the proof of Proposition 6.
3
3.1
Reduction of the proof of main results
Reduction to the zero momentum case
Our aim in this subsection is to reduce using Galilean invariance the proof of Theorem 2
and Theorem 3 to the one of a similar result under additional assumption (21) on u0 :
Z
Im
∇u0 u0 = 0.
15
Proposition 7 (Reduction to the zero momentum case) Let u0 ∈ Bα∗ satisfying
(21) and assume the corresponding solution u(t) to (1) blows up in finite time 0 < T <
+∞. Let
1
x − x(t) iγ(t)
u(t, x) = N (Qb(t) + ε) t,
e
λ(t)
λ 2 (t)
be the decomposition of Lemma 2 or Lemma 3 depending on the blow regime, and where
Qb is correspondingly Q̃b or Q̂b . Let
x − x(t) iγ(t)
ũ(t, x) = N ε t,
e
,
λ(t)
λ 2 (t)
1
(47)
then there exist u∗ ∈ L2 , x(T ) ∈ RN , such that:
and
u∗
ũ(t) → u∗ as t → T,
(48)
x(t) → x(T ) as t → T,
(49)
satisfies estimates of Theorem 3.
Proof of Theorem 2 and Theorem 3 assuming Proposition 7
Let u0 ∈ Bα∗ such that the corresponding solution u(t) to (1) blows up in finite time
0 < T < +∞. Let
R
β
Im( ∇u0 u0 )
R
β = −2
and uβ (0, x) = u0 ei 2 ·x ,
2
|u0 |
(50)
then the corresponding solution to (1) is from Galilean invariance
β
uβ (t, x) = u(t, x − βt)ei 2 ·(x−βt)
(51)
which satisfies (21) from the choice of β, and blows up at T . We thus may apply Proposition 7 and denote
!
x − xβ (t) iγβ (t)
uβ (t, x) = N (Qbβ (t) + εβ ) t,
e
λβ (t)
λ 2 (t)
1
β
its geometrical decomposition for which:
ũβ (t, x) =
1
N
λβ2 (t)
!
εβ
x − xβ (t) iγβ (t)
t,
e
→ u∗β in L2 as t → T,
λβ (t)
(52)
and xβ (t) → xβ (T ) ∈ RN . Let now
x(t) = xβ (t) − βT, γ(t) = γβ (t) −
16
β
· x(t),
2
(53)
then:
!
β
x − x(t) iγ(t)
u(t) −
e
= ũβ (t, x + βt)e−i 2 ·x + Rβ (t, x)
N Q
λβ (t)
λβ (t) 2
1
with
β
o x − (x (t) − βt)
eiγβ (t) e−i 2 ·x n −iλβ (t) β ·y
β
2 Q
e
Rβ (t, x) =
−
Q
.
bβ (t)
N
λ
2
β (t)
λβ (t)
!
From L2 scaling,
β
|Rβ |L2 = e−iλβ (t) 2 ·y Qbβ (t) − Q
L2
→ 0 as t → T
from λβ (t) → 0, Qb → Q as b → 0 in some strong sense and asymptotic stability b(t) → 0
as t → T . Thus from (52):
!
β
x − x(t) iγ(t)
u(t) −
e
→ u∗ (x) = u∗β (x + βT )e−i 2 ·x in L2 as t → T, (54)
N Q
λβ (t)
λβ (t) 2
1
and blow up point is finite:
x(T ) = xβ (T ) − βT ∈ RN .
β
Moreover, from (54), u∗ (x + x(T )) = u∗β (x + xβ (T ))e−i 2 ·(x+x(T )) so that local behavior in
H 1 of u∗ at x(T ) is the same like the one of u∗β at xβ (T ), and Theorem 3 follows.
This concludes the proofs of Theorem 2 and Theorem 3 assuming Proposition 7.
3.2
Universal structure of the singularity in the log-log regime
This subsection is devoted to the proof of Proposition 1.
Proof of Proposition 1 assuming Proposition 7
From the previous subsection, it is enough to prove it under additional assumption
(21).
Let then u0 ∈ O satisfying (21) and (λ(t), x(t), γ(t)) the parameters associated to geometrical decomposition of Lemma 2. Let λ0 (t) given by (16), we have:
λ(t)
→ 1 as t → T,
λ0 (t)
(55)
|x(T ) − x(t)|
→ 0 as t → T.
λ(t)
(56)
Indeed, (55) follows from (9). For (56), we have from (24), (25), (28), (29) and (30):
xs 1
≤
λ
|log(T − t)|C
1 xs 1
and thus |xt | = ≤
λ λ
|log(T − t)|C
17
s
log|log(T − t)|
.
T −t
Integrating this in time, we conclude:
s
log|log(T − t)| T
1
≤ C
C
T −t
t |log(T − τ )|
1
≤
C → 0 as t → T.
|log(T − t)| 2
|x(T ) − x(t)|
λ(t)
Z
s
log|log(T − τ )|
dτ ≤
T −τ
We now have:
=
2
Z 1
x − x(t)
x − x(T ) 1
−
Q
dx
N Q
λ(t) N2
λ(t)
λ0 (t)
λ0 (t) 2
N 2
Z 2
λ
(t)
λ
(t)
x(T
)
−
x(t)
0
0
Q(y) −
dy → 0 as t → T.
Q
y−
λ(t)
λ(t)
λ(t)
This concludes the proof of (56) and of Proposition 1.
The rest of this paper is devoted to the proof of Proposition 7. We thus let an initial
data u0 ∈ Bα∗ satisfying (21) with blow up time 0 < T < +∞, and assume the corresponding solution to (1) admits a geometrical decomposition on [0, T ) as in Lemma 2 or
Lemma 3 depending on the blow regime, and where Qb is correspondingly Q̃b or Q̂b , which
we denote:
u(t, x) =
1
N
λ 2 (t)
(Qb(t) + ε) t,
x − x(t) iγ(t)
e
= QS (t, x) + ũ(t, x)
λ(t)
(57)
with:
QS (x, t) =
1
N
λ(t) 2
4
Qb(t)
x − x(t) iγ(t)
1
x − x(t) iγ(t)
e
, ũ(t, x) = N ε t,
e
.
λ(t)
λ(t)
λ 2 (t)
(58)
Existence of L2 profile u∗
This section is devoted to the proof of (48) and (49) of Proposition 7.
4.1
Convergence of the blow up point
In this subsection, we prove the existence and finiteness of blow up point x(T ).
Proposition 8 (Existence and finitness of the blow up point)
x(t) → x(T ) ∈ RN as t → T.
18
(59)
Remark 6 This result has already been proved in the log-log regime, see [13], as a direct
consequence of the log-log upper bound on blow up speed. In the S(t) regime, it will follow
from dispersive controls on ε.
Proof of Proposition 8
(59) follows from:
Z T
|xt |dt < +∞.
(60)
0
log-log regime: If u0 ∈ O, then from (24) and (9),
1 xs |xt | = ≤ C|∇u(t)|L2 ≤ C
λ λ
s
log|log(T − t)|
T −t
(61)
and (60) follows.
S(t) regime: In this case,
down. From (38), we have:
RT
0
|∇u(t)|L2 dt = +∞ so that previous argument breaks
1 xs 1
≤
λλ λ
|xt | =
R
Z
|∇ε(t)|2 +
Z
− |y|
|ε|2 e−2
1
2
.
Now (44) yields 0T λ2dt(t)
|∇ε(t)|2 + |ε(t)|2 e−|y| < +∞ and (60) follows.
This concludes the proof of Proposition 8.
R
4.2
R
Existence of a L2 limit outside blow up point
We now claim existence of a L2 profile outside blow up point x(T ):
Proposition 9 (Existence of a L2 profile outside blow up point) There exists u∗ ∈
L2 such that for all R > 0,
ũ(t) → u∗ in L2 (|x − x(T )| > R).
(62)
Proof of Proposition 9
step 1 Space time control of u outside blow up point.
We claim: for all R > 0,
Z T
0
Z
!
|∇u(t)|
|x−x(T )|>R
19
2
dt < +∞.
(63)
Remark
7 From local well posedness of the Cauchy problem in L2 , blow up occurs in L2
RT R
if 0 |∇u(t)|2 dt = +∞. (63) thus means a space time gain of regularity on the solution
outside blow up point which will yield some L2 control.
Proof of (63): Fix R > 0. From (57), we have:
1
2
Z T
!
Z
2
|∇u(t)|
Z T
dt ≤
|x−x(T )|>R
0
!
Z
|∇ũ(t)|
2
Z T
S
|∇Q (t)|
dt+
|x−x(T )|>R
0
!
Z
|x−x(T )|>R
0
From convergence (59) and uniform exponential decay on Qb , we have for t ∈ [t(R), T ):
Z
|∇QS (t)|2 ≤ e
R
−C λ(t)
≤ C(R).
|x−x(T )|>R
For the ũ term, we argue differently depending on the blow up regime:
log-log regime: Using (59), we have for t ∈ [t(R), T ):
Z T
!
Z
2
|∇ũ(τ )|
Z T
dτ
=
|x−x(T )|>R
0
0
dτ
λ2 (τ )
Z +∞
≤
0
!
Z
2
|∇ε(τ )|
|λ(t)y+x(t)−x(T )|>R
!
Z
R
|y|> 2λ
|∇ε(s)|
2
ds.
Now observe from choice (32) of cut-off parameter A(t) and (30) that:
1
C0
A(t) ≤ Γ−C
<< e Γb ≤
b
1
.
λ
Since cut radiation ζ̃(y) given by (33) is zero for |y| ≥
dispersive control (35) now yields for s large enough:
Z +∞
s
!
Z
2
R
|y|> 2λ(s)
|∇ε(s)|
Z +∞
ds =
s
R
2λ ,
and from definition (34),
!
Z
2
R
|y|> 2λ(s)
|∇ε̃(s)|
ds < +∞.
S(t) regime: From (44),
Z T
|∇ũ(t)|2 dt =
0
Z T
0
dt
2
λ (t)
This concludes the proof of (63).
step 2 Existence of the L2 limit.
20
Z
|∇ε(t)|2 < +∞.
2
dt.
Fix a parameter R > 0. We claim that u(t) satisfies Cauchy criterion as t → T in
L2 (|x| > R). Indeed, pick a small ε0 > 0. We may assume from (63) that t(R) is close
enough to T so that:
!
Z T
Z
t(R)
R
|x−x(T )|> 10
|∇u(t)|2 dt < ε0 .
(64)
Next, given a fixed parameter τ > 0, let
v τ (t, x) = u(t + τ, x) − u(t, x).
u(t) is strongly continuous in L2 at time t(R), so there exists τ0 > 0 such that
Z
∀τ ∈ [0, τ0 ],
|v τ (t(R))|2 < ε0 .
(65)
We now claim:
∀τ ∈ [0, τ0 ], ∀t ∈ [t(R), T − τ ),
Z
|x−x(T )|≥ R
4
|v τ (t)|2 < Cε0 ,
(66)
for some universal constant C > 0. This indeed implies that u(t) is a Cauchy sequence in
L2 (|x| ≥ R) and yields the claim.
Proof of (66): v τ (t, x) satisfies:
4
4
ivtτ + ∆v τ = − u|u| N (t + τ ) − u|u| N (t) .
Let a cut off function φ(x) = 1 for |x| ≥ 2, φ(x) = 0 for |x| ≤ 1, we compute:
1
x − x(T )
x − x(T )
|v τ |2 = Im
φ
∇φ
R
R
R
t
Z 4
4
x − x(T ) τ v u|u| N (t + τ ) − u|u| N (t) .
+ Im
φ
R
1
2
Z
Z
·
∇v τ v τ
From Cauchy-Schwarz and the conservation of the L2 norm:
!
Z
Z
∇φ x − x(T ) · ∇v τ v τ ≤ C(R) + C(R)
R
2
R
|x−x(T )|≥ 10
2
|∇u(t)| + |∇u(t + τ )|
.
For the second term, we first have by homogeneity:
4
4
|v τ | |u(t)|1+ N + |u(t + τ )|1+ N
4
4
≤ C(|u(t)|2+ N + |u(t + τ )|2+ N ).
4
Then, we may assume cut off function φ writes φ = φ̃2+ N and φ̃ regular, and thus from
Gagliardo-Nirenberg inequality and the conservation of the L2 norm:
Z
4
x − x(T )
φ(
)|u(t)|2+ N =
R
Z
≤ C
2+ 4
Z x − x(T )
N
φ̃(
)u(t)
R
x − x(T )
|∇ φ̃(
)u(t) |2
R
Z
x − x(T )
|φ̃(
)u(t)|2
R
21
2
N
≤ C(R) 1 +
Z
R
|x−x(T )|≥ 10
!
2
|∇u(t)|
.
We thus conclude: ∀τ ∈ [0, τ0 ), ∀t with [t, t + τ ] ∈ [t(R), T ),
Z
τ 2
φR |v |
t
(
Z
≤ C(R) 1 +
)
R
|x−x(T )|> 10
(|∇ũ(t)|2 + |∇ũ(t + τ )|2 ) .
Integrating this in time with controls (64) and (65) now yields (66). This concludes the
proof of Proposition 9.
4.3
Non concentration of ũ at blow up point
In this subsection, we conclude the proof of L2 convergence (48) of ũ of Proposition 7,
what from Proposition 9 amounts proving that ũ does not concentrate any L2 mass at
blow up point.
Proof of L2 convergence (48) of Proposition 7
Remark that (48) is implied by
Z
|u∗ |2 =
Z
|u0 |2 −
Z
Q2 .
(67)
Indeed, ũ * u∗ in L2 . Moreover,
Z
|u0 |2 =
Z
|u(t)|2 =
Z
|ũ(t) + QS (t)|2 ,
and from asymptotic stability:
Z
S
2
|Q (t)| →
Z
2
Q
Thus (67) implies
R
1
Z
2
S
2 −C|y|
and (ũ(t), Q (t)) ≤
|ε(t, y)| e
dy
→ 0 as t → T. (68)
|ũ(t)|2 →
R
|u∗ |2 what concludes the proof.
Proof of (67): As for the asymptotic stability of the blow up profile, the proof requires
some work in the log-log regime, but is straightforward in the S(t) regime.
log-log case: Let
R(t) = A(t)λ(t)
with A(t) given by (32). Note from (32) and (29) that there holds for some constant
C > 0:
1
A(t) ≤
,
(69)
|log(T − t)|Ca
22
a given by (31). Let a cut-off function χ(x) = 1 for |x| ≥ 2, χ(x) = 0 for |x| ≤ 1. Compute
from (1) using (61):
Z
Z
1 x − x(τ )
Im
∇χ
· ∇uu
=
R(t) R(t)
τ
Z
1
Z
2
C
x − x(τ )
xτ
· ∇χ
|u(τ )|2 ≤
|∇u(τ )|2 .
2
R(t)
A(t)λ(t)
1 2
−
x − x(τ )
χ
|u(τ )|2
R(t)
Integrating this in time, we get using (48) and (69):
≤
Z Z Z T
C
χ x − x(T ) |u∗ |2 − χ x − x(t) |u(t)|2 ≤
|∇u(τ )|L2 dτ
R(t)
R(t)
A(t)λ(t) t
s
s
Z T
C
|log(T − t)|Ca
log|logT − t|
T −t
log|logT − τ |
1
dτ ≤
Ca .
T −τ
|log(T − t)| 2
t
Z
Z
Letting t → T , we have:
|u∗ |2 = lim
χ
t→T
x − x(t)
|u(t)|2 .
R(t)
(70)
We then have:
Z
lim
t→T
x − x(t)
χ
|u(t)|2 = lim
t→T
R(t)
x − x(t)
χ
|ũ(t)|2 = lim
t→T
R(t)
Z
Z
|ũ(t)|2 dt,
(71)
which concludes the proof.
Proof of (71): Recall first the Sobolev type estimate observed in [14]: there exists a
universal constant C > 0 such that
∀D ≥ 1, ∀v ∈ H 1 ,
Z
|v|2 ≤ CD2
Z
|∇v|2 +
Z
|v|2 e−|y| .
(72)
|y|≤D
Then from (28):
Z
|ũ(t)|2 =
Z
|x−x(t)|≤2R(t)
≤ CA(t)2
Z
|∇ε(t)|2 +
Z
2
|y|≤ A(t)
|ε(t, y)|2 dy
1
2
|ε(t)|2 e−|y| ≤ Γb(t)
→ 0 as t → T
provided η > 0 small enough, what concludes the proof of (71).
S(t) case: From Proposition 9, ũ → u∗ in L2 (|x − x(T )| > 1) as t → T . Moreover,
from (45), there exists a sequence tn → T such that
Z
|∇ũ(tn )|2 =
1
λ2 (tn )
Z
23
|∇ε(tn )|2 ≤ CE0 ,
and thus by uniqueness of the weak limit in H 1 ,
u∗ ∈ H 1 and ũ(tn ) * u∗ in H 1 as tn → T.
(73)
From compact embedding of H 1 into L2loc ,
ũ(tn ) → u∗ L2 (|x − x(T )| ≤ 1),
and thus:
Z
|ũ(tn )|2 →
Z
|u∗ |2 .
Now (67) follows from the conservation of the L2 norm and (68).
This concludes the proof of L2 convergence (48) of Proposition 7.
5
Asymptotic of the L2 profile at blow up point
In this section, we conclude the proof of Proposition 7 by proving that L2 profile u∗ given
by (48) satisfies estimates of Theorem 3.
To estimate the size of u∗ at blow up point is equivalent to understand the mass ejection phenomenon which couples the singular part with the regular part of the solution.
In the S(t) regime, these two parts are decoupled what leads to a regular and flat profile
u∗ at blow up point as expressed by (20). On the contrary, a characteristic of the log-log
regime is the strong coupling between the singularity formation and dispersion outside
blow up point. The mass ejection process is indeed the main mechanism in the blow up
dynamic. The proof of estimate (18) involves two different arguments:
1.Localization of the L2 mass in the rescaled variables: In [14], an explicit study of
dispersive effects in L2 in the rescaled variables has allowed us to derive the very strong
localization estimate (36) which should be formally understood as:
Z 2A
A
|ε(t)|2 ∼ Γb(t) .
(74)
2.Control of the L2 flux: To derive estimates on u∗ , we propagate this information in
original variables by proving that this amount of mass is frozen in time up to T .
Remark 8 In the log-log case, we will in fact prove that given any R > 0, there is a time
t(R) such that for t ∈ [t(R), T ), the mass ejection phenomenon at R becomes negligible,
and then the L2 flux around R is frozen at its asymptotic value given by the equivalent
(18) for u∗ . In time averaging sense, we will then have: ∀t ∈ [t(R), T ),
Z
2
R
≤|x−x(T )|≤R
2
|u(t)| ∼
Z
R
R
≤|y|≤ λ(t)
2λ(t)
24
|ε(t)|2 ∼
C
.
log(R) (log|log(R)|)3
(75)
The behavior in R of the right hand side is a consequence of (74), or more specifically
as exhibited in [14] of the fact that in the region |y| ≤ A(t), a good approximation of the
solution ε(t) is the universal radiation ζb(t) of Lemma 1. From (75), we now may estimate
the size of the region in space for which this approximation is good, which will turn out to
be much smaller than the one suggested at a formal level in [19]. Indeed, assuming ε ∼ ζb
for |y| ≤ B(t), we get from explicit computation on ζb(t) and (75):
Z
R
|y|∼ |λ(t)
=B(t)
|ε(t)|2 = Γb(t) ∼
C
log(R) (log|log(R)|)3
R
|ε(t)|2 ∼
λ(t)
C
.
log(λ(t)B(t)) (log|log(λ(t)B(t))|)3
=
This provides us with a control from above and below on B(t) which implies in particular:
B(t) ≤
1
, 0 < δ < 1.
λ(t)1−δ
In original variables, this means that the radiative zone does not escape the focusing point,
and that the regular part of the solution indeed corresponds to a different regime.
S(t) case: First observe that u∗ ∈ H 1 has been proved in the proof of Proposition 7 as
a straightforward consequence of (45), see (73). It remains to prove degeneracy estimate
at blow up point (20).
Indeed, let tn the sequence such that (45) holds. From L2 convergence (14),
Z
∗
2
|u (x)| dx = lim
tn →T
|x−x(T )|<R
Z
|x−x(tn )|<R
|ũ(tn , x)|2 dx.
Now for the sequence tn :
Z
|x−x(tn )|<R
|ũ(tn , x)|2 dx =
≤
Z
|y|≤ λ(tR )
n
CR2
λ2 (tn )
|ε(tn , y)|2 dy
Z
2
|∇ε(tn )| +
Z
2 −|y|
|ε(tn )| e
≤ CE0 R2 ,
where we used Sobolev type estimate (72). This concludes the proof of (20) and of Proposition 7 in the S(t) regime.
log-log regime:
step 1: Control of the L2 flux.
For all t0 close enough to T , let A(t0 ) given by (32), we define:
R(t0 ) = λ(t0 )A(t0 ).
25
(76)
For a given radial cut off function ψ(r) supported in {1 ≤ r ≤ 2} with ψ(r) = 1 in a
neighborhood of r = 32 , let for t ∈ [t0 , T ):
Z x − x(t) |ũ(t)|2 .
m(ψ, t) = ψ
R(t ) 0
Then:
Z
m(ψ, t0 ) =
ψ
x − x(t0 )
|ũ(t0 , x)|2 dx =
R(t0 )
Z
ψ
y
|ε(t0 , y)|2 dy,
A(t0 )
and thus estimate (36) with (27) writes:
1
≤
C(ψ)log|log(T − t)|
Z T
m(ψ, t0 )
t
λ2 (t0 )
dt0 ≤
C(ψ)
.
log|log(T − t)|
(77)
We now claim:
Lemma 4 (Control on the flux of the L2 norm) There holds for η > 0 small enough
and t0 close enough to T : ∀t ∈ [t0 , T ),
Z
Z
ψ x − x(t) |ũ(t)|2 − ψ x − x(t0 ) |ũ(t0 )|2 R(t )
R(t )
0
≤ C(ψ)Γ
1+a
2
b(t0 )
0
Z 1
2
x − x(t) 2
sup
∇ψ
|ũ(t)|
.
R(t ) (78)
0
t∈[t0 ,T )
Remark 9 From now on, parameter η is fixed small enough so that above estimates hold,
√
and so is a = η.
Proof of Lemma 4
We first claim from support property of ψ:
∀t ∈ [t0 , T ), QS (t, x) = 0 ie u(t, x) = ũ(t, x) for
x − x(t)
∈ Supp(ψ).
R(t0 )
(79)
Indeed, first remark that
b(t)
λ(t)
t
1
= 3
λ
λs
Cb2
bs − b ≥ 3 > 0
λ
λ
b(t)
where we used (24) and (28) in the last step. Thus A(t0 )λ(t0 ) λ(t)
≥ A(t0 )b(t0 ) ≥ 100 for
t0 close to T from (32). Consequently:
∀t ∈ [t0 , T ),
R(t0 )
A(t0 )λ(t0 )
100
=
≥
.
λ(t)
λ(t)
b(t)
26
Let
x−x(t)
R(t0 )
∈ Supp(ψ) ⊂ [1, 2], then:
x − x(t) λ(t) =
x − x(t) R(t ) ≥ 1 implies
0
x − x(t) R(t0 )
100
R(t ) λ(t) ≥ b(t) .
0
Now from Proposition 2, Q̃b has support in the ball |y| ≤ 4b , so that (58) yields (79).
We now compute the flux of the L2 norm using (79): ∀t ∈ [t0 , T ),
1
2
Z
x − x(t)
|ũ(t)|2
ψ
R(t0 )
xt
x − x(t)
= −
|ũ|2
· ∇ψ
2R(t0 )
R(t0 )
Z
Z
1
x − x(t)
+
· ∇ũũ .
Im
∇ψ
R(t0 )
R(t0 )
t
Z
From (24), (28) and (47),
Z
|xt | +
2
|∇ũ(t)|
1
2
1
=
λ(t)
Z
2
|∇ε(t)|
1
2
1
−Cη
Γ2
≤ b
.
λ(t)
Thus: ∀t ∈ [t0 , T ),
1
Z
−Cη Z 1
2
Γ
2
x − x(t)
x − x(t) b(t)
2
2
ψ
|ũ(t)| ≤ C(ψ)
∇ψ
|ũ(t)|
R(t0 )
R(t0 )λ(t)
R(t0 )
t
1
−Cη
1
2
∇ψ x − x(t) |ũ(t)|2
.
R(t )
Z
2
Γb(t)
1
≤ C(ψ)
sup
A(t0 ) λ(t0 )λ(t) t∈[t0 ,T )
0
Integrating this in time, we first estimate from (25) and (29):
1
1
λ(t0 )
Z t Γ 2 −Cη
b(τ )
t0
λ(τ )
s
dτ
≤ C
≤
log|log(T − t0 )|
T − t0
1
1
|log(T − t0 )| 2 −3Cη
Z T
s
dτ
|log(T − τ )|
t0
1
−4Cη
2
b(t0 )
≤Γ
1
−2Cη
2
log|log(T − τ )|
T −τ
,
and thus:
Z
Z
ψ x − x(t) |ũ(t)|2 − ψ x − x(t0 ) |ũ(t0 )|2 R(t )
R(t )
0
≤ C(ψ)
1
−Cη
2
b(t0 )
Γ
A(t0 )
0
Z 1
2
x − x(t) 2
sup
|ũ(t)|
∇ψ
R(t )
0
t∈[t0 ,T )
Now from (32),
1
−Cη
2
Γb(t
0)
A(t0 )
1
−Cη a(1−Cη)
2
≤ Γb(t
Γb(t0 )
0)
27
1+a
≤ Γb(t20 ) ,
where we used (31) for η > 0 small enough in the last step. This concludes the proof of
(78) and of Lemma 4.
step 2 Iteration of the flux control.
In integration in time, the right hand side of (78) is much larger than size estimate
(77). Our aim now is to iterate this flux control to get on the contrary in time averaging
sense and for some suitable ψ:
Z
1+a
x − x(t0 )
ψ
|ũ(t0 )|2 >> Γb(t20 ) sup
R(t0 )
t∈[t0 ,T )
1
Z 2
∇ψ x − x(t) |ũ(t)|2
.
R(t ) 0
Fix an integer L = L(a) such that:
a
1
≤ (1 + a) 1 − L .
4
2
1+
For 0 ≤ k ≤ L, we consider a family of radial cut-off functions ψk with the following
properties:
(
ψk (x) =
1
1
≤ |x| ≤ 1 + 12 + 2k+2
1 for 1 + 21 − 2k+2
1
1
0 for |x| ≤ 1 + 2 − 2k+1 and |x| ≥ 1 +
1
2
+
1
,
2k+1
∀k ≥ 0, ∀x ∈ RN , |∇ψk+1 (x)| ≤ 3ψk (x).
We claim: ∀t0 close enough to T ,
Z
a
ψL x − x(T ) |u∗ |2 − m(ψL , t0 ) ≤ CΓ 4
b(t0 )
R(t )
0
sup
0≤k≤L−1
1+ a
m(ψk , t0 ) + CΓb(t08) .
(80)
Proof of (80): For a given cut off ψ supported in {1 ≤ |x| ≤ 2}, we note M (ψ, t0 ) =
supt∈[t0 ,T ) m(ψ, t). Applying flux control (78), we have: ∀0 ≤ k ≤ L, ∀t ∈ [t0 , T ),
1+a
1+a
q
q
|m(ψk+1 , t) − m(ψk+1 , t0 )| ≤ CΓb(t20 ) M (|∇ψk+1 |, t0 ) ≤ CΓb(t20 ) M (ψk , t0 ),
(81)
and in particular:
1+a
q
M (ψk+1 , t0 ) ≤ m(ψk+1 , t0 ) + CΓb(t20 ) M (ψk , t0 ).
(82)
We claim: there holds for some universal constant C > 0,
!
∀0 ≤ k ≤ L, M (ψk , t0 ) ≤ C
sup m(ψp , t0 ) +
0≤p≤k
28
k
Γpb(t
0)
,
(83)
with pk = (1 + a)(1 − 2−(k+1) ). Indeed, we argue by induction on k. For k = 0, (78) yields
(83) with p0 = 1+a
2 . We now assume (83) for k and prove it for k + 1. Indeed, from (82):
1+a
q
M (ψk+1 , t0 ) ≤ m(ψk+1 , t0 ) + CΓb(t20 ) M (ψk , t0 )
1+a
≤ m(ψk+1 , t0 ) + CΓb(t20 )
≤ m(ψk+1 , t0 ) + CΓ
≤ C
sup
r
k
sup m(ψp , t0 ) + Γpb(t
0)
0≤p≤k
1+a
2
b(t0 )
pk
2
r
!
sup m(ψp , t0 ) + Γb(t0 )
0≤p≤k
m(ψp , t0 ) +
0≤p≤k+1
1+a+pk
2
b(t0 )
!
Γ1+a
b(t0 )
+Γ
sup
m(ψk , t0 ) + CΓb(t04) ,
≤C
sup
m(ψp , t0 ) +
0≤p≤k+1
pk+1
Γb(t
0)
k
from pk+1 = 1+a+p
, and (83) follows.
2
For k = L − 1, (83) writes:
M (ψL−1 , t0 ) ≤ C
1+ a
0≤k≤L−1
what injected into (81) for k = L − 1 yields: ∀t ∈ [t0 , T ),
1+a
2
b(t0 )
|m(ψL , t) − m(ψL , t0 )| ≤ CΓ
q
1+a
2
b(t0 )
M (ψL−1 , t0 ) ≤ CΓ
a
4
≤ CΓb(t
0)
r
sup
m(ψk , t0 ) + Γ
0≤k≤L−1
1
+ a8
2
b(t0 )
1+ a
sup
0≤k≤L−1
m(ψk , t0 ) + CΓb(t08) .
Letting t → T according to strong L2 convergence (48) yields (80).
step 3 Proof of estimate (18).
We claim for t close to T :
1
≤
Clog|log(T − t)|
Z T
t
dt0
λ2 (t0 )
Z
ψL
x − x(T )
C
|u∗ |2 ≤
.
R(t0 )
log|log(T − t)|
(84)
Proof of (84): From (77),
1
≤
Clog|log(T − t)|
Z T
m(ψL , t0 )
t
λ2 (t
0)
dt0 ≤
C
,
log|log(T − t)|
so that using (80), (84) is equivalent to prove:
log|log(T − t)|
Z T
t
a
dt0
1+ a8
4
Γ
sup
m(ψ
,
t
)
+
Γ
0
k
b(t
)
b(t0 )
0
λ2 (t0 )
1≤k≤L−1
29
!
→ 0 as t → T.
(85)
!
!
,
This is a consequence of (35) and (77). Indeed, first rewrite (35) with (27):
Z T
t
dt0
Γ
=
2
λ (t0 ) b(t0 )
Z +∞
s
Γb(s) ds ≤
C
C
≤
.
log(s)
log|log(T − t)|
Using this and (77), we conclude:
log|log(T − t)|
Z T
t
a
8
a
dt0
1+ a8
4
sup
m(ψ
,
t
)
+
Γ
Γ
0
k
b(t0 )
b(t0 )
λ2 (t0 )
1≤k≤L−1
(
≤ log|log(T − t)|Γb(t)
Z T
sup
1≤k≤L−1 t
dt0
m(ψk , t0 ) +
λ2 (t0 )
Z T
t
!
dt0
Γ
λ2 (t0 ) b(t0 )
)
a
8
≤ CΓb(t) → 0 as t → T.
This is (85) and thus (84) follows.
We now change variables in (84). Let
R0 = R(t0 ) = λ(t0 )A(t0 ) = λ(t0 )e
we claim:
a b(tπ
0)
,
log|log(R0 )|
1 dt0
Clog|log(R0 )|
≤ 2
≤
.
CR0
λ (t0 ) dR0
CR0
(86)
Indeed, from (24) and pointwise control (28), we estimate:
dR0
bt
= A(t0 ) −λt0 − aπλ 20
dt0
b
0<
A(t0 )
λs
bs
=
b − ( 0 + b) − aπ 20
λ(t0 )
λ
b
,
dR0
b(t0 )A(t0 )
b(t0 )A(t0 )
≤
,
≤C
Cλ(t0 )
dt0
λ(t0 )
1
1
1 dt0
C
C
=
≤ 2
≤
=
.
Cb(t0 )R0
Cb(t0 )A(t0 )λ(t0 )
λ (t0 ) dR0
b(t0 )A(t0 )λ(t0 )
b(t0 )R0
(87)
Moreover, we estimate from (76), (26) and (25):
1
1
C
≤ log|log(T − t0 )| ≤ log|log(R0 )| ≤ Clog|log(T − t0 )| ≤
,
Cb(t0 )
C
b(t0 )
what with (87) yields (86).
With (86), (84) writes for R > 0 small enough:
1
≤
Clog|log(R)|
Z R
dR0
0
log|log(R0 )|
R0
Z
ψL
30
x − x(T )
C
|u∗ (x)|2 dx ≤
.
R0
log|log(R)|
We now have from support property of ψL : ∀y ∈ RN , 1a1 ≤|y|≤b1 ≤ ψL (y) ≤ 1a2 ≤|y|≤b2 for
some a1 , b1 , a2 , b2 > 0. From Fubbini:
Z R
x − x(T )
ψL
|u∗ (x)|2 dx
R0
0
Z
Z
log|log(R0 )|
≤
dxdR0 |u∗ (x)|2
R0
a2 R0 ≤|x−x(T )|≤b2 R0 0≤R0 ≤R
log|log(R0 )|
dR0
R0
Z
∗
Z
2
|u (x)| dx
=
|x−x(T )|≤b2 R
≤ C
Z
|x−x(T )|
a2
Z
|x−x(T )|
b2
log|log(R0 )|
dR0
R0
log|log(|x − x(T )|)|u∗ (x)|2 dx,
|x−x(T )|≤b2 R
and we conclude:
Z
log|log(|x − x(T )|)|u∗ (x)|2 dx ≥
|x−x(T )|≤b2 R
1
.
Clog|log(R)|
Arguing similarly for the upper bound, we thus have proved: for all R > 0 small,
1
≤
Clog|log(R)|
Z
log|log(|x − x(T )|)|u∗ (x)|2 dx ≤
|x−x(T )|≤R
C
.
log|log(R)|
(88)
Upper bound in (18) follows. For the lower bound, letting C > 0 the constant involved in
(88) and f (R) = e−|log(R)|
10C 2
so that:
C
1
=
,
log|log(f (R))|
10Clog|log(R)|
(89)
we have from control (88):
1
≤
Clog|log(R)|
Z
Z
log|log(|x − x(T )|)|u∗ (x)|2 dx +
=
log|log(|x − x(T )|)|u∗ (x)|2 dx
|x−x(T )|≤R
Z
log|log(|x − x(T )|)|u∗ (x)|2 dx
|x−x(T )|≤f (R)
f (R)≤|x−x(T )|≤R
C
≤ log|log(f (R))|
|u∗ (x)|2 dx +
log|log(f (R))|
|x−x(T )|≤R
Z
1
≤ 10C 2 log|log(R)|
|u∗ (x)|2 +
10Clog|log(R)|
|x−x(T )|≤R
Z
where we used (89) in the last step, and thus:
Z
|u∗ (x)|2 ≥
|x−x(T )|≤R
This concludes the proof of estimate (18).
31
1
20C 3 (log|log(R)|)2
.
step 4 u∗ ∈
/ H 1.
To conclude the proof of Proposition 7, we remark that ∀p > 2, u∗ ∈
/ Lp . Indeed,
assume by contradiction u∗ ∈ Lp for some p > 2, then by Hölder:
Z
∗
2
|u (x)| dx ≤
Z
∗ p
|u |
N
N +2
|x−x(T )|≤R
Z
r
N −1
p−2
dr
p
≤ C(u∗ )R
(p−2)N
p
0≤r≤R
what contradicts lower bound in (18) in the limit R → 0.
This concludes the proof of estimates of Proposition 7.
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