STABLE SELF SIMILAR BLOW UP DYNAMICS FOR SLIGHTLY
L2 SUPERCRITICAL NLS EQUATIONS
FRANK MERLE, PIERRE RAPHAËL, AND JEREMIE SZEFTEL
Abstract. We consider the focusing nonlinear Schrodinger equations i∂t u +
∆u + u|u|p−1 = 0 in dimension 1 ≤ N ≤ 5 and for slightly L2 supercritical
nonlinearities pc < p < (1 + ε)pc with pc = 1 + N4 and 0 < ε 1. We prove the
existence and stability in the energy space H 1 of a self similar finite time blow
up dynamics and provide a qualitative description of the singularity formation
near blow up time
1. Introduction
1.1. Setting of the problem. We consider in this paper the nonlinear Schrödinger
equation
iut = −∆u − |u|p−1 u, (t, x) ∈ [0, T ) × RN
(N LS)
(1)
u(0, x) = u0 (x), u0 : RN → C
in dimension 1 ≤ N ≤ 5 with
N +2
1<p<
.
N −2
From a result of Ginibre and Velo [6], (1) is locally well-posed in H 1 = H 1 (RN )
and thus, for u0 ∈ H 1 , there exists 0 < T ≤ +∞ and a unique solution u(t) ∈
C([0, T ), H 1 ) to (1) and either T = +∞, we say the solution is global, or T < +∞
and then limt↑T |∇u(t)|L2 = +∞, we say the solution blows up in finite time.
(1) admits the following conservation laws in the energy space H 1 :
R
R
L2 − norm :
|u(t, x)|2 dx
=
|u0 (x)|2 dx;
R
R
1
1
Energy : E(u(t, x)) = 2 |∇u(t, x)|2 dx − p+1
|u(t, x)|p+1 dx = E(u0 ).
R
R
Momentum : Im( ∇u(t, x)u(t, x)dx) = Im( ∇u0 (x)u0 (x)dx)
2
The scaling symmetry λ p−1 u(λ2 t, λx) leaves the homogeneous Sobolev space Ḣ σc
invariant with
N
2
σc =
−
.
(2)
2
p−1
From the conservation of the energy and the L2 norm, the equation is subcritical
for σc < 0 and all H 1 solutions are global and bounded in H 1 . The smallest power
for which blow up may occur is
4
pc = 1 +
N
which corresponds to σc = 0 and is referred to as the L2 critical case. The case
0 < σc < 1 is the L2 super critical and H 1 subcritical case.
Even though the existence of finite time blow up dynamics for σc ≥ 0 has been
known since the 60’ using standard global obstructive arguments based on the virial
identity, [24], the explicit descritpion of the singularity formation and of the different
possible regimes is mostly open.
1
2
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
1.2. The L2 critical case and the stable log-log blow up. In the L2 critical case
p = pc , a specific blow up regime of "log-log" type has been exhibited by Perelman
[16] in dimension N = 1 and further extensively studied by Merle and Raphael in
the series of papers [10], [11], [17], [12], [13], [14] where a complete description of
this stable blow up dynamics is given together with sharp classification results in
dimension N ≤ 5. The ground solitary wave solution to (1) which is the unique
nonnegative radially symmetric solution to
∆Qp − Qp + Qpp = 0, Q ∈ H 1 ,
(3)
see [5], [9], plays a distinguished role in the analysis as it provides the blow up
profile of log-log solutions.
Let Λ be the generator of the scaling symmetry as given by (22) and recall the following Spectral Property which was proved in [10] in dimension N = 1 and cheked
numerically in dimensions N ≤ 5 in [4]:
Spectral Property Let N ≤ 5 and p = pc . Consider the two real Schrödinger
operators
4
4
2 4 −1
2
−1
+ 1 QpNc y · ∇Qpc , L2 = −∆ + QpNc y · ∇Qpc , (4)
L1 = −∆ +
N N
N
and the real valued quadratic form for ε = ε1 + iε2 ∈ H 1 :
H(ε, ε) = (L1 ε1 , ε1 ) + (L2 ε2 , ε2 ).
(5)
Then there exists a universal constant δ1 > 0 such that ∀ε ∈ H 1 , if (ε1 , Qpc ) =
(ε1 , ΛQpc ) = (ε1 , yQpc ) = (ε2 , ΛQpc ) = (ε2 , Λ2 Qpc ) = (ε2 , ∇Qpc ) = 0, then
Z
Z
H(ε, ε) ≥ δ1
|∇ε|2 + |ε|2 e−|y| .
We then have the following:
Theorem 1 (Existence of a stable log-log regime, [10], [11], [12], [17], [13], [14],
[4]). Let N ≤ 5 and p = pc . There exists a universal constant α∗ > 0 such that the
following holds true. For any initial data u0 ∈ H 1 with small super-critical mass
|Qp |L2 < |u0 |L2 < |Qp |L2 + α∗
(6)
and nonpositive Hamiltonian E(u0 ) < 0, the corresponding solution to (1) blows up
in finite time 0 < T < +∞ according to the following blowup dynamics: there exist
geometrical parameters (λ(t), x(t), γ(t)) ∈ R∗+ × RN × R and an asymptotic residual
profile u∗ ∈ L2 such that:
1
x − x(t) iγ(t)
u(t) − N Qp
e
→ u∗ in L2 .
λ(t)
λ 2 (t)
The blowup point converges at blowup time:
x(t) → x(T ) ∈ RN as t → T,
the blowup speed is given by the log-log law
r
√
log|log(T − t)|
λ(t)
→ 2π as t → T,
T −t
and the residual profile satisfies:
(7)
u∗ ∈ L2 but u∗ ∈
/ Lp , ∀p > 2.
More generally, the set of initial data satisfying (6) and such that the corresponding
solution to (1) blows up in finite time with the log-log law (7) is open in H 1
3
In other words, the stable log-log regime corresponds to an almost self similar
regime where the blow up solution splits into a singular part with a universal blow
up speed and a universal blow up profile given by the exact ground state Qpc , and
a regular part which remains in the critical space u∗ ∈ L2 but looses any regularity
above scaling u∗ ∈
/ H σ for σ > 0.
1.3. On the super critical problem. The explicit description of blow up dynamics in the super critical setting is mostly open. In fact, the only rigorous description
of a blow up dynamics in a super critical setting is for p = 5 in any dimension
N ≥ 2, see Raphael [18] for N = 2, Raphael, Szeftel [19] for N ≥ 3. Note that
this includes energy super critical problems. In this setting, the existence and radial
stability of self similar solutions blowing up on an asympotic blow up sphere -and
not a blow up point- is proved. These solutions reproduce on the blow up sphere
the one dimensional quintic and hence L2 critical blow up dynamic and the blow
up speed is indeed given by the log log law (7).
In the formal and numerical work [3], Fibich, Gavish and Wang propose in dimension N = 2, 3 and for pc < p < 5 a generalization of the standing ring blow up
solutions and investigate a blow up dynamic where the solution concentrates on
spheres with radius collapsing to zero. Such dynamics are clearly exhibited numerically and seem to be stable by radial perturbations. A striking feature moreover is
that these solutions form a Dirac mass in L2 like in the L2 critical case:
|u|2 * M δx=0 + |u∗ |2
(8)
for some universal quantum of mass M > 0, and with a specific predicted blow up
speed:
5−p
1
α=
.
(9)
|∇u(t)|L2 ∼
1 ,
(N − 1)(p − 1)
(T − t) 1+α
The rigorous derivation of such collapsing ring solutions is an important open problem in the field.
While ring solutions display a stability with respect to radial perturbations, they
are widely believed to be unstable by non radial perturbations, [3]. In fact, it has
long been conjectured according to numerical simulations, see [22] and references
therein, that the generic blow up dynamics in the super critical setting -at least for
p near pc - should be of self similar type:
x − xT iγ(t)
1
)e
u(t, x) ∼
2 P(
λ(t)
[λ(t)] p−1
for some blow up point xT ∈ RN and where λ is in the self similar regime:
√
λ(t) ∼ T − t.
The delicate issue here is the profile P which does not seem to be given by the
ground state solution Qp to (3) anymore. In fact, explicit self similar solutions may
be computed using the following standard procedure. Let explicitely
p
1
x − xT − log(T −t)
2b
, b > 0, λ(t) = 2b(T − t),
u(t, x) =
)e
2 Qb (
λ(t)
[λ(t)] p−1
then u solves (1) iff Qb,p = Qb satisfies the nonlinear stationary elliptic PDE:
∆Qb − Qb + ibΛQb + Qb |Qb |p−1 = 0.
(10)
Exact zero energy solutions to (10) have been exhibited by Koppel and Landman,
[7], for slightly super critical exponent pc < p < (1 + ε)pc , ε 1, using geometrical
4
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
ODE techniques. The existence of such solutions is related to a nonlinear eigenvalue
problem and for all p ∈ (pc , (1 + ε)pc ), a unique value b(p) is found such that (10)
admits a zero energy solution. An asymptotic law is derived which confirms previous
fomal computations, see [22] p147 and references therein:
π
σc = e− b (1+o(1)) as p → pc .
(11)
Moreover, the self similar profile converges to Qpc locally:
1
in Hloc
as p → pc .
Qb → Qpc
However the self similar solutions belong to Ḣ 1 ∩ Lp+1 but always miss the critical
Sobolev space due to a logarithmic divergence at infinity:
|Qb |(y) ∼
1
|y|
N
2
−σc
and hence Qb ∈
/ Ḣ σc ,
and thus they in particular miss L2 and hence the physically relevant space H 1 .
Moreover, the construction of the self similar solution is delicate enough that it
is not clear at all how this object should generate a stable self similar blow up
dynamics.
1.4. Statement of the result. The law for the nonlinear eigenvalue (11) is deeply
related to the log-log law (7) which can be rewritten in the following form:
π
1
λs
ds
= 2 , b = − , bs = −e− b (1+o(1)) as s → +∞.
ds
λ
λ
This intimate connection between the log-log law and the nonlinear eigenvalue problem underlying the self similar equation is at the heart of formal heuristics which
first predicted (11), we refer to the monograph [22] for a complete introduction to
the history of the problem.
Our main claim in this paper is that the log-log analysis [13] which allowed
Merle and Raphael to derive the sharp log-log law for a large class of initial data
provides a framework to somehow bifurcate from the critical value p = pc and prove
the existence of a stable self similar regime in the energy space for slighlty super
critical exponents, with a blow up speed asymptotically satisfying the nonlinear
eigenvalue relation (11). In fact, we will show that it is enough to construct a crude
and compactly supported approximation of the self similar profile (10) and then the
intuition and technical tools inherited from the log-log analysis imply new rigidity
properties and a dynamical trapping of a self similar regime. The outcome is a
suprisingly robust proof of the existence of a H 1 stable self similar blow up regime:
Theorem 2 (Existence and stability of a self similar blow up regime). Let 1 ≤
N ≤ 5. There exists p∗ > pc such that for all p ∈ (pc , p∗ ), there exists δ(p) > 0 with
δ(p) → 0 as p → pc , there exists b∗ (p) > 0 with
σc = e
− b∗π(p) (1+δ(p))
(12)
and an open set O in H 1 of initial data such that the following holds true. Let u0 ∈
O, then the corresponding solution to (1) blows up in finite time 0 < T < +∞ according to the following dynamics: there exist geometrical parameters (λ(t), x(t), γ(t)) ∈
R∗+ × R∗+ × RN × R and an excess of mass ε(t) ∈ H 1 such that:
1
x − x(t) iγ(t)
∀t ∈ [0, T ), u(t, x) =
[Q
+
ε(t)]
e
(13)
2
λ(t)
λ p−1 (t)
5
with
|∇ε(t)|L2 ≤ δ(p).
The blowup point converges at blowup time:
x(t) → x(T ) ∈ RN as t → T,
and the blow up speed is self similar:
p
p
λ(t)
∀t ∈ [0, T ), (1 − δ(p)) 2b∗ (p) ≤ √
≤ (1 + δ(p)) 2b∗ (p).
T −t
Moreover, there holds the strong convergence:
∀σ ∈ [0, σc ), u(t) → u∗ in H σ as t → T,
(14)
(15)
(16)
(17)
and the asympotic profile u∗ displays a singular behavior on the blow up point:
∃R(u0 ), C(p) > 0 such that:
Z
1
|u∗ |2 ≤ C(p)(1 + δ(p)).
(18)
∀R ∈ (0, R(u0 )), C(p)(1 − δ(p)) ≤ 2sc
R
|x|≤R
In particular:
u∗ ∈ H σ for 0 ≤ σ < σc but u∗ ∈
/ Ḣ σc .
Comments on Theorem 2
1. Asymptotic dynamics: Let us stress onto the fact that (13) does not give a
sharp asymptotics on the singularity formation, and in particular the question of
the possible asymptotic stability in renormalized variables in open. In fact, we will
construct a rough aproximate self similar profile b → Qb and show that the solution
decomposes into
1
x − x(t) iγ(t)
u(t, x) =
[Q
+
ε]
e
b(t)
2
λ(t)
λ p−1 (t)
with −λt λ ∼ b(t) and |ε(t)|Ḣ 1 1. The key will be to prove a dynamical trapping
on the projection parameter b(t) and a uniform control on the radiation ε:
∀t ∈ [0, T ), (1 − δ(p))b∗ (p) ≤ b(t) ≤ (1 + δ(p))b∗ (p) and |ε(t)|Ḣ 1 1,
but this does not exclude possible oscillations of both b(t) and ε(t). In some
sense, this confirms the analysis in [10], [11], see also Rodnianski, Sterbenz [21]
and Raphael, Rodnianski, Sterbenz [20], where the key observation was that is is
not necessary to obtain a complete description of the dipsersive strucutre of the
problem in renormalized variables to prove finite time blow up. The main difference
however with these works is that we are here in a situation where there holds no a
priori orbital stability bound on ε(t), and a spectacular feature is that rough profiles
are enough to capture the self similar blow up speed. This somehow confirms the
robustness of the log-log analysis developped in [11], [12], [13].
2. On the behavior of the critical norm: In [15], Merle and Raphael showed in
the range of parameters N ≥ 3, 0 < σc < 1, that any radially symmetric finite time
blow up solution in H 1 must leave the critical space at blow up time with a lower
estimate:
|u(t)|Ḣ σc ≥ |log(T − t)|α(p) as t → T.
(19)
The self similar solutions constructed from Theorem 2 satisfy a logarithmic upper
bound –see Remark 8–:
p
|u(t)|Ḣ σc . |log(T − t)| p+1 as t → T
(20)
6
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
which proves the sharpness in the logarithmic scale of the lower bound (19). A logarithmic lower bound could also be derived after some extra work. The singularity
(18) in fact sharpens for this specific class of initial data the divergence (20), and
is according to (18) the major difference between the L2 critical blow up where the
L2 conservation law forces the radiation to remain in the critical L2 space, and the
super critical blow up where the radiation leaves aymptotically Ḣ σc .
3. Weak blow up: On the contrary to the ring solutions which concentrate in L2 at
blow up time according to (8), the self similar solutions constructed from Theorem
2 do not concentrate mass according to (17), a situation which is referred to as weak
blow up, see [22]. A striking feature of the analysis is that the L2 conservation law,
even though below scaling, plays a fundamental role in the proof of the stabilization
of the self similar blow up. Eventually, let us say that the size of the self similar
solutions we construct measured in the critical space Ḣ σc is lower bounded but
arbirtrarily large.
1.5. Strategy of the proof. Let us briefly state the main steps of the proof of
Theorem 2.
step 1 Construction of an approximate self similar profile.
Let a small parameter b > 0 and recall that the equation for self similar profiles
(10) does not admit solutions in H 1 . The first step is to construct an approximate
solution Qb which is essentially compactly supported and satisfies an approximate
self similar equation:
∆Qb − Qb + ibΛQb + Qb |Qb |p−1 = Ψb + O(σc2 ),
see Proposition 2. This profile incorporates the leading order O(σc ) deformation
with respect to the L2 critical profiles constructed in [11], [13], while the error Ψb
is a far away localized error which is inherited from the space localization of the
profile to avoid the slowly decaying tails.
step 2 Dynamical trapping of b.
We now chose initial data such that on a short time, the solution admits a decomposition
1
u(t, x) =
λ
2
p−1
(Qb(t) + ε)(t,
(t)
x − x(t) iγ(t)
− π
)e
with |∇ε(t)|2L2 e b(t)
λ(t)
and aim at deriving a dynamical trapping for the parameter b(t) and the deformation
1
ε(t). More precisely, introducing the global rescaled time ds
dt = λ2 , the dynamical
system driving λ is
λs
−λλt = − ∼ b(t)
λ
and hence finite time blow up in the self similar regime will follow from
π
− b(t)
(1−c)
b(t) ∼ b0 > 0, |∇ε(t)|2L2 e
for some small constant c > 0 in the maximum time interval of existence, Proposition
3. In order to derive such dynamical controls, we run the log-log analysis developped
in [13] by keeping track of the leading order O(σc ) deformation. The outcome is the
derivation of two structural monotonicity formulae for the parameter b. The first
7
one is inherited from the local virial control first derived in [10] and roughly leads
to
π
σc + |∇ε|2L2 − e− b . bs ,
see Proposition 4. The positive term +σc in the above LHS is a non trivial supercritical effect and is a consequence of the structure of the self similar profiles in the
local range |y| ≤ 1. The second monotonicity formula is a consequence of the L2
conservation law and the control of the mass ejection phenomenon:
π
bs . σc − e− b ,
π
where the nonpositive term −e− b is computed from a flux computation in the far
away zone |y| >> 1 which is based on the presence of the slow decaying tails of self
similar solutions. The outcome is the derivation of the dynamical system for b:
π
bs ∼ σc − e− b
which traps b around the value
π
b ∼ b∗ with σc ∼ e− b∗ .
Note that this shows that the self similar blow up speed is derived from the constraints both on compact sets and at infinity where the dispersive mass ejection
process is submitted to the global constraint of the L2 conservation law. Note also
that the L2 critical log-log law corresponds to the dynamical system
π
bs ∼ −e− b
and hence the supercritical self similar blow up appears as directly branching from
the L2 critical degenerate log-log blow up. The nontrivial pointwise control on ε
π
(1−c)
− b(t)
|∇ε(t)|2L2 e
will also follow from the obtained Lyapounov controls. Here a difficulty will occur
with respect to the L2 critical case to control the nonlinear terms in ε due in particular to the unboundedness of the scaling invariant critical Sobolev norm. A new
strategy inspired from [19] is derived which relies on the control of Sobolev norms
strictly above scaling in the self similar regime, see section 4.1.
The conclusions of Theorem 2 are now a simple consequence of this dynamical trapping of the solution.
This paper is organized as follows. In section 2, we construct approximate self
similar solutions, Proposition 2. We then describe the set of initial data leading to
self similar blow up, Definition 1, and set up the bootstrap argument, Proposition
3. In section 3, we derive the key dynamical controls and the two monotonicity
formulae, Proposition 4 and Proposition 5. In section 4, we close the bootstrap
argument as a consequence of the obtained Lyapounov type controls and conclude
the proof of Theorem 2.
Acknowledgements. F.M. is supported by ANR Projet Blanc OndeNonLin. P.R
and J.S are supported by ANR jeunes chercheurs SWAP.
σ f (ξ) = |ξ|σ fb(ξ). We let Q be
d
Notations We let Dσ be the Fourier multiplier D
p
the unique radially symmetric nonnegative solution in H 1 to
∆Qp − Qp + Qp+1
= 0.
p
8
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
We introduce the error to L2 criticality:
pc = 1 +
4
N
2
N (p − pc )
, σc =
−
=
,
N
2
p−1
2(p − 1)
(21)
where 0 < σc < 1 is the Sobolev critical exponent. We introduce the associated
scaling generators:
Λf =
N
2
f + y · ∇f, Df = f + y · ∇f = Λf + σc f.
p−1
2
We denote the L2 (RN ) scalar product
Z
(f, g) =
(22)
f (x)g(x)dx
RN
and observe the integration by parts formula:
(Df, g) = −(f, Dg), (Λf, g) = −(f, Λg + 2σc g).
(23)
We let L = (L+ , L− ) be the linearized operator close to Qp :
p−1
L+ = −∆ + 1 − pQp−1
p , L− = −∆ + 1 − Qp .
(24)
2. Description of the blow up set of initial data
This section is devoted to the description of the open H 1 set O of initial data
leading to the self similar blow up solutions described by Theorem 2 which relies on
the construction of approximate self similar profiles.
2.1. Construction of approximate self similar solutions. Our aim in this
section is to construct suitable approximate solutions to the self similar equation
(10). Let us make the following general ansatz:
1
u(t, x) =
λ
2
p−1
v(t,
(t)
x − x(t) iγ(t)
)e
λ(t)
and introduce the rescaled time
ds
1
= 2 ,
dt
λ (t)
then u is a solution to (1) iff v solves:
i∂s v + ∆v − v − i
λs
xs
Λv + v|v|p−1 = (γs − 1)v + i · ∇v.
λ
λ
Let us fix
γs = 1, xs = 0, −
λs
= b(s)
λ
and look for solutions of the form:
v(s, y) = Qb(s) (y)
where the unkowns are the mappings
s → b(s),
b → Qb .
This corresponds a to a slow variable formulation of a generalized self similar equation which goes back to previous formal works, see [22], and was first rigorously
used in [11], see also [8], [20] for related transformations in different settings. To
prepare the computation, we let
bs = σc µb
9
so that the generalized self similar equation becomes:
∂Qb
+ ∆Qb − Qb + ibΛQb + Qb |Qb |p−1 = 0.
∂b
Let us perform the conformal change of variables
iσc µb
P b = Qb e −
ib|y|2
4
(25)
,
then a simple algebra leads to:
∂Pb
1
+ ∆Pb − Pb − iσc Pb + (b2 + σc µb )|y|2 Pb + Pb |Pb |p−1 = 0.
(26)
∂b
4
Our aim is to find µb so as to be able to construct an approximate solution to (26)
with an error of order formally σc2 in the region |y| ≤ 1b . Our construction is elementary and relies on the computation of the first term in Taylor expansion of (µb , Qb )
in σc near the L2 critical value σc = 0.
iσc µb
Let a small parameter 0 < η << 1 to be fixed later, a non zero number b, and set
p
2p
1 − η and Rb− = 1 − ηRb .
(27)
Rb =
|b|
Denote BRb = {y ∈ RN , |y| < Rb } and ∂BRb = {y ∈ RN , |y| = Rb }. We introduce
a regular radially symmetric cut-off function
0 for |x| ≥ Rb ,
0 ≤ φb (x) ≤ 1,
(28)
φb (x) =
1 for |x| ≤ Rb− ,
such that:
|φ0b |L∞ + |∆φb |L∞ → 0 as |b| → 0.
(29)
kf (k) (r)k
We also consider the norm on radial functions kf kC j = max0≤k≤j
L∞ (R+ ) .
Let us start with the construction of the profile for σc = 0 where we view σc and p
as independent parameters and leave p supercritical.
(0)
Proposition 1 (Qb profiles). There exists p∗ > pc and C, η ∗ > 0 such that for all
pc ≤ p < p∗ , for all 0 < η < η ∗ , there exists b∗ (η), ε∗ (η) > 0 going to zero as η → 0
such that for all |b| ≤ b∗ (η), the following holds true:
(0)
(i) Existence of a unique Pb profile: there exists a unique radial solution Pb to

(0) p+1
(0)
(0)
b2 |y|2 (0)

= 0,
 ∆Pb − Pb + 4 Pb + (Pb )
(0)
(30)
Pb > 0 in BRb ,

 (0)
(0)
∗
∗
Pb (0) ∈ (Qp (0) − ε (η), Qp (0) + ε (η)), Pb (Rb ) = 0.
Moreover, let
(0)
(0)
P̃b (r) = Pb (r)φb (r),
then
(0)
P̃b
is twice differentiable with respect to b2 with uniform estimate:
(1−η)
θ(|b|r)
|b|
ke
(0)
(P̃b
(1−η)
− Qp )kC 3 + ke
ke
(1−η)
θ(|b|r)
|b|
θ(|b|r)
|b|
(31)
(0)
(
∂ P̃b
− ρ)kC 2 → 0 as b → 0,
∂b2
(32)
(0)
∂ 2 P̃b
k 3 ≤C
∂ 2 (b2 ) C
(33)
where
Z
w
r
1−
θ(w) = 10≤w≤2
0
z2
θ(2)
dz + 1w>2
w,
4
2
(34)
10
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
1
and ρ is the unique solution in Hrad
to
1
L+ ρ = |y|2 Q.
4
(0)
(ii) Properties of the Q̃b
(0)
(0)
(0)
profile: Q̃b = eib
(0)
(0)
|y|2
4
(35)
(0)
P̃b
satisfies:
(0)
(0)
(0)
∆Q̃b − Q̃b + ibDQ̃b + Q̃b |Q̃b |p−1 = −Ψb = −Ψ̃b eib
(0)
(0)
4
1+ N
(0)
−Ψ̃b = 2∇φb · ∇Pb + Pb (∆φb ) + (φb
and for any polynomial f (y) and integer k = 0, 1,
|y|2
4
(0)
− φb )(Pb )p
(36)
(37)
(0)
|f (y)
C
∂ Ψ̃b
− P
|L∞ ≤ e |b| .
k
∂y
(iii) Computation of the momentum and the L2 mass: there holds
Z
Z
b
(0) (0)
(0) (0)
(0)
= 0, Im
y · ∇Q̃b Q̃b
= − |y Q̃b |22
Im
∇Q̃b Q̃b
2
and the supercritical mass property:
Z
d2
(0) 2
|Q̃b |
= c0 (p) with c0 (p) → c0 (pc ) > 0 as p → pc .
db2
|b=0
(38)
(39)
The proof of Proposition 1 is parallel to the one of Proposition in [11] and Proposition [12] and relies on standard elliptic techniques and the knowledge of the kernel
of the linearized operator close to Q, explicitely:
Ker(L+ ) = span(∇Q), Ker(L− ) = span(Q),
(40)
see [23], [2], and the fact that we are working here before the turning point 2b and
hence with uniformly elliptic operators. The detailed proof is left to the reader.
Remark 1. The question of the value of the energy of the modified profile is an
important issue. It will indeed be computed in Proposition 2 for the full approximate
profile, see (57).
After the turning point 2b , leading order phenomenons are of linear type and a
natural prolongation of the approximate blow up profile is given by the so called
outgoing radiation.
Lemma 1 (Linear outgoing radiation). See Lemma 15 in [12]. There exist universal
constants C > 0 and η ∗ > 0 such that ∀0 < η < η ∗ , there exists b∗ (η) > 0 such that
(0)
∀0 < b < b∗ (η), the following holds true: let Ψb be given by (36), there exists a
unique radial solution ζb to
∆ζ
R b − ζ2b + ibDζb = Ψb
(41)
|∇ζb | < +∞.
Moreover, let θ be given by (34), and consider
Γb =
lim |y|N |ζb (y)|2 ,
then there holds:
N
2
|y| (|ζb | + |y||∇(ζb )|)
L∞ (|y|≥R
Z
(42)
|y|→+∞
1
b)
≤ Γb2
|∇ζb |2 ≤ Γb1−Cη .
−Cη
< +∞,
(43)
(44)
11
For |y| large, we have more precisely:
∀|y| ≥ Rb2 , e−2(1−Cη)
θ(2)
b
θ(2)
4
≥ |y|N |ζb (y)|2 ≥ Γb ≥ e−2(1+Cη) b ,
5
(45)
1
∀|y| ≥
Rb2 ,
|∇ζ(y)| ≤
Γb2
.
|b|
C
(46)
N
|y|1+ 2
For |y| small, we have: ∀σ ∈ (0, 5), ∃η ∗∗ (σ) such that ∀0 < η < η ∗∗ (σ), ∃b∗∗ (η)
such that ∀0 < b < b∗∗ (η), there holds:
1
θ(b|y|) +1σ
≤ Γb2 10 .
(47)
ζb (y)e−σ b 2
C (|y|≤Rb )
Last, ζb is differentiable with respect to b with estimate
1
∂ζb ≤ Γ 2 −Cη .
b
∂b 1
(48)
C
Remark 2. Recall from (34) that θ(2) = π2 . Moroever it is enough for our analsyis
to compute the radiation with the L2 scaling generator D in (41) instead of Λ as the
error will generate lower order terms.
The proof of Lemma 1 is completely similar to the one of Lemma 15 in [12] -see
also Appendix A in [13]- and hence left to the reader. In particular, the nondegen(0)
eracy (45) is a consequence of the nonlinear construction of the profile P̃b .
(0)
One should think of Q̃b as being an approximate solution to the generalized self
similar equation (25) with µb = 0 -self similar law- and an error of order σc in the
elliptic zone |y| ≤ 1b . We now claim that there exists a -locally unique- non trivial
µb (p) > 0 which allows one to construct an approximate solution to the generalized
self similar equation (25) of order O(σc2 + Γ2b ) in the elliptic zone |y| ≤ 2b :
Proposition 2 (Approximate generalized self similar profiles). There exists p∗ > pc
and C, c1 , η ∗ > 0 such that for all pc < p < p∗ and 0 < η < η ∗ , there exists c0 (p) > 0,
b∗ (η) > 0 such that for all |b| < b∗ (η), the following holds true:
(i) Construction of the modified profile: there exist µb = µ(b, p) > 0 and a radially
symmetric complex valued function Tb = T (b, p) with
θ(|b|r)
θ(|b|r)
∂µb (1−η) |b| ∂Tb (r)
(1−η) |b|
≤ C as b → 0,
Tb (r)kC 3 + ke
kC 2 + (49)
ke
∂b
∂b 8|Qp |2L2
as b → 0
(1 + 2σc )|yQp |2L2
with the following properties. Let
µb →
(0)
Pb = P̃b
+ σc Tb , Qb = Pb ei
b|y|2
4
(50)
,
then
Ψb = −iσc µb
(0)
with Ψb
∂Qb
(0)
(1)
− ∆Qb + Qb − ibΛQb − Qb |Qb |p−1 = Ψb + Ψb
∂b
(51)
given by (36) and for k = 0, 1:
ke
(1−η)
ke
θ(|b|r)
|b|
(1−η)
(1)
∂ k Ψb
1+c1
k ∞
,
− ≤ σc
∂y k L (|y|≤Rb )
θ(|b|r)
|b|
(52)
(1)
∂ k Ψb
k ∞
− ≤ Cσc .
∂y k L (|y|≥Rb )
(53)
12
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
(ii) Estimate of the invariants on Qb : there holds
Z
Im
∇Qb Qb = 0,
Z
Im
Z
(54)
b
y · ∇Qb Qb = − |yQ|22 (1 + O(|b| + σc )) as b → 0,
2
Z
|Qb |2 = |Qp |2 + c0 (p)(1 + o(1))b2 + O(σc ) as b → 0.
(55)
(56)
and the degeneracy of the Hamiltonian:
|E(Qb )| ≤ Γb1−Cη + Cσc .
(57)
Proof of Proposition 2
The proof relies on a Taylor expansion in σc of formal solutions to (25). The
choice of µb is dictated by the presence of a non trivial kernel for the operator L−
driving the imaginary part of (25) near Q as given by (40), while L+ on the real
part is invertible in the radial sector. The computation of the invariants and in
particular the energy degeneracy (57) follow from Pohozaev.
step 1 Construction of Tb .
Let Qb = Pb eib
−Ψ̃b = iσc µb
(0)
Let Ψ̃b
|y|2
4
and Ψb given by (51) , then: Ψb = Ψ̃b eib
|y|2
4
with:
∂Pb
1
+ ∆Pb − Pb − iσc bPb + (b2 + σc µb )|y|2 Pb + Pb |Pb |p−1 = 0. (58)
∂b
4
be given by (37), equivalently:
(0)
(0)
−Ψ̃b = ∆P̃b
(0)
We expand Pb = P̃b
compute:
(0)
− P̃b
+
b2 2 (0)
(0)
(0)
|y| P̃b + P̃b |P̃b |p−1 .
4
(59)
+ σc Tb . Let φb be the cut off function given by (28). We
∂Tb
1
1
− iσc2 bTb + σc2 µb |y|2 Tb + σc b2 (1 − φb )|y|2 Tb
∂b
4
4
(0)
(0)
(0)
(0)
(0)
+ (P̃b + σc Tb )|P̃b + σc Tb |p−1 − (P̃b )p − σc p(P̃b )p−1 Re(Tb ) − iσc (P̃b )p−1 Im(Tb )
"
#
(0)
h
∂ P̃b
µb 2 (0) i
(0)
+ σc −(Lb )+ Re(Tb ) + |y| P̃b
+ iσc −(Lb )− Im(Tb ) + µb
− bP̃b
(60)
4
∂b
(0)
−Ψ̃b = −Ψ̃b + iσc2 µb
(0)
where we introduced the linearized operators close to P̃b :
b2
b2
(0)
(0)
φb |y|2 − p(P̃b )p−1 , (L+ )b = −∆ + 1 − φb |y|2 − (P̃b )p−1 .
4
4
We thus aim at finding (µb , Tb ) so as to cancel the O(σc ) in the RHS of (60):
(L+ )b = −∆ + 1 −
(0)
∂ P̃
µb 2 (0)
(0)
|y| P̃b = 0, −(Lb )− Im(Tb ) + µb b − bP̃b = 0. (61)
4
∂b
Recall that in the limit b → 0, (L+ )0 = L+ is an elliptic invertible operator
in the radial sector and (L− )0 = L− is definite positive on (Span(Q))⊥ with
Ker(L− ) = Span(Q). The following lemma is a standard consequence of the LaxMilgram theorem and the perturbative theory of uniformly elliptic Schrodinger operators, and its proof is left to the reader:
−(Lb )+ Re(Tb ) +
13
Lemma 2 (Invertibility of (L+ )b , (L− )b ). Given η > 0 small, there exists b∗ (η) > 0
such that for all |b| < b∗ (η), the operator (L+ )b is invertible in the radial sector with
(1−η)
ke
θ(|b|r)
|b|
(L+ )−1
b f kH 2 . Cη ke
(1−Cη)
θ(|b|r)
|b|
f kL2 .
(62)
Moreover, (L− )b admits a lowest eigenvalue λb with eigenvector ξb which are C 1
functions of b with:
(1−η)
|λb | + ke
θ(|b|r)
|b|
(ξb − Q)kH 1 → 0 as b → 0
(63)
θ(|b|r)
∂λb (1−η) |b| ∂ξb
kL∞ ≤ C as b → 0.
∂b + ke
∂b
(64)
and
Moreover,
(1−η)
∀f ∈ (Span(ξb ))⊥ , ke
θ(|b|r)
|b|
(1−Cη)
(L− )−1
b f kH 2 . Cη ke
θ(|b|r)
|b|
f kL2 .
(65)
From (65), the solvability of the second equation in (61) imposes the choice of
µb :
!
(0)
(0)
(bP̃b , ξb )
∂ P̃b
(0)
− bP̃b , ξb = 0 ie µb =
µb
.
(0)
∂b
∂ P̃b
( ∂b , ξb )
We now observe the crucial non degeneracy of µb as b → 0 from (32), (63) and (35):
(0)
µb =
(bP̃b , ξb )
(0)
(
∂ P̃b
∂b
→
, ξb )
8|Qp |2L2
|Qp |2L2
=
as b → 0,
2(ρ, Qp )
(1 + 2σc )|yQp |2L2
(66)
where we used (32) and the computation from (35) and L+ (ΛQp ) = −2Qp :
2(ρ, Qp ) = −(L+ ρ, ΛQp ) = −(
|y|2
2
1 + 2σc
,
Qp + y · ∇Qp ) =
|yQp |2L2 .
4 p−1
8
The non degeneracy of the denominator as b → 0 and the uniform differentiability
properties (33), (64) ensure that µb is a C 1 function of b with
∂µb (67)
∂b ≤ C as b → 0.
Hence from (62), (65) and the uniform bounds (32), (66), (67) we may find Tb solution to (61) which is a C 1 function of b satisfying the uniform bounds (49).
step 2 Estimate on the error.
We now turn to the proof of the estimate of the error (52) which amounts estimating the remaining terms in the RHS of (60). The nonlinear term is estimated
thanks to the homogeneity estimate:
(0)
(0)
(0) p
(0) p−1
(0) p−1
p−1
(
P̃
+
σ
T
)|
P̃
+
σ
T
|
−
(
P̃
)
−
σ
p(
P̃
)
Re(T
)
−
iσ
(
P̃
)
Im(T
)
b
c b
c b
c
c b
b
b b
b
b
p
σc |Tp |p for p < 2,
.
(0)
σcp |Tp |p + σc2 |P̃b |p−2 |Tp |2 for p ≥ 2,
and (49) now yields (52), (53).
step 3 Computation of the invariants.
14
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
(54) still holds because Qb is radially symmetric. (55) and (56) follow from (38),
|y|2
(0)
(39) and the decomposition Qb = Q̃b + σc Tb eib 4 . To compute the energy, we use
the Pohozaev multiplier ΛQb on (51). We first integrate by parts to get the general
formula:
Z
Re(∆Qb − Qb + ibΛQb + Qb |Qb |p−1 , ΛQb ) = −2E(Qb ) + σc (2E(Qb ) + |Qb |2 ),
and hence from (51):
2E(Qb ) − Re(ΛQb , Ψb + iσc µb
∂Qb
)
∂b
∂Θ
∂Σ
= 2E(Qb ) − (Re(Ψb ), ΛΣ) − (Im(Ψb ), ΛΘ) + σc µb [(
, ΛΣ) − (
, ΛΘ)]
∂b
∂b
Z
= σc (2E(Qb ) + |Qb |2 ).
(68)
This together with (56) and the estimates on Ψb (52) yields the degeneracy (57).
This concludes the proof of Proposition 2.
2.2. Setting of the bootstrap. We are now on position to describe the set of initial data O leading to the self similar blow up and to set up the bootstrap argument.
Pick a Sobolev exponent
N
1
})
σ ∈ (σc , min{ ,
2 N +2
(69)
independent of p and close enough to 0.
Definition 1 (Geometrical description of the set O). Pick a number ν0 > 0 small
enough. Then for p ∈ (pc , p∗ (ν0 )) with p∗ (ν0 ) close enough to pc , we let O be the
set of initial data u0 ∈ H 1 of form:
1
x − x0
u0 (x) =
eiγ0
2 (Qb0 + ε0 )
λ
0
λ0p−1
for some (λ0 .b0 , x0 , γ0 ) ∈ R∗+ × R∗+ × RN × R with the following controls:
(i) b0 is in the self similar asymptotics (11):
1−ν0
0
Γ1+ν
b(0) ≤ σc ≤ Γb(0) ;
(70)
(ii) Smallness of the scaling parameter:
0 < λ0 ≤ Γ100
b0 ;
(iii) Degeneracy of the energy and the momentum:
Z
2(1−σc )
1−2sc 50
λ0
|E0 | + λ0
Im( ∇ψ · ∇u(0)u(0)) < Γb0 ;
(iv) Ḣ 1 ∩ Ḣ σ smallness of the excess of L2 mass:
Z
Z
Z
0
|Dσ ε0 |2 + |∇ε0 |2 +
|ε0 |2 e−|y| ≤ Γ1−ν
.
b0
(71)
(72)
(73)
|y|≤ 100
b
0
Remark 3. Observe that the set O is non empty open set in H 1 . Indeed, pick a
small paramater ν0 and p close enough to pc . Pick b0 > 0 such that (70) holds, and
pick then λ0 > 0 such that (71) holds. Let f (y) be smooth, radial and compactly
15
supported in the ball |y| ≤ 1 and such that (f, Q) = 1, and let ε0 = µ0 f with µ0 to
be chosen. From
d
E(Qb0 + µ0 f )|µ0 =0 = −(f, Q)(1 + o(1)) = −1 + o(1) as b0 → 0,
dµ0
0
) such that
and the degeneracy of the Hamiltonian (57), we may find µ0 = O(Γ1−2ν
b0
100
|E(Qb0 + ε0 | . Γb0 so that (72) holds. (73) now follows from the size of µ0 .
Let now u0 ∈ O and u(t) be the corresponding solution to (1) with maximum
life time interval [0, T ), 0 < T ≤ +∞. Using the regularity u ∈ C([0, T ), H 1 ) and
standard modulation theory, we can find a small interval [0, T ∗ ) such that for all
t ∈ [0, T ∗ ), u(t) admits a unique geometrical decomposition
1
u(t, x) =
λ
2
p−1
(Qb(t) + ε)(t,
(t)
x − x(t) iγ(t)
)e
λ(t)
(74)
where uniqueness follows from the freezing of orthogonality conditions: ∀t ∈ [0, T ∗ ],
ε1 (t), |y|2 Σ + ε2 (t), |y|2 Θ = 0,
(75)
(ε1 (t), yΣ) + (ε2 (t), yΘ) = 0,
ε2 (t), Λ2 Σ − ε1 (t), Λ2 Θ = 0,
(76)
(ε2 (t), ΛΣ) − (ε1 (t), ΛΘ) = 0,
(78)
(77)
where we have denoted:
ε = ε1 + iε2 , Qb = Σ + iΘ
in terms of real and imaginary parts. See [10], [11] for related statements. Moroever,
the parameters (λ(t), b(t), x(t), γ(t)) ∈ R∗+ × R∗+ × RN × R are C 1 functions of time
and ε ∈ C([0, T ∗ ), H 1 ) with a priori bounds: ∀t ∈ [0, T ∗ ),
0
0
Γ1+5ν
≤ σc ≤ Γ1−5ν
b(t)
b(t) ,
(79)
0 < λ(t) ≤ Γ10
b(t) ,
Z
2(1−σc )
1−2sc < Γ10 ,
u
)
[λ(t)]
|E0 | + [λ(t)]
Im(
∇ψ
·
∇u
0
0
b(t)
(80)
Z
Z
2
|∇ε(t)| +
100
|y|≤ b(t)
Z
1−20ν0
|ε(t)|2 e−|y| < Γb(t)
,
0
|Dσ ε(t)|2 ≤ Γb1−50ν
.
t)
(
(81)
(82)
(83)
Remark 4. The strict Ḣ 1 subcriticality of the problem implies:
σc =
2
N
N
N
−
<
−
.
2
p−1
2
p+1
Hence from Sobolev embedding, the Ḣ σ control (83) ensures for σ close enough to
σc :
Z
h
i p+1
2
1−C(p)ν0
0
|ε|p+1 . |ε|p+1
≤
Γ
. Γ1+z
(84)
N− N
b
b(s0 )
Ḣ
2
p+1
for some universal constant z0 > 0 independent of p for p close enough to pc .
Our main claim is that the above regime is a trapped regime:
16
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
Proposition 3 (Bootstrap). There holds: ∀t ∈ [0, T ∗ ),
0
0
≤ σc ≤ Γ1−3ν
Γ1+3ν
b(t) ,
b(t)
(85)
0 < λ(t) ≤ Γ20
b(t) ,
Z
2(1−σc )
1−2sc < Γ20 ,
[λ(t)]
|E0 | + [λ(t)]
Im(
∇ψ
·
∇u
u
)
0
0
b(t)
Z
Z
1−10ν0
,
|ε(t)|2 e−|y| < Γb(t)
|∇ε(t)|2 +
(86)
(87)
(88)
100
|y|≤ b(t)
Z
0
|Dσ ε(t)|2 ≤ Γ1−20ν
b t)
(
(89)
and hence
T ∗ = T.
The next section is devoted to the derivation of the key dynamical controls at
the heart of the proof of the bootstrap Proposition 3 which is proved in section 4.
Theorem 2 will be a simple consequence of Proposition 3.
3. Control of the self similar dynamics
In this section, we exhibit the two Lyapounov type functionals which will allow
us to lock the self similar dynamics and prove the bootstrap Proposition 3. The
key is the dynamical lock (85) of the geometrical parameter b(t) which controls the
selfsimilarity of blow up from the modulation equation
λs
b ∼ − = −λλt .
λ
This corresponds to an upper bound and a lower bound of b. The proof will follow by somehow bifurcating from the log-log analysis and by tracking the effect
of the leading order σc deformation in the Qb profile. The lower bound on b is
a consequence of the local viriel estimate type of control first derived in [11], see
Proposition 4, the upper bound follows from the sharp log-log analysis derived in
[13] and uses very strongly the L2 conservation law, see Proposition 5. Remember
also that we have no a priori orbital stablity bound on neither b or ε, and indeed
the upper control (88) on ε requires both monotonicity properties, see step 1 of the
proof of Proposition 3 in section 4.1.
3.1. Preliminary estimates on the decomposition. Let us recall the geometrical decomposition (74):
u(t, x) =
1
2
p−1
(Qb(t) + ε)(t,
x − x(t) iγ(t)
)e
λ(t)
λ (t)
and derive the modulation equations on the geometrical parameters and preliminary
estimates inehrited from the conservation laws. We let the rescaled time
Z t
dτ
s(t) =
, s∗ = s(T ∗ ) ∈ (0, +∞],
2
0 λ (τ )
and compute the equation of ε in terms of real and imaginary parts on [0, s∗ ):
∂Σ
λs
xs
(bs − σc µb )
+ ∂s ε1 − M− (ε) + bΛε1 =
+ b ΛΣ + γ̃s Θ +
· ∇Σ
∂b
λ
λ
λs
xs
+
+ b Λε1 + γ̃s ε2 +
· ∇ε1
λ
λ
+ Im(Ψb ) − R2 (ε)
(90)
17
(bs − σc µb )
∂Θ
+ ∂s ε2 + M+ (ε) + bΛε2
∂b
λs
xs
=
+ b ΛΘ − γ̃s Σ +
· ∇Θ
λ
λ
λs
xs
+
+ b Λε2 − γ̃s ε1 +
· ∇ε2
λ
λ
− Re(Ψb ) + R1 (ε),
(91)
with γ̃(s) = −s − γ(s) and Ψb given by (51). Here we also denoted M = (M+ , M− )
the linear operator close to Qb , explicitely:
Σ2
|Qb |p−1 ε1 − (p − 1)ΣΘ|Qb |p−3 ε2 ,
M+ (ε) = −∆ε1 + ε1 − 1 + (p − 1)
|Qb |2
Θ2
|Qb |p−1 ε2 − (p − 1)ΣΘ|Qb |p−3 ε1 .
M− (ε) = −∆ε2 + ε2 − 1 + (p − 1)
|Qb |2
The non linear interaction terms are explicitly:
R1 (ε) = (ε1 + Σ)|ε + Qb |p−1 − Σ|Qb |p−1
Σ2
−
1 + (p − 1)
|Qb |p−1 ε1 − (p − 1)ΣΘ|Qb |p−3 ε2 ,
|Qb |2
(92)
R2 (ε) = (ε2 + Θ)|ε + Qb |p−1 − Θ|Qb |p−1
Θ2
−
1 + (p − 1)
|Qb |p−1 ε2 − (p − 1)ΣΘ|Qb |p−3 ε1 .
|Qb |2
(93)
We now claim the following preliminary estimates on the decomposition:
Lemma 3. There holds for some universal constants C > 0, δ(p) → 0 as p → pc
and for all s ∈ [0.s∗ ):
(i)Estimates induced by the conservation of energy and momentum:
Z
Z
2
0
|2(ε1 , Σ) + 2(ε2 , Θ)| ≤ C( |∇ε| + |ε|2 e−|y| ) + Γ1−11ν
(94)
b
Z
Z
1
0
|(ε2 , ∇Σ)| ≤ δ(p)( |∇ε|2 + |ε|2 e−|y| ) 2 + Γ1−50ν
.
b
(95)
(ii) Estimates on the modulation parameters:
Z
Z
λs
2
2 −1|y|
0
| + b| + |bs | ≤ C
|∇ε| + |ε| e
+ Γ1−11ν
,
(96)
b
λ
Z
Z
θ(b|y|)
γ̃s − 1 (ε1 , L+ D2 Q) + xs ≤ δ(p)( |∇ε|2 e−2(1−η) b + |ε|2 e−|y| ) 12
λ
|DQ|22
Z
0
+ C |∇ε|2 + Γ1−11ν
.
(97)
b
Remark 5. In all what follows, we will use the fact that given a universal constant
C > 0, we have the control
0
Γb1−Cη ≤ Γ1−ν
b
provided η > 0 has been chosen small enough with respect to ν0 . We may for example
take
η = ν0100 .
Now such kind of controls arise from the construction of Qb in Proposition 1 and
hence pushing η → 0 requires pushing b → 0 or equivalently p → pc from (70). This
is how the asymptotics (12) follows.
18
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
Proof of Lemma 3
It relies as in [13] on the expansion of the conservation laws, the choice of orthogonality conditions for ε and the bootstrapped controls (70), (80), (81), (82), (83).
step1 Expansion of the conservation laws.
(94) and (95) follow from the expansion of the momentum and the energy using
the decomposition (74) and the estimates of Proposition 2.
For the momentum:
Z
Z
2Im(ε, ∇Qb ) = Im ∇εε − λ1−σc Im( ∇u0 u0 )
from which using (81):
Z
Z
1
2
|(ε2 , ∇Q)| ≤ δ(p)( |∇ε| + |ε|2 e−|y| ) 2 + C|ε|2 1 + Γ10
b
Ḣ 2
Z
Z
1
≤ δ(p)( |∇ε|2 + |ε|2 e−|y| ) 2 + Γb1−50ν0
1
where we interpolated Ḣ 2 between Ḣ σ and Ḣ 1 and used (82), (83).
For the energy:
2(ε1 , Σ + bΛΘ − Re(Ψb ) + σc µb
∂Θ
∂Σ
) + 2(ε2 , Θ − bΛS − Im(Ψb ) − σc µb
)
∂b
∂b
Z
= −2λ2(1−σc ) E0 + 2E(Qb ) + (M+ (ε), ε1 ) + (M− (ε), ε2 ) −
Z
2
−
F (ε)
p+1
|ε|2
(98)
where F (ε) is the formally cubic part of the potential energy:
F (ε) = |Qb + ε|p+1 − |Qb |p+1 − (p + 1)Re(ε, Qb |Qb |p−1 )
Θ2
Σ2
p−1 2
|Q
|
ε
−
1
+
(p
−
1)
|Qb |p−1 ε22
−
1 + (p − 1)
b
1
|Qb |2
|Qb |2
− 2(p − 1)ΣΘ|Qb |p−3 ε2 ε1 ,
(99)
We then used standard homegeneous estimates and Sobolev embeddings similarily
like in [10], [11] to estimate the nonlinear term F (ε). Indeed, pick 0 < µ < p − 1,
then by homogeneity:
|F (ε)| . |ε|p+1 + |Qb |p−1µ |ε|2+µ ,
1
and thus from Holder with 2+µ
= α2 + 1−α
p+1 and the bootstrap controls (82), (84):
Z
Z
α(2+µ) (1−α)(2+µ)
|F (ε)| .
|ε|p+1 + |εe−C|y| |L2
|ε|Lp+1
2+µ
1−20ν0
0
0
. Γ1+z
+
Γ
. Γ1+z
(100)
b
b
b
for some z0 > 0 independent of p. Injecting this into (98) together with the degeneracy estimate (57), the bootstrap bound (70) and the orthogonality condition (78)
yields (94).
Remark 6. Note that the algbebraic formula (98) does not use the orthogonality
conditions on ε.
19
step 2 Computation of the modulation parameters.
The estimates (96), (97) are obtained by computing the geometrical paramaters
from the choice of orthogonality conditions (75), (76), (77), (78). The computation
is the same like the one performed in [13] up to O(σc ) = O(p − pc ) terms which are
brutally estimated in absolute value using the bootstrap bound (70). The detail of
this is left to the reader.
This concludes the proof of Lemma 3.
3.2. Local viriel identity. We now proceed through the derivation of the local
viriel control. The corresponding monotonicity property was first discovered in [10]
and will yield a strict lower bound on b.
Proposition 4 (Local viriel identity). There holds for some universal constant
c1 > 0 the lower bound:
Z
Z
1−ν 2
(101)
∀s ∈ [0, s∗ ), bs ≥ c1 σc + |∇ε|2 + |ε|2 e−|y| − Γb 0 .
Remark 7. The super critical effect relies in the presence of the term σc in the
RHS of (101). The positive sign is crucial and structural and is the reason why we
had to push the construction of Qb to an order σc2 .
Proof of Proposition 4
step1 Algebraic derivation of the bs law.
The first step is a careful derivation of the modulation equations for b. For further use, we shall exhibit first a general formula which does not rely on the specific
choice of orthogonality conditions in ε. Once the formula is derived, the use of the
orthogonality conditions and suitable coercivity properties inherited from the Spectral Property stated in the introduction will yield the claim.
Take the inner product of (96) with (−ΛΘ) and (97) with ΛΣ and sum the
obtained identities using (23) to get:
∂Θ
∂Σ
∂Σ
∂Θ
bs (
, ΛΣ) − (
, ΛΘ) − (ε2 , Λ
) + (ε1 , Λ
) + {(ε2 , ΛΣ) − (ε1 , ΛΘ)}s =(102)
∂b
∂b
∂b
∂b
− (M+ (ε) + bΛε2 , ΛΣ) − (M− (ε) − bΛε1 , ΛΘ) − γ̃s {(ε1 , ΛΣ) + (ε2 , ΛΘ)}
xs
λs
−
+ b (ε2 , Λ2 Σ + 2σc ΛΣ) − (ε1 , Λ2 Θ + 2σc ΛΘ) −
· {(ε2 , ∇ΛΣ) − (ε1 , ∇ΛΘ)}
λ
λ
+ Re(ΛQb , Ψb + iσc µb
∂Qb
) − σc γ̃s |Qb |2L2 + (R1 (ε), ΛΣ) + (R2 (ε), ΛΘ).
∂b
We compute the linear term in ε in (102). For this, consider the Qb equation (51),
2
compute the equation satisfied by µ p−1 Q(µy) and differentiate the obtained identity
with respect to µ at µ = 1. This yields:
M+ (ΛQb ) + bΛ2 Θ = −2[Σ + bΛΘ − Re(Ψb ) + σc µb
∂Θ
∂Θ
] + Re(ΛΨb ) − σc µb Λ
,
∂b
∂b
M− (ΛQb ) − bΛ2 Σ = −2[Θ − bΛΣ − Im(Ψb ) − σc µb
∂Σ
∂Σ
] + Im(ΛΨb ) + σc µb Λ
.
∂b
∂b
20
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
Integrating by parts and using the conservation of the energy (98), we obtain the
following identity:
−(M+ (ε) + bΛε2 , ΛΣ) − (M− (ε) − bΛε1 , ΛΘ)
= −(ε1 , M+ (ΛQb ) − bΛ2 Σ) − (ε2 , M− (ΛQb ) − bΛ2 Θ)
∂Θ
∂Θ
= (ε1 , 2[Σ + bΛΘ − Re(Ψb ) + σc µb
] − Re(ΛΨb ) + σc µb Λ
)
∂b
∂b
∂Σ
∂Σ
+ (ε2 , 2[Θ − bΛΣ − Im(Ψb ) − σc µb
] − Im(ΛΨb ) − σc µb Λ
)
∂b
∂b
∂Θ
∂Σ
= −(ε1 , Re(ΛΨb ) − (ε2 , Im(ΛΨb )) + σc µb [(ε1 , Λ
) − (ε2 ,
)]
∂b
Z ∂b
− 2λ
2(1−σc )
E0 + 2E(Qb ) + (M+ (ε), ε1 ) + (M− (ε), ε2 ) −
2
|ε| −
p+1
2
Z
F (ε).
We now inject this into (102). A key here is to use the Pohozaev identity (68):
Z
∂Qb
2E(Qb ) − Re(ΛQb , Ψb + iσc µb
) = σc (2E(Qb ) + |Qb |2 )
∂b
to generate a nonnegative term in the RHS of (102):
∂Θ
∂Σ
∂Σ
∂Θ
, ΛΣ) − (
, ΛΘ) − (ε2 , Λ
) + (ε1 , Λ
) + {(ε2 , ΛΣ) − (ε1 , ΛΘ)}s =
bs (
∂b
∂b
∂b
∂b
σc (2E(Qb ) + |Qb |2L2 − γ̃s |Qb |2L2 ) − 2λ2(1−σc ) E0 − (ε1 , Re(ΛΨb ) − (ε2 , Im(ΛΨb ))
Z
+ (M+ (ε), ε1 ) + (M− (ε), ε2 ) − |ε|2 + (R1 (ε), ΛΣ) + (R2 (ε), ΛΘ)
(103)
xs
· {(ε2 , ∇ΛΣ) − (ε1 , ∇ΛΘ)}
− γ̃s {(ε1 , ΛΣ) + (ε2 , ΛΘ)} −
λ
Z
λs
2
−
+ b (ε2 , Λ2 Σ + 2σc ΛΣ) − (ε1 , Λ2 Θ + 2σc ΛΘ) −
F (ε).
λ
p+1
It remains to extract the formally cubic term in ε in the RHS (103). We let:
p−1
p−1
|Qb |p−5 (pΣ3 + 3ΣΘ2 )ε21 −
Qb |p−5 (Σ3 + (p − 2)ΣΘ2 )ε22
2
2
− (p − 1)Qb |p−5 (Θ3 + (p − 2)Σ2 Θ)ε1 ε2 ,
(104)
G1 (ε) = R1 (ε) −
p−1
p−1
|Qb |p−5 (Θ3 + 3ΘΣ2 )ε21 −
Qb |p−5 (pΘ3 + 3ΘΣ2 )ε22
2
2
− (p − 1)Qb |p−5 (Σ3 + (p − 2)Θ2 Σ)ε1 ε2 ,
(105)
G2 (ε) = R2 (ε) −
and eventually arrive at the following algbebraic viriel identity:
∂Θ
∂Σ
∂Σ
∂Θ
bs (
, ΛΣ) − (
, ΛΘ) − (ε2 , Λ
) + (ε1 , Λ
) + {(ε2 , ΛΣ) − (ε1 , ΛΘ)}s =
∂b
∂b
∂b
∂b
σc (2E(Qb ) + |Qb |2L2 − γ̃s |Qb |2L2 ) − 2λ2(1−σc ) E0 − (ε1 , Re(ΛΨb ) − (ε2 , Im(ΛΨb ))
+ Hp (ε, ε) − γ̃s {(ε1 , ΛΣ) + (ε2 , ΛΘ)}
(106)
xs
−
· {(ε2 , ∇ΛΣ) − (ε1 , ∇ΛΘ)} + E(ε, ε) + (G1 (ε), ΛΣ) + (G2 (ε), ΛΘ)
λ
Z
λs
2
−
+ b (ε2 , Λ2 Σ + 2σc ΛΣ) − (ε1 , Λ2 Θ + 2σc ΛΘ) −
F (ε).
λ
p+1
21
where the virial quadratic form Hp can be expressed in the form:
Z
Z
p(p − 1)
2
(σc Qp + y · ∇Qp )Qp−2 ε21
(107)
Hp (ε, ε) =
|∇ε| +
2
Z
(p − 1)
(σc Qp + y · ∇Qp )Qp−2 ε22
+
2
and the error E(ε, ε) simply comes from the error made by replacing Qb by Q in
the potential terms and is easily estimated thanks to Proposition 2 by:
Z
Z
2
2 −|y|
|E(ε, ε)| ≤ δ(p)
|∇ε| + |ε| e
(108)
with δ(p) → 0 as p → pc .
step 2 Estimates of nonlinear terms and coercivity of the quadratic form.
Observe from the construction of Qb that
∂Θ
∂Σ
∂Qb
, ΛΣ) − (
, ΛΘ) = Im(ΛQb ,
)
∂b
∂b
∂b
|yQpc |2L2
∂Pb
|y|2
|y|2
Pb ,
−i
Pb ) =
+ O(|b| + σc ). (109)
= Im(ΛPb − ib
2
∂b
4
4
We then inject the orthogonality conditions (75), (76), (77), (78) and (109) into
(106) to derive the algebraic identity:
(
|yQpc |2L2
(1 + δ(p))bs
(110)
4
= σc (2E(Qb ) + |Qb |2L2 − γ̃s |Qb |2L2 ) − 2λ2(1−σc ) E0 − (ε1 , Re(ΛΨb ) − (ε2 , Im(ΛΨb ))
xs
+ Hp (ε, ε) − γ̃s {(ε1 , ΛΣ) + (ε2 , ΛΘ)} −
· {(ε2 , ∇ΛΣ) − (ε1 , ∇ΛΘ)}
λ
Z
2
+ E(ε, ε) + (G1 (ε), ΛΣ) + (G2 (ε), ΛΘ) −
F (ε)
p+1
with δ(p) → 0 as p → pc . Let us now estimate all the terms in the RHS of (110).
First observe from (56), (57) and the smallness of ε the non degeneracy:
σc |Qpc |2L2
.
(111)
2
The nonlinear terms are estimated as in [11], [13] using homogeneity estimates,
Sobolev embeddings and the bootstrap bounds (82), (84) as for the proof of (100):
Z
E(ε, ε) + (G1 (ε), ΛΣ) + (G2 (ε), ΛΘ) − 2
F
(ε)
p+1
Z
Z
0
|∇ε|2 + |ε|2 e−|y| + Γ1+z
≤ δ(p)
.
b
σc (2E(Qb ) + |Qb |2L2 − γ̃s |Qb |2L2 ) ≥
We now focus onto the quadratic terms (110). We inject the estimates of the scaling
parameter (96), (97). Using the simple bound
c|y|
e [|Qp − Qpc | + |∇Qp − ∇Qpc |] ∞ → 0 as p → pc ,
L
we may view the obtained quadratic form together with Hp as a perturbation of the
one obtained in the L2 critical case:
xs
Hp (ε, ε) − γ̃s {(ε1 , ΛΣ) + (ε2 , ΛΘ)} −
· {(ε2 , ∇ΛΣ) − (ε1 , ∇ΛΘ)}
λ
= H̃(ε, ε) + E1 (ε, ε)
22
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
with
Z
H̃(ε, ε) =
−
Z
Z
4
4
2
2
4
−1
2
−1
N
|∇ε| + (1 + ) Q
y · ∇Qpc ε1 +
QpNc y · ∇Qpc ε21
N
N
N
1
(ε1 , L+ D2 Qpc )(ε1 , DQpc ),
|DQpc |2L2
Z
Z
2
2 −|y|
+ Γb2−30ν0 .
|E1 (ε, ε)| ≤ δ(p)
|∇ε| + |ε| e
2
We now recall from [12] the following coercivity property which is a consequence of
the spectral property stated in the introduction: ∀ε = ε1 + iε2 ∈ H 1 ,
Z
Z
2
2 −|y|
|∇ε| + |ε| e
H̃(ε, ε) ≥ c0
− (ε1 , Qpc )2 + (ε1 , |y|2 Qpc )2 + (ε1 , yQpc )2 + (ε2 , DQpc )2 + (ε2 , D2 Qpc )
for some universal constant c0 > 0, and hence our choice of orthogonality conditions,
(23) and the degeneracy estimates (94), (95), (81) ensure:
Z
Z
3
1
c0
2
2
2 −|y|
2
H̃(ε, ε) −
(ε
,
L
D
Q
)(ε
,
DQ
)
≥
|∇ε|
+
|ε|
e
−
Γ
1
+
p
1
p
c
c
b
2
|DQpc |2L2
(112)
for ν0 = ν0 (p) > 0 small enough. We now inject (109), the orthogonality condition
(78), the nondegeneracy estimate (111), the estimates on nonlinear terms (112) and
the coercivity property (112) into (110) to derive
Z
Z
c0
1−ν 2
2
2 −|y|
σc + |∇ε| + |ε| e
− Γb 0
bs ≥
4
− |(ε1 , Re(ΛΨb ) + (ε2 , Im(ΛΨb )|.
A crude bound for the remaining linear term is derived from (52):
Z
Z
c0
1−Cη
2
2 −|y|
|(ε1 , Re(ΛΨb ) + (ε2 , Im(ΛΨb )| ≤ Γb
+
σc + |∇ε| + |ε| e
, (113)
10
and (101) follows for η < η(ν0 ) chosen small enough.
This concludes the proof of Proposition 4.
3.3. Refined local viriel identity and introduction of the radiation. We
now proceed through a refinement of the local viriel estimate (101) and adapt the
analysis in [13].
We start with introducing a localized version of the radiation ζ introduced in Lemma
1 due to the non-L2 slowly decaying tail (46). Let a radial cut off function χA (r) =
χ Ar with χ(r) = 1 for 0 ≤ r ≤ 1 and χ(r) = 0 for r ≥ 2 with the choice:
θ(2)
A = A(t) = e
2a b(t)
− a2
so that Γb
− 3a
2
≤ A ≤ Γb
,
(114)
for some parameter a > 0 small enough to be chosen later and which depends on η.
Let
ζ̂b = χA ζb = ζ̂1 + iζ̂2 .
ζ̂b still satisfies size estimates of Lemma 1 and is moreover in H 1 with estimate:
2
∂ ζ̂b 10
2
10 ∂ ζ̂b
|(1 + |y|) (|ζ̂b | + |∇ζ̂b |)|L2 + (1 + |y|) (|
| + |∇
|) ≤ Γ1−Cη
.
(115)
b
∂b
∂b 2
L
From (41), the equation satisfied by ζ̂ is now:
(0)
∆ζ̂b − ζ̂b + ibDζ̂b = Ψb + F
23
with
F = (∆χA )ζb + 2∇χA · ∇ζb + iby · ∇χA ζb .
(116)
We then consider the new profile and dispersion:
Q̂b = Qb + ζ̂b , ε̂ = ε − ζ̂b
and claim the following refined local viriel estimate for ε̂:
Lemma 4 (Refined viriel estimate for ε̂). There holds for some universal constant
c2 > 0:
Z
2
|∇ε̂| +
{f1 (s)}s ≥ c2
Z
2 −|y|
|ε̂| e
+ Γb
1
−
c2
Z
σc +
2A
2
|ε|
,
(117)
A
with
f1 (s) = (Θ, ΛΣ̂) − (Σ, ΛΘ̂) + (ε2 , Λζ̂1 ) − (ε1 , Λζ̂2 ).
(118)
Proof of Lemma 4
The proof is parallel to the one of the local viriel estimate (101) up to the estimate of the remaining leading order liner term (113) for which will use a sharp flux
computation based on (45).
step 1 Estimate on Ψ̂b
Let
Ψ̂b = −iσc µb
∂ Q̂b
(2)
− ∆Q̂b + Q̂b − ibΛQ̂b − Q̂b |Q̂b |p−1 = Ψ̂b − F
∂b
(119)
with explicitely:
(2)
Ψ̂b = −iσc µb
i
h
∂ ζ̂b
(1)
+ ibσc ζb + Ψb − (Qb + ζ̂b )|Qb + ζ̂b |p−1 − Qb |Qb |p−1 ,
∂b
(1)
where Ψb is given by (51). Observe from (52), (53), (115) and the degeneracy on
compact sets (47) that:
(2)
(2)
|(1 + |y|)10 (|Ψ̂b | + |∇Ψ̂b |)|2L2 ≤ Γ1+c
+ Cσc2
b
(120)
for some universal constants c, C > 0.
step 2 Rerunning the local virial estimate bound.
We now turn to the proof of (117) and propose a small short cut with respect to
the analysis in [13]. Let us indeed rerun exactly step 1 of the proof of Proposition
4 with the new profile Q̂b and variable ε̂. Recall indeed that the whole algebra is
completely intrinsic to the equation of the profile and relies only on the definition
of respectively (51), (119). Moreover, the algebraic formula (110) does not rely on
24
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
the choice of orthogonality conditions for ε –which no longer hold for ε̂–, and hence:
!
n
o
∂ Θ̂
∂ Σ̂
∂ Σ̂
∂ Θ̂
bs (
, ΛΣ̂) − (
, ΛΘ̂) − (ε̂2 , Λ
) + (ε1 , Λ
) + (ε̂2 , ΛΣ̂) − (ε̂1 , ΛΘ̂) =
∂b
∂b
∂b
∂b
s
σc (2E(Q̂b ) + |Q̂b |2L2 − γ̃s |Q̂b |2L2 ) − 2λ2(1−σc ) E0 − (ε̂1 , Re(ΛΨ̂b ) − (ε̂2 , Im(ΛΨ̂b ))
n
o
+ Hp (ε̂, ε̂) − γ̃s (ε̂1 , ΛΣ̂) + (ε̂2 , ΛΘ̂)
(121)
n
o
xs
−
· (ε̂2 , ∇ΛΣ̂) − (ε̂1 , ∇ΛΘ̂) + E(ε̂, ε̂) + (G1 (ε̂), ΛΣ̂) + (G2 (ε̂), ΛΘ̂)
λ
n
Z
o
2
λs
−
+b
F (ε̂).
(ε̂2 , Λ2 Σ̂ + 2σc ΛΣ̂) − (ε̂1 , Λ2 Θ̂ + 2σc ΛΘ̂) −
λ
p+1
We first observe after an intergration by parts that:
Z
∂ Θ̂
∂ Θ̂
∂ Σ̂
1 d
∂ Σ̂
(
, ΛΣ̂) − (
, ΛΘ̂) = −
Im
y · ∇Q̂b Q̂b − σc [(
, Σ̂) − (
, Θ̂)].
∂b
∂b
2 db
∂b
∂b
Using the orthogonality relation (78), we compute:
Z
1
− Im( y · ∇Q̂b Q̂b ) + (ε̂2 , ΛΣ̂) − (ε̂1 , ΛΘ̂) = (Θ, ΛΣ̂) − (Σ, ΛΘ̂) + (ε2 , Λζ̂1 ) − (ε1 , Λζ̂2 )
2
= f1 .
Next, all the terms in (121) are treated like for the proof of (101) except the linear
term involving Ψ̂b . Arguing as in [13] and using in particular the degeneracy of
ζb on compact sets (47) to treat the scalar products terms in ε̂, we arrive at the
following preliminary estimate:
Z
Z
2
2 −|y|
0
{f1 }s ≥ c0
|∇ε̂| + |ε̂| e
− Cλ2(1−σc ) E0 − Cσc − Γ1+z
b
− (ε̂1 , Re(ΛΨ̂b ) − (ε̂2 , Im(ΛΨ̂b ))
(122)
for some universal constants C, z0 > 0 and with f1 given by (118). It remains to
estimate the leading order linear term. We first estimate from (119), (120):
−(ε̂1 , Re(ΛΨ̂b ) − (ε̂2 , Im(ΛΨ̂b )) ≥ (ε̂1 , Re(DF ) − (ε̂2 , Im(DF ))
(123)
Z
Z
c0
0
|∇ε̂|2 + |ε̂|2 e−|y| − Γ1+c
− Cσc
−
b
10
To estimate the remaining linear term, we proceed as in [13] and split ε̂ = ε − ζ̂b :
(ε̂1 , Re(DF ))+(ε̂2 , Im(DF )) = (ε1 , Re(DF ))+(ε2 , Im(DF ))−(ζ̂1 , Re(DF )−(ζ̂2 , Im(DF ).
The last term is the flux term for which the following lower bound can be derived
from (45):
−(ζ̂1 , Re(DF )) − (ζ̂2 , Im(DF )) ≥ c0 Γb .
(124)
The other term is estimated from Cauchy Schwarz and a sharp estimate on F from
(116) and (45), (46):
Z 2A
21 Z 2A
12
|ε|2
|F |2
|(ε1 , Re(DF )) − (ε2 , Im(DF ))| .
A
A
Z
2A
c0
10
Γb +
|ε|2 .
(125)
10
c0 A
We refer to [13], step 4 of the proof of Lemma 6, for a detailed proof of (124), (125).
Injecting (124), (125) into (123) and (122) now yields (117).
This concludes the proof of Lemma 4.
.
25
3.4. Computation of the L2 flux. We now turn to the computation of L2 fluxes
which are the key to get upper bounds on the far away localized L2 term which
appears in the RHS of (117). The obtained identity diplays new features with respect
to the analsyis in [13] which reflect the L2 super critical nature of the problem. We
introduce a radial non negative cut off function φ(r) such that φ(r) = 0 for r ≤ 21 ,
φ(r) = 1 for r ≥ 3, 14 ≤ φ0 (r) ≤ 12 pour 1 ≤ r ≤ 2, φ0 (r) ≥ 0. We then let
r
,
φA (s, r) = φ
A(s)
A(s) given by (114).
Lemma 5 (L2 fluxes). There holds for some universal constant c3 , z0 > 0 and
s ≥ 0:
Z
Z 2A
Z
a
1
1+z0
2
2σc
2
2
− Γb
|ε| − Γb
λ
φA |ε|
≥ c3 b
|∇ε|2 .
(126)
λ2σc
A
s
Proof of Lemma 5
Take a smooth cut off function χ(t, x). We integrate by parts on (1) to compute
the flux of L2 norm:
Z
Z
1
1
2
χ(t, x)|u(t, x)| dx =
∂t χ(t, x)|u(t, x)|2 dx + Im (∇χ · ∇uu) .
2
2
t
We apply this with χ(t, x) = φA ( x−x(t)
λ(t) ) and inject the decomposition (74). Recall
by construction that Qb (y) = σc Tb (y) for |y| ≥ 2b which is uniformly exponentially
decreasing and hence using (70), its contribution near A generates terms which are
negligible with repsect to the leading order Γb . We get after a bit of algebra using
also the bootstrap estimates:
Z
Z
Z
Z
b
1
1
∂φA 2
2
2σc
2
≥
φA |ε|
λ
y · ∇φA |ε| +
|ε| + Im
∇φA · ∇εε
2λ2σc
2
2
∂s
s
Z
Z
1 λs
1 xs
2
0
−
+b
y · ∇φA |ε| −
· ∇φA |ε|2 − Γ1+z
(127)
b
2 λ
2 λ
for some universal constant z0 > 0. From the choice of φ:
Z
Z
Z
Z 2A
y
1
1
1
0 y
2
2
0 y
2
10 φ
|ε| ≥
y · ∇φ
|ε| ≥
φ
|ε| ≥
|ε|2 , (128)
A
A
A
10
A
40 A
and also from the choice of A and the control of the geometrical parameters:
Z
Z
Z
Z 2A
xs
∂φA 2 λs
1
2
2
+
b
y
·
∇φ
|ε|
+
·
∇φ
|ε|
+
|ε|
.
|ε|2 .
A
A
λ
1000
λ
∂s
A
(129)
Moreover:
Z
Z
y
1
Im
∇φA · ∇εε = Im
∇φ
· ∇εε A
A
Z
Z
Z
Z
y
1
1
1
40
y
b
≤
( |∇ε|2 ) 2 ( φ0
|ε|2 ) 2 ≤
|∇ε|2 +
φ0
|ε|2 ,
A
A
bA
40
A
Z
Z
a
b
0 y
2
2
≤
φ
|ε| + Γb
|∇ε|2 .
(130)
100
A
Injecting (128), (129) and (130) into (127) yields (126) and concludes the proof of
Lemma 5.
26
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
3.5. L2 conservation law and second monotonicity formula. We now couple
the estimates (117) and (126) together with the L2 conservation law to derive a new
monotonicity formula which completes the dynamical information given by (101).
Proposition 5 (Second monotonicity formula). There holds for some universal
constant c4 > 0:
Z
Z
Z 2A
b
2
2 −1|y|
2
(131)
− {J }s ≥ c4 b Γb + |∇ε̃| + |ε̃| e
+
|ε| − σc ,
c4
A
with
Z
J (s) =
− c3 c2
Z
2
|Qb | −
Q
2
Z
b
bf˜1 (b) −
Z
+ 2(ε1 , Σ) + 2(ε2 , Θ) +
(1 − φA )|ε|2 (132)
˜
f1 (v)dv + b{(ε2 , Λζ̂1 ) − (ε1 , Λζ̂2 )} ,
0
where c3 , c2 are the universal small constants involved in (117), (126), and:
f˜1 (b) = (Θ, ΛΣ̂) − (Σ, ΛΘ̂).
(133)
Proof of Proposition 5
step 1 Coupling (117) and (126).
Let us multiply (117) by bc2 :
Z
Z
Z
2
2
2 −|y|
bc2
|∇ε̃| + |ε̃| e
+ Γb ≤ b{c2 f1 }s + b
2A
|ε|2 + bσc .
(134)
A
We the integrate by parts in time using (118):
Z b
˜
˜
f1 (v)dv + b(ε2 , Λζ̂1 ) − b(ε1 , Λζ̂2 )
b{c2 f1 }s = c2 bf1 (b) −
0
s
n
o
− c2 bs (ε̂2 , Λζ̂2 ) − (ε̂1 , Λζ̂1 )
and estimate from (96):
Z
Z
n
o
bc22
1+z0
2
2 −|y|
+
|∇ε̃| + |ε̃| e
+ Γb .
c2 bs (ε̂2 , Λζ̂2 ) − (ε̂1 , Λζ̂1 ) ≤ Γb
10
Injecting this into (134) yields:
Z
Z
Z b
bc22
2
2 −|y|
˜
˜
|∇ε̃| + |ε̃| e
+ Γb
≤ c2 bf1 (b) −
f1 (v)dv + b(ε2 , Λζ̂1 ) − b(ε1 , Λζ̂2 )
2
0
s
Z 2A
+ b
|ε|2 + bσc .
A
L2
We now inject the control of
fluxes (126) and obtain for a > Cη:
Z
Z
Z b
bc3 c22
2
2 −|y|
˜
˜
f1 (v)dv + b(ε2 , Λζ̂2 ) − b(ε1 , Λζ̂1 )
|∇ε̃| + |ε̃| e
+ Γb
≤ c3 c2 bf1 (b) −
4
0
s
Z
1
+
λ2σc φA |ε|2 + bc3 σc
(135)
λ2σc
s
step 2 Injection of the L2 conservation law.
27
We now rewrite the L2 conservation law as follows:
Z
Z
Z
2
2sc
2
2
|u0 | = λ
|Qb | + 2Re(ε, Qb ) + |ε| ,
and inject this into (135) to get:
Z
Z
Z
1
1
2
2
2σc
2σc
2
φA |ε|
= − 2σc λ
(1 − φA )|ε| + 2Re(ε, Qb ) + |Qb |
λ
λ2σc
λ
s
s
Z
Z
Z
Z
λ
s
2
2
2
2
=
(1 − φA )|ε| + 2Re(ε, Qb ) + |Qb |
(1 − φA )|ε| + 2Re(ε, Qb ) + |Qb | .
− 2σc
λ
s
Using the Hardy type bound:
Z
Z
Z
2
2 −|y|
2
3
|∇ε| + |ε| e
(1 − φA )|ε| ≤ CA
(136)
and the control (96), we derive the rough bound:
Z
Z
λs
2
2 σc
(1 − φA )|ε| + 2Re(ε, Qb ) + |Qb | . Cbσc .
λ
We may thus rewrite (135):
Z
Z
bc3 c22
|∇ε̃|2 + |ε̃|2 e−|y| + Γb ≤ Cbσc
4
Z
Z
Z
2
2
+
−( |Qb | − Qp ) − 2Re(ε, Qb ) − (1 − φA )|ε|2
+
c3 c2
bf˜1 (b) −
Z
b
˜
,
f1 (v)dv + b(ε2 , Λζ̂2 ) − b(ε1 , Λζ̂1 )
0
s
this is (131). This concludes the proof of Proposition 5.
4. Existence and stability of the self similar regime
This section is devoted to the proof of the main Theorem 2. We first show how
the coupling of the monotonicity formula (101), (131) implies a dynamical trapping
of b and a uniform bound on ε which allows us to close the bootstrap Proposition 3.
We then conclude the proof of Theorem 2 as a simple consequence of these uniform
bounds.
4.1. Closing the bootstrap. We are now in position to close the bootstrap and
conclude the proof of Proposition 3.
Proof of Proposition 3
step 1 Pointwise bound on ε.
Let us start with the proof of the pointwise bound on ε (88). We argue by
contradiction and assume that there exists s2 ∈ [s0 , s∗ ] such that:
Z
Z
1−9ν0
2
|∇ε(s2 )| + |ε(s2 )|2 e−|y| > Γb(s
.
2)
A simple continuity argument based on the initialization of the bootstrap estimate
(73) implies that there exists [s3 , s4 ] ⊂ [s0 , s∗ ] such that:
Z
Z
Z
Z
1−7ν0
1−9ν0
2
|∇ε(s3 )|2 + |ε(s3 )|2 e−|y| = Γb(s
,
|∇ε(s
)|
+
|ε(s4 )|2 e−|y| = Γb(s
,
4
3)
4)
(137)
28
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
and
Z
∀s ∈ [s3 , s4 ],
|∇ε(s)|2 +
Z
1−7ν0
.
|ε(s)|2 e−|y| ≥ Γb(s)
(138)
From (138) and the first viriel monotonicity (101), we have: ∀s ∈ [s3 , s4 ],
1−ν02
bs ≥ c1 (Γb1−7ν0 − Γb
)>0
and hence
b(s4 ) ≥ b(s3 ).
(139)
On the other hand, using the lower bound
Z
Z
Z
Z
1
2
2 −|y|
2
2 −|y|
0
|∇ε̂| + |ε̂| e
≥
− Γ1−ν
|∇ε| + |ε| e
b
2
together with (138), (70) and the second monotonicity formula (131), there holds:
∀s ∈ [s3 , s4 ],
−Js ≥ bc4 (Γb1−7ν0 − Γb1−6ν0 ) ≥ 0
and hence
J (s4 ) ≤ J (s3 ).
(140)
We now claim the following upper and lower control of J :
R
R
1
2 + |ε|2 e−|y| − Cσ ,
+
≥ −Γ1−Ca
|∇ε|
c
b
C
R
R
J (s) − f2 (b(s))
≤ CA3
|∇ε|2 + |ε|2 e−|y| + Γb1−Ca + Cσc ,
(141)
where f2 given by
Z
f2 (b, σc ) =
2
|Qb | −
Z
2
Q
− c3 c2
bf˜1 (b) −
Z
b
f˜1 (v)dv
(142)
0
satisfies
∀b∗ > b2 > b1 , f2 (b2 ) ≥ f2 (b1 ) − Cσc ,
1 2
(b + σc ) ≤ f2 (b1 ) ≤ C(b21 + σc ). (143)
C 1
Let us assume (141), (143). Then (140), (141) imply:
Z
Z
1
2
2 −|y|
f2 (b(s4 )) − Γ1−Ca
+
|∇ε(s
)|
+
|ε(s
)|
e
− Cσc ≤ J (s4 ) ≤ J (s3 )
4
4
b(s4 )
C
Z
Z
1−Ca
2
2
2
≤ f2 (b(s3 )) + CA (s3 )
|∇ε(s3 )| + |ε(s3 )| + Γb(s
+ Cσc
3)
and hence from the monotonicity (139), (143) and the controls (137), (70):
Z
Z
1−7ν0 −Ca
2
0
0
Γ1−9ν
=
|∇ε(s
)|
+
|ε(s4 )|2 e−|y| ≤ Γ1−Ca
+ Cσc ≤ Γ1−8ν
4
b(s4 )
b(s4 ) + Γb(s3 )
b(s4 )
for a = Cη > 0 and η chosen small enough, a contradiction which concludes the
proof of (88).
Proof of (141), (143): It is a standard consequence of the coercivity of the linearized
energy with our choice of orthogonality conditions, [13]. Indeed, we rewrite J given
by (132) using the conservation of energy (98):
J
∂Θ
∂Σ
) + 2(ε2 , Im(Ψb ) − σc µb
) − c3 c2 b{(ε2 , Λζ̂1 ) − (ε1 , Λζ̂2 )}
∂b
∂b
Z
Z
2
2(1−σc )
2
+ 2E(Qb ) − 2λ
E0 + (M+ (ε), ε1 ) + (M− (ε), ε2 ) − φA |ε| −
F (ε).
p+1
= f2 (b, σc ) + 2(ε1 , Re(Ψb ) − σc µb
29
The upper bound in (141) now follows from the Hardy bound (136) and the degeneracy (57). For the lower bound, we recall the following coercivity of the linearized
energy which holds true forA large anough ie |b| ≤ b∗ small emough:
Z
Z
Z
(M+ (ε), ε1 ) + (M− (ε), ε2 ) − φA |ε|2 ≥ c3 ( |∇ε|2 + |ε|2 e−|y| )
−
1 (ε1 , Q)2 + (ε1 , |y|2 Q)2 + (ε1 , yQ)2 + (ε2 , Q2 )2 ,
c3
for some universal constant c3 > 0, see Appendix D fin [13]. The choice of orthogonality conditions together with the degeneracy 94, (95) now yield (141). (143) is a
direct consequence of (56) and the smallness (115).
step 2 Dynamical trapping of b.
We now turn to the core of the argument which is the dynamical trapping of b
(85).
1−3ν0
, then from (70) and
Assume that there exists s5 ∈ [s0 , s∗ ] such that σc ≥ Γb(s
5)
a simple continuity argument, consider s6 ∈ [s0 , s5 ) the largest time such that
0
σc = Γ1−2ν
b(s6 ) , then bs (s6 ) ≤ 0 by construction while from (101):
1−2ν0
1−ν0
0
bs (s6 ) ≥ c1 σc − Γ1−ν
≥
c
Γ
−
Γ
1
b(s6 )
b(s6 )
b(s6 ) > 0
and a contradiction follows.
Similarily, assume that there exists s7 ∈ [s0 , s∗ ] such that
0
σc ≤ Γ1+3ν
b(s7 ) .
(144)
From (132) and the poinwise bounds (73), (70), there holds:
|J − d0 b2 | . ν010 b2
(145)
for some universal constant d0 > 0. Hence (70), (144) imply:
1+ 5 ν0

1+2ν0

4
, σc ≤ Γr J (s ) 
σc ≥ Γr J (s ) 
0
d0
.
7
d0
1+ 3 ν0

We then consider the largest time s8 ∈ [s0 , s7 ] such that σc =
Γr
2
J (s8 )
d0

,
then (J )s (s8 ) ≥ 0 by definition while from (131), (145):
 
1+ ν0

1+2ν0 
4
1  r


− {J }s (s8 ) ≥ b(s8 ) c4 Γr J (s ) 
−
Γ J (s ) 
 > 0,
8
8
2c4
d
d
0
0
and a contradiction follows.
step 3 Control of the scaling parameter.
We now turn to the control of the scaling paramater λ(t). From (70), the dynamical trapping of b (85) implies:
∀t ∈ [0, T ∗ ), (1 − 4ν0 )b0 ≤ b(t) ≤ (1 + 4ν0 )b0 .
(146)
30
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
Hence the upper bound (96), the control (88) and (146) ensure:
λs
= −λt λ ≤ (1 + ν0 )b0 .
(147)
λ
In particular, λ is nonincreasing while b(t) is trapped from (146) and thus (71), (72)
imply (86), (87).
∀t ∈ [0, T ∗ ), 0 < (1 − ν0 )b0 ≤ −
step 4 Control of the solution in Ḣ σ .
It remains to close the Ḣ σ estimate (89) which is the key to the control of the
nonlinear term (84). We use here the fact that the blow up is self similar and strictly
H 1 subcritical so that
N
N
2
N
−
> σc =
−
.
σp =
2
p+1
2
p−1
In other words, norms above scaling can be controlled dynamically in the bootstrap
as was for example observed by Rodnianski and Sterbenz [21].
It is more convenient here to work in original variables. Consider the decomposition
1
u(t, x) = Qsing (t, x) + ũ(t, x) =
λ
2
p−1
(Qb + ε) (t,
(t)
x − x(t) iγ(t)
)e
,
λ(t)
then first oberve by rescaling that (84) is equivalent to:
Γb1−20ν0
.
(148)
λ2(σ−σc )
To prove (148), we write down the equation for ũ and use standard Strichartz
estimates, see [1], for the linear Schrödinger flow. Indeed, the equation for ũ is:
|ũ|2Ḣ σ ≤
i∂t ũ + ∆ũ = −E − f (ũ)
with:
= i∂t Qsing + ∆Qsing + Qsing |Qsing |p−1
∂Qb
λs
xs
x − x(t) iγ(t)
1
p−1
ibs
+ ∆Qb − Qb + Qb |Qb |
− i ΛQb − i · ∇Qb + γ̃s Qb t,
e
=
2+ 2
∂b
λ
λ
λ(t)
λ p−1 1
∂Qb
λs
xs
x − x(t) iγ(t)
=
−Ψb + i(bs − σc µb )
−i
+ b ΛQb − i · ∇Qb + γ̃s Qb t,
e
,
2+ 2
∂b
λ
λ
λ(t)
λ p−1
and
f (ũ) = (Qsing + ũ)|Qsing + ũ|p−1 − Qsing |Qsing |p−1 .
(149)
E
Let t ∈ [0, T ∗ ), we write down the Duhamel formula on [0, t]. Following [1], we
consider the Strichartz pair:
r=
4(p + 1)
2
N
N
N (p + 1)
, γ=
,
=
−
N + σ(p − 1)
(p − 1)(N − 2σ) γ
2
r
(150)
and estimate from Strichartz estimates:
|Dσ ũ|L∞
[0,t]
L2x
. |Dσ ũ0 |L2 + |Dσ E|L1
[0,t]
. 1 + |Dσ E|L1
[0,t]
We claim:
L2x
|D E|L1
[0,t]
L2x
≤
+ |Dσ f (ũ)|Lγ 0
[0,t]
+ |Dσ f (ũ)|Lγ 0
[0,t]
1
σ
L2x
Lrx0
Lrx0
(151)
(1−17ν0 )
Γb20
[λ(t)](σ−σc )
,
(152)
31
1
σ
|D ũ(t)|L2 ≤
(1−18ν0 )
2
Γb(t)
(153)
[λ(t)](σ−σc )
which together with (151) yield (148).
Proof of (152): From the estimates on the geometrical paramaters (96), (97), the
degeneracy estimates (52), (53), the pointwise bounds (88) and (85), there holds:
∀t ∈ [0, t∗ ],
1
∂Qb
λs
xs
σ
|D E(t)|L2 .
Ψ
+
i(b
−
σ
µ
)
−
i
+
b
ΛQ
−
i
·
∇Q
+
γ̃
Q
s
c
s
b
b
b
b
b
1
∂b
λ
λ
[λ(t)]2+(σ−σc ) H
1
Z
Z
1
1
−11ν0 2
2
2 −|y|
2
|∇ε|
+
|ε|
e
+
Γ
≤
b
[λ(t)]2+(σ−σc )
1
(1−15ν0 )
Γb20
≤
(154)
[λ(t)]2+(σ−σc )
We now observe from the self similar blow up speed estimate (147): ∀q > 2,
Z t
Z
dτ
C t
λt
C
≤
dτ ≤
.
(155)
−
q
q−1
q−2
[λ(τ
)]
b
[λ(τ
)]
(q
−
2)b(t)[λ(t)]
0 0
0
Integrating (154) in time and using (155) yields (152).
Proof of (153): This estimate follows in the bootstrap using the fact that the blow
up is self similar and that ũ is small in Ḣ σ for σ > σc after renormalization. Indeed,
we first claim from standard nonlinear estimates in Besov spaces:
σ
|D f (ũ)|L . |D
r0
σ
ũ|pLr
+
Z
1
λp(σ̃−σc )
2
|∇ε| +
Z
2 −|y|
1
2
|ε| e
.
(156)
Assume (156). Then form further Sobolev embeddings:
|Dσ ũ|Lr . |Dσ̃ ũ|L2 with σ̃ = σ +
N
2
N
−
=σ+ .
2
r
γ
(157)
From direct check, σ < σ̃ < 1 providing σ has been chosen close enough to the
critical scaling exponent σc himself close enough to 0. We may thus interpolate
between σ and 1 and use the bootstrap assumptions (73), (70), to estimate:
1
σ̃
|D u|L2 ≤
Γb2
(1−50ν0 )
.
λ1−σ̃
Injecting this together with (88) into (156) yields:
σ
|D f (ũ)|Lγ 0
Lr0
[0,t] x
p
Z
1
(1−50ν0 )
(1−10ν0 )
2
2
+ Γb
. Γb0
0
t
10
dτ
[λ(τ )](σ̃−σc
γ
)pγ 0
.
(158)
Now from (150), (157), there holds:
2
(σ̃ − σc )p − 0
γ
2p
1
p+1
−2 1−
= p(σ − σc ) − 2 + 2
= p(σ − σc ) +
γ
γ
γ
N − 2σ
= (σ − σc ) + (p − 1) σ − σc +
−2
2
= σ − σc .
(159)
32
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
In particular, (σ̃ − σc )pγ 0 = 2 + γ 0 (σ − σc ) > 2, and we may thus inject (155) into
(158) to conclude:
1
σ
|D f (ũ)|Lγ 0
[0,t]
Lrx0
≤
(1−11ν0 )
2
Γb(t)
(160)
[λ(t)]σ−σc
for ν0 > 0 small enough thanks to p > 1, this is (153).
Proof of (156): Recall -see [1]- the equivalent expression of homogeneous Sobolev
norms: ∀0 < σ̃ < 1, ∀1 < q < +∞,

+∞
Z
σ̃
|D u|Lq
#2
"
∼
−σ̃
t
0
sup |u(· − y) − u(·)|Lq
|y|≤t
1
2
dt 
.
t
(161)
Using the homogeneity estimate:
∀u, v, |f (u) − f (v)| . |u − v|(|Qsing |p−1 + |u|p−1 + |v|p−1 ),
we first estimate from Holder, Sobolev embbeddings and the choice of the Strichartz
pair (150) :
|f (ũ)(· − y) − f (ũ)(·)|Lr0
. (ũ(· − y) − ũ(·))[|u(· − y)|p−1 + |u(·)|p−1 ]Lr0 + |Qsing |p−1 (ũ(· − y) − ũ(·))Lr0
p−1
(ũ(· − y) − ũ(·))Lr0
. |ũ(· − y) − ũ(·)|Lr |Dσ u|p−1
Lr + |Qsing |
and hence from (161):
σ
|D f (ũ)|Lr0 . |D
σ
1
ũ|pLr +
2
p(σ+ p−1
)− N
r0
λ

Z

+∞
t
0
#2
"
−σ
sup ||Qb |
p−1
|y|≤t
[ε(· − y) − ε(·)]|Lr0
1
2
dt 
.
t
To estimate the second term in the above RHS, we argue differently depending on
σ with
the dimension. For N ≥ 2, we use the Sobolev embedding Ḣ 1 ,→ Ḃk,2
r0 < 2 < k =
2N
,
N − 2 + 2σ
so that from Holder, the decay in space of Qb and (161):
#2
#2
Z +∞ "
Z +∞ "
dt
dt
−σ
−σ
p−1
t sup |ε(· − y) − ε(·)|Lk
t sup ||Qb | [ε(· − y) − ε(·)]|Lr0
.
t
t
0
0
|y|≤t
|y|≤t
. |ε|2Ḃ σ . |∇ε|2L2
k,2
and
p (156) follows. For N = 1, we use the crude bound |ε(· − y) − ε(·)|L∞ .
|y||∇ε|L2 and the decay of Q to get:
#2
Z +∞ "
Z +∞ h √
i2 dt
dt
−σ
p−1
t sup ||Qb | [ε(· − y) − ε(·)]|Lr0
.
t−σ t|∇ε|L2
t
t
0
0
|y|≤t
. |∇ε|2L2 ,
which concludes the proof of (156).
This concludes the proof of the bootstrap Proposition 3.
33
4.2. Proof of Theorem 2. We are now in position to prove Theorem 2.
Proof of Theorem 2
Pick ν0 > 0, p ∈ (pc , p∗ (ν0 )) and u0 ∈ O corresponding to b0 = b∗ (p) as given by
Definition 1. Note that (12) follows from (70). Let u(t) be the corresponding solution to (1) with maximum lifetime interval on the right [0, T ), then from Proposition
3, u(t) admits on [0, T ) a geometrical decomposition
1
u(t, x) =
λ
2
p−1
(Qb(t) + ε)(t,
(t)
x − x(t) iγ(t)
)e
λ(t)
which satisfies the estimates of Proposition 3. This implies in particular (14).
step 1 Finite time blow and self similar blow up speed.
Recall (147):
∀t ∈ [0, T ), (1 − ν0 )b0 ≤ −λt λ ≤ (1 + ν0 )b0 .
Integrating this in time first from 0 to t yields:
λ20
1
∀t ∈ [0, T ), (1 − ν0 )b0 t ≤ λ20 and hence T ≤
2
2(1 − ν0 b0 )
and hence the solution blows up in finite time. From the H 1 Cauchy theory,
|∇u(t)|H 1 → +∞ as t → T and hence from (88), λ(t) → 0 as t → T . We thus
integrate (147) from t to T to get:
∀t ∈ [0, T ], (1 − ν0 )b0 (T − t) ≤
λ2 (t)
≤ (1 + ν0 )b0 (T − t)
2
which implies (16).
step 2 Convergence of the blow up point.
From (97) and Proposition 3, we have the rough bound:
1
Γ4
1 xs
|xt | = | | . b0
λ λ
λ
and thus from (16):
1
Z
T
Z
|xt |dt .
0
0
T
Γb40
p
< +∞,
b0 (T − t)
and (15) follows. We moreover get the convergence rate:
1
Z T
1
x(t) − x(T ) Γb40
dτ
8
. p
p
.
Γ
b0 .
λ(t)
2b0 (T − t) t
2b0 (T − τ )
(162)
step 3 Strong convergence in H s for 0 ≤ s < σc .
We now turn to the proof of (17). Pick 0 ≤ s < σc . Let 0 < τ 1 and
0 < t < T − τ , let uτ (t) = u(t + τ ) and v(t) = uτ (t) − u(t), then v satisfies:
ivt + ∆v = u|u|p−1 − uτ |uτ |p−1 .
(163)
34
F. MERLE, P. RAPHAËL, AND J. SZEFTEL
Consider the Strichartz pair
r=
N (p + 1)
4(p + 1)
2
N
N
, γ=
,
=
− ,
N + s(p − 1)
(p − 1)(N − 2s) γ
2
r
then from standard nonlinear estimates in Sobolev spaces –[1]–, we have:
s
D u|u|p−1 − uτ |uτ |p−1 r0 . |Ds u|p r + |Ds uτ |p r . |Dσ̃ u|p 2 + |Dσ̃ uτ |p 2
L
L
L
L
L
with
N
N
2
−
=s+ .
2
r
γ
σ̃ = s +
Now observe that σ̃ → NN+2 > 0 as p → pc and hence σc < σ < σ̃ < 1 from (69) for
p close enough to pc . We thus estimate from the geometrical decomposition (74)
and the bounds (88), (89):
1
|Dσ̃ u|L2 .
λσ̃−σc
|Dσ̃ (Qb + ε)|L2 .
1
λσ̃−σc
.
We conclude using (159):
Z
s
|D u|Lγ 0
[t,T )
Lr0
T
|D
.
t
Z
.
t
T
σ̃
0
u|pγ
L2
γ10
Z
.
t
T
γ10
dτ
[λ(τ )]pγ 0 (σ̃−σc )
γ10
dτ
[λ(τ )]2+γ 0 (s−σc )
→ 0 as t → T
(164)
p
from the scaling law λ(t) ∼ 2b0 (T − t) and s − sc < 0. By running the standard
Strichartz estimates -[1]- on (163), we conclude that:
|Ds v|L∞
[t,T −τ )
L2
. |Ds v(t)|L2 +
T
Z
dτ
[λ(τ )]2+γ 0 (s−σc )
t
γ10
,
and the continuity u ∈ C([0, T ), Ḣ s ) now implies that u(t) is Cauchy sequence in
Ḣ s as t → T , and (17) follows.
Remark 8. Note that the case s = sc in (164) leads to the logarithmic upper bound
on the critical norm (20).
step 4 Behavior of u∗ on the blow up point.
It remains to prove (18) which follows by adpating the argument in [14].
Let smooth radially symmetric cut off function χ(r) = 1 for r ≤ 1 and χ(r) = 0 for
r ≥ 2. Fix t ∈ [0, T ) and let
R(t) = A0 λ(t)
(165)
with A0 given by
A0 = e
A(t)
2
2a
θ(2)
b0
.
(166)
From (146),
≤ A0 ≤ 2A(t) where A(t) is given by (114). We then compute the
flux of L2 norm:
Z
Z
d
2
x
−
x(T
)
x
−
x(T
)
2
χ(
)|u(τ )| = Im
∇χ(
) · ∇u(τ )u(τ ) dτ
R(t)
R(t)
R(t)
1
1
1
.
|u(τ )|2 1 .
Ḣ 2
R(t)
R(t) [λ(τ )]1−2sc
35
where we used (88), (89). We integrate this from t to T , divide by R2sc (t) and get
from (17):
Z
Z
Z T
1
x
−
x(T
)
x
−
x(T
)
dτ
1
1
∗
2
2
R2sc (t) χ( R(t) )|u | − R2sc (t) χ( R(t) )|u(t)| . R2sc +1 (t)
1−2sc
t [λ(τ )]
Z T
1
1
dτ
1
1
(167)
.
. 2sc +1 ≤ 1+sc
2sc +1 [λ(t)]2sc +1 (t)
1−2s
c
A0
A0
b0
A0
t [λ(τ )]
where we used the self similar speed (16) and (166). On the other hand, we have
from (74):
Z
Z 1
x − x(T )
λ(t)
λ2sc (t)
x(t) − x(T )
2
χ(
χ
)|u(t)| =
(y +
) |Qb + ε|2 (y)dy
R2sc (t)
R(t)
R2sc (t)
R(t)
λ(t)
Z 1
x(t) − x(T )
1
χ
(y +
=
) |Qb + ε|2 (y)dy.
2sc
A
λ(t)
A0
0
Now observe from the Hardy type bound (136), (166) and the bound (88) that:
Z
Z
Z
1
2
3
2
2 −|y|
|ε| . A0
. Γb40
|∇ε| + |ε| e
|y|≤10A0
and hence (162) ensures:
Z Z
1
x(t) − x(T )
2
χ
(y +
) |Qb + ε| = |Q|2 (1 + δ(p))
R(t)
λ(t)
with δ(p) → 0 as p → pc . Injecting this into (167) yields:
Z
Z
x − x(T ) ∗ 2
1
1
1
χ(
)|u | = 2sc
|Q|2 (1 + δ(p)) + O( 1+sc ).
R2sc (t)
R(t)
A0
A0
We now let t → T ie R(t) → 0 from (165) and (18) follows.
This concludes the proof of Theorem 2.
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Université de Cergy Pontoise and IHES, France
E-mail address: [email protected]
IMT, Université Paul Sabatier, Toulouse, France
E-mail address: [email protected]
CNRS, France, and Princeton University, USA
E-mail address: [email protected]