ON THE SINGULARITY FORMATION FOR THE NON LINEAR
SCHRÖDINGER EQUATION
PIERRE RAPHAËL
These notes are an introduction to the qualitative description of singularity formation
for the nonlinear Schrödinger equation. Part of the material was presented
during the 2008 Clay summer school on Nonlinear Evolution Equations at the ETH
Zurich. The manuscript has been enriched with additions in 2012 in order to give a
more accurate view on this very active research field and present a number of open
problems.
We consider the semi linear Schrödinger equation
(NLS)
iut = −∆u − |u|p−1u, (t, x) ∈ [0, T) × RN
u(0, x) = u0(x), u0 : RN → C
(0.1)
with u0 ∈ H1 = {u, u ∈ L2(RN )} in dimension N ≥ 1 and for energy subcritical
nonlinearities:
1 < p < 2∗
− 1 with 2∗
=
+∞ for N = 1, 2
2N
N−2 for N ≥ 3
. (0.2)
where 2∗ is the Sobolev exponent of the injection ˙H1 → L2∗
. The case p = 3 appears
in various areas of physics: for the propagation of waves in non linear media
and optical fibers for N = 1, the focusing of laser beams for N = 2, the BoseEinstein
condensation phenomenon for N = 3, see the monograph [106] for a more
systematic introduction to this physical aspect of the problem.
Our aim is to develop tools for the qualitative description of the flow for data in
the energy space H1, and this includes long time existence, scattering or formation of
singularities. The possibility of finite time blow up corresponding to a self focusing
of the nonlinear wave and the concentration of energy will be of particular interest
to us. Note that (NLS) is an infinite dimensional Hamiltonian system without any
space localization property and infinite speed of propagation. It is in this context
together with the critical generalized (gKdV) equation1
one of the few examples
where blow up is known to occur. For (NLS), an elementary proof of existence of
blow up solutions is known since the 60’s but is based on energy constraints and is
not constructive. In particular, no qualitative information of any type on the blow
up dynamics is obtained this way. In fact, the theory of global existence or blow
up for (NLS) as known up to now is intimately connected to the theory of ground
states or solitons which are special periodic in time solutions to the Hamiltonian
system. A central question is the stability of these solutions and the description of
the flow around them which has attracted a considerable amount of work for the
past thirty years.
These notes are organized as follows.
1see (4.22).
1
2 P. RAPHAËL
In the first section, we recall the main standard results about subcritical non
linear Schrödinger equations and in particular the existence and orbital stability of
soliton like solutions which relies on nowadays standard variational tools. In section
2, we introduce the blow up problem and present some of the very few general
results known on the singularity formation in this case, and this includes old results
from the 50’ and very recent ones. Section 3 focuses onto the mass critical
problem p = 1 + 4
N and we extend in the critical blow up regime the subcritical
variational theory of ground states. In section 4, we present the state of the art
on the question of description of the flow near the ground state for mass critical
problems, including recent complete answers for the generalized (gKdV) problem.
In section 5, we present a detailed proof of the pioneering result obtained in collaboration
with F.Merle in [71], [72] on the derivation of the sharp log-log upper bound
on blow up rate for a suitable class of initial data near the ground state solitary wave.
We expect the presentation to be essentially self contained provided the prior
knowledge of standard tools in the study of non linear PDE’s.
1. The subcritical problem
We recall in this section the main classical facts regarding the global well posedness
in the energy space of (NLS), and the main variational tools at the heart of the
proof of the existence and stability of special periodic solutions: the ground state
solitary waves.
1.1. Global well posedness in the subcritical case. Let us consider the general
non linear Schrödinger equation:
iut = −∆u − |u|p−1u
u(0, x) = u0(x) ∈ H1 (1.1)
with p satisfying the energy subcriticality assumption (0.2). The local well posedness
of (1.1) in H1 is a result of Ginibre, Velo, [23], see also [31]. Thus, for u0 ∈ H1,
there exists 0 < T ≤ +∞ such that u(t) ∈ C([0, T), H1). Moreover, the life time
of the solution can be proved to be lower bounded by a function depending on the
H1 size of the solution only, T(u0) ≥ f( u0 H1 ), and hence there holds the blow
up alternative:
T < +∞ implies lim
t→T
u(t) H1 = +∞. (1.2)
We refer to [11] for a complete introduction to the Cauchy theory. To prove the
global existence of the solution, it thus suffices to control the size of the solution
in H1. This is achieved in some cases using the invariants of the flow. Indeed, the
following H1 quantities are conserved:
• L2-norm:
|u(t, x)|2
= |u0(x)|2
; (1.3)
• Energy -or Hamiltonian-:
E(u(t, x)) =
1
2
| u(t, x)|2
−
1
p + 1
|u(t, x)|p+1
= E(u0); (1.4)
• Momentum:
Im uu(t, x) = Im u0u0(x) . (1.5)
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION 3
Note that the growth condition on the non linearity (0.2) ensures from Sobolev
embedding that the energy is well defined, and this is why H1 is referred to as the
energy space. These invariants are related to the group of symmetry of (1.1) in H1:
• Space-time translation invariance: if u(t, x) solves (1.1), then so does u(t +
t0, x + x0), t0 ∈ R, x0 ∈ RN .
• Phase invariance: if u(t, x) solves (1.1), then so does u(t, x)eiγ, γ ∈ R.
• Scaling invariance: if u(t, x) solves (1.1), then so does uλ(t, x) = λ
2
p−1 u(λ2t, λx),
λ > 0.
• Galilean invariance: if u(t, x) solves (1.1), then so does u(t, x−βt)ei β
2
·(x− β
2
t)
,
β ∈ RN .
Let us point out that this group of H1 symmetries is the same like for the linear
Schrödinger equation -up to the conformal invariance to which we will come back
later-.
The critical space is a fundamental phenomenological number for the analysis and
is defined as the number of derivatives in L2 which are left invariant by the scaling
symmetry of the flow:
|uλ(t)| ˙Hsc = |u(λ2
t)| ˙Hsc for sc =
N
2
−
2
p − 1
. (1.6)
Observe that sc < 1 from (0.2).
A direct consequence of the Cauchy theory, the conservation laws and Sobolev
embeddings is the celebrated global existence result:
Theorem 1.1 (Global wellposedness in the subcritical case). Let N ≥ 1 and 1 <
p < 1 + 4
N -equivalently sc < 0-, then all solutions to (1.1) are global and bounded
in H1.
Proof of Theorem 1.1. By L2 conservation: |u(t)|L2 = |u0|L2 . Moreover, the GagliardoNirenberg
interpolation estimate:
∀v ∈ H1
, |v|p+1
≤ C(N, p) | v|2
N(p−1)
4
|v|2
p+1
2
−
N(p−1)
4
. (1.7)
applied to v = u(t) implies using the conservation of the energy and the L2 norm:
∀t ∈ [0, T), E0 ≥
1
2

 | v|2
− C(u0) | v|2
N(p−1)
4

 .
The subcriticality assumption p < 1 + 4
N now implies an a priori bound on the H1
norm which concludes the proof of Theorem 1.1.
The critical exponent
p = 1 +
4
N
ie sc = 0
arises from this analysis and corresponds to the so-called L2 or mass critical case.
It is the smallest power nonlinearity for which blow up can occur and corresponds to
an exact balance between the kinetic and potential energies under the constraint of
conserved L2 mass. The L2 supercritical -and energy subcritical cases- correspond
to
1 +
4
N
< p < 2∗
− 1 ie 0 < sc < 1.
4 P. RAPHAËL
1.2. The solitary wave. A fundamental feature of the focusing (NLS) problem is
the existence of time periodic solutions. Indeed,
u(t, x) = φ(x)eit
is an H1 solution to (1.1) iff φ solves the nonlinear elliptic equation:
∆φ − φ + φ|φ|p−1
= 0, φ ∈ H1
(RN
) (1.8)
There are various ways to construct solutions to (1.8), the simplest one being to
look for radial solutions via a shooting method, [4].
Proposition 1.2 (Existence of solitary waves). (i) For N = 1, all solutions to (1.8)
are translates of
Q(x) =

 p + 1
2 cosh2 (p−1)x
2


p−1
. (1.9)
(ii) For N ≥ 2, there exist a sequence of radial solutions (Qn)n≥0 with increasing
L2 norm such that Qn vanishes n times on RN .
The exact structure of the set of solutions to (1.8) is not known in dimension
N ≥ 2. An important rigidity property however which combines nonlinear elliptic
techniques and ODE techniques is the uniqueness of the nonnegative solution to
(1.8).
Proposition 1.3 (Uniqueness of the ground state). All solutions to
∆φ − φ + φ|φ|p−1
= 0, φ ∈ H1
(RN
), φ(x) > 0 (1.10)
are a translate of an exponentially decreasing C2 radial profile Q(r) ([22]) which is
the unique nonnegative radially symmetric solution to (1.8) ([42]). Q is the so called
ground state solution.
The uniqueness is thus the consequence of two facts. A positive decaying at infinity
solution to (1.10) is necessarily radially symmetric with respect to a point, this
is a very deep and non trivial result due to Gidas, Ni, Nirenberg [22] which relies
on the maximum principle. And then there is uniqueness of the radial decaying
positive solution in the ODE sense. The original -and delicate- proof of this last
fact by Kwong [42] has been revisited by MacLeod [52] and is very nicely presented
in the Appendix of Tao [107]. We also refer to [48] for a beautiful extension of
uniqueness methods to nonlocal problems where the ODE approach fails.
Let us now observe that we may let the full group of symmetries of (1.1) act on
the solitary wave u(t, x) = Q(x)eit to get a 2N + 2 parameters family of solitary
waves: for (λ0, x0, γ0, β) ∈ R∗
+ × RN × R × RN ,
u(t, x) = λ
2
p−1
0 Q(λ0(x + x0) − λ2
0βt)eiλ2
0t
eiγ0
ei β
2
·(λ0(x+x0)−λ2
0βt)
.
These waves are moving according to the free Galilean motion and oscillating at a
phase related to their size: the larger the λ0, the wilder the oscillations in time. An
explicit computation reveals that the solitary wave can be made arbitrarily small in
H1 in the subcritical regime sc < 0 only.
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION 5
1.3. Orbital stability of the ground states in the subcritical case. We address
in this section the question of the stability of the ground state solitary wave
u(t, x) = Q(x)eit, Q > 0, as a solution to (1.1) in the mass subcritical case
1 < p < 1 +
4
N
, sc < 0. (1.11)
Let us first observe that two trivial instabilities are given by the symmetries of the
equation:
• Scaling instability: ∀λ > 0, the solution to (1.1) with initial data u0(x) =
λ
2
p−1 Q(λx) is u(t, x) = λ
2
p−1 Q(λx)eiλ2t.
• Galilean instability: ∀β > 0, the solution to (1.1) with initial data u0(x) =
Q(x)eiβ is u(t, x) = Q(x − βt)eit+β
2
·(x−β
2
t)
.
In both cases,
sup
t∈R
|u(t, x) − Q(x)eit
| > |Q(x)|
and thus the solution does not stay uniformly close to Q. Cazenave and Lions [12]
proved that these trivial instabilities are the only ones in the energy subcritical
setting: this is the celebrated orbital stability of the ground state solitary wave.
Theorem 1.4 (Orbital stability of the ground state, [12]). Let N ≥ 1 and p satisfy
(1.11). For all ε > 0, there exists δ(ε) such that the following holds true. Let
u0 ∈ H1 with
u0 − Q H1 < δ(ε),
then there exist a translation shift x(t) ∈ C0(R, RN ) and a phase shift γ(t) ∈ C0(R, R)
such that:
∀t ∈ R, u(t, x) − Q(x − x(t))eiγ(t)
H1 < ε.
The strength -and the weakness- of the proof is that it relies only on the conservation
laws and the variational characterization of the ground state solitary wave.
This study falls into the classical sets of concentration compactness techniques as
introduced by Lions in [50],[51]. Given λ > 0, we let
Qλ(x) = λ
2
p−1 Q(λx).
The following variational result immediately implies Theorem 1.4:
Proposition 1.5 (Description of the minimizing sequences). Let N ≥ 1 and p
satisfy (1.11). Let M > 0 be fixed.
(i) Variational characterization of Q: The minimization problem
I(M) = inf
|u|L2 =M
E(u) (1.12)
is attained on the family
Qλ(M)(· − x0)eiγ0
, x0 ∈ RN
, γ0 ∈ R,
where λ(M) is the unique scaling such that |Qλ(M)|L2 = M.
(ii) Description of the minimizing sequences: Any minimizing sequence vn to (1.12)
is relatively compact in H1 up to translation and phase shifts, that is up to a subse-
quence:
vn(· + xn)eiγn
→ Qλ(M) in H1
.
The fact that Proposition 1.5 implies Theorem 1.4 is now a simple consequence
of the conservation laws and is left to the reader. The next section is devoted to
the proof of Proposition 1.5.
6 P. RAPHAËL
1.4. The concentration compactness argument. The first key to the proof of
Proposition 1.5 is the description of the lack of compactness in RN of the Sobolev
injection H1 → Lp+1, 2 ≤ p + 1 < 2∗. This description is a consequence of Lions’
concentration compactness Lemma. Let us recall that the injection is compact
on a smooth bounded domain. Note also that the injection is still compact when
restricted to radial functions in dimension N ≥ 2. Here one uses the estimate:
u2
(r) = −
+∞
r
u(s)u (s)ds and thus |u|L∞(r≥R) ≤
C
R
N−1
2
| u|
1
2
L2 |u|
1
2
L2
so that any H1 bounded sequence of radially symmetric functions is Lp+1 compact.
This would considerably simplify the proof of Proposition 1.5 when restricting the
problem to radially symmetric functions. In general, there holds the following:
Proposition 1.6 (Description of the lack of compactness of H1 → Lq). Let a
sequence un ∈ H1 with
|un|L2 = M, | un|L2 ≤ C, (1.13)
Then there exists a subsequence unk
such that one of the following three scenari
occurs:
(i) Compactness: ∃yk ∈ RN such that
∀2 ≤ q < 2∗
, unk
(· + yk) → u in Lq
. (1.14)
(ii) Vanishing:
∀2 < q < 2∗
, unk
→ 0 in Lq
. (1.15)
(iii) Dichotomy: ∃vk, wk, ∃0 < α < 1 such that ∀2 ≤ q < 2∗:



Supp(vk) ∩ Supp(wk) = ∅, dist(Supp(vk), Supp(wk)) → +∞,
vk H1 + wk H1 ≤ C,
vk L2 → αM, wk L2 → (1 − α)M,
limk→+∞ |unk
|q − |vk|q − |wk|q| = 0,
lim infk→+∞ | unk
|2 − | vk|2 − | wk|2 ≥ 0.
(1.16)
Remark 1.7. The key in the dichotomy case is that there is no loss of potential
energy during the splitting in space of unk
into two bumps vk, wk which support go
away from each other, while on the other hand only a lower semi continuity bound
can be derived for the kinetic energy.
Remark 1.8. The case dichotomy corresponds to the localization of the first bubble
of concentration. One can then continue the extraction iteratively and obtain the
profile decomposition of the sequence un, see P. Gerard [21], Hmidi, Keraani [28]
for a very elegant proof.
The proof of Proposition 1.6 is given in Appendix A. We now show how the
description of the lack of compactness of the Sobolev injection is a powerful tool for
the study of variational problems.
Proof of Proposition 1.5. step1 Computation of I(M). Let I(M) be given by
(1.12). We claim that
−∞ < I(M) = M
2(1−sc)
|sc| I(1) < 0. (1.17)
Indeed, I(M) > −∞ follows directly form the Gagliardo-Nirenberg inequality (1.7)
and the subcriticality condition (1.11). The computation of the nonpositive value of
the infimum follows from the scaling properties of the problem. First, given u ∈ H1
with u L2 = 1, we use the L2 scaling
vλ(x) = λ
N
2 u(λx)
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION 7
to get:
E(vλ) = λ2 1
2
| u|2
−
1
(p + 1)λ(p−1)|sc|
|u|p+1
.
Letting λ → 0 yields I(1) < 0. The homogeneity in M of I(M) is derived using the
scaling of the equation
vλ(x) = λ
2
p−1 u(λx), vλ L2 = λ|sc|
u L2 , E(vλ) = λ2(1−sc)
E(u),
which yields the claim.
Let now un be a minimizing sequence for I(M). Then un is bounded in H1 from
(1.7) and satisfies the assumptions of Proposition 1.5, and we now examine the various
scenario:
step 2 Vanishing cannot occur. Otherwise, from (1.15):
I(M) = lim
k→+∞
E(unk
) ≥ lim inf
k→+∞
1
2
| unk
|2
≥ 0
which contradicts (1.17).
step 3 Dichotomy cannot occur. Otherwise, from (1.16), we have sequences
vk, wk and 0 < α < 1 such that
vk L2 = αM, wk L2 = (1 − α)M
and
I(M) ≥ lim inf
k→+∞
E(vk) + lim inf
k→+∞
E(wk).
In particular, this implies:
I(M) ≥ I(αM) + I((1 − α)M) (1.18)
and thus from (1.17):
1 ≤ α
2(1−sc)
|sc| + (1 − α)
2(1−sc)
|sc| for some 0 < α < 1.
Now a straightforward convexity argument implies from 2(1−sc)
|sc| > 1 that α = 0 or
α = 1, a contradiction.
step 4 Conclusion. We conclude that only compactness occurs ie
unk
(· + xk) → u in Lp+1
.
Observe then from the strong Lp+1 convergence and the lower semicontinuity of the
˙H1 norm that u attains the infimum:
u L2 = M, E(u) = I(M).
It thus remains to characterize the infimum. We claim that:
u(x) = Qλ(M)(· + x0)eiγ0
(1.19)
which concludes the proof of Proposition 1.6.
Proof of (1.19): First observe from | |u||2 ≤ | u|2 that v = |u| is a minimizer
with v ≥ 0. From standard Euler Lagrange theory, v solves
∆v + v|v|p−1
= µv, v ∈ H1
.
8 P. RAPHAËL
The Lagrange multiplier, which a priori depends on v, can be computed by multiplying
the equation by v and then y · v (Pohozaev integration) leading to:
µ = µ(M) =
N + 2 − p(N − 2)
2M N(p−1)
4 − 1
I(M) > 0.
We now observe by rescaling that w(x) = λ
2
p−1 v(λx) with λ =
√
µ satisfies
∆w − w + w|w|p−1
= 0, w ∈ H1
(RN
), w ≥ 0,
and w non zero. From the uniqueness statement of Proposition 1.3, this yields:
w(x) = Q(x − x0),
and hence v(x) = Qλ(M)(x − x0). This implies in particular that v does not vanish
which together with | u|2 = | |u||2 -because they both are minimizers- implies2
u(x) = |u(x)|eiγ0
= Qλ(M)(x − x0)eiγ0
,
and (1.19) is proved.
Furthers comments
1. More general nonlinearities: The proof we have presented reproduces the original
argument by Cazenave, Lions [12] and heavily relies onto the specific scaling
properties of the nonlinearity. The advantage of this argument is to completely
avoid the linearization near the ground state, but the prize to pay is the proof of
global estimates like (1.18) which may be non trivial in the absence of symmetries.
Another approach to stability proceeds by brute force linearization and the derivation
of suitable coercivity properties of the linearized operator close to the ground
state as for example done in Grillakis, Shatah, Strauss [26] to treat more general
nonlinearities. We also refer to [44], [46], [47] for analogue results for gravitational
kinetic equations which display a similar structure.
2. Asymptotic stability: An important question is to know whether, when stability
holds, asymptotic stability also holds, that is do solutions asymptotically converge
to the ground state in some local norm in space as t → +∞? This kind of property
corresponds to a form of asymptotic irreversibility of the flow. This is an extremely
delicate problem which has attracted a considerable amount of work for the past
ten years. For some specific type of nonlinearities, asymptotic stability holds due
to a fine tuning mechanism known as the "Fermi Golden Rule", see Soffer, Weinstein
[105], Rodnianski, Soffer, Schlag [102], Sulem, Buslaev [10], Sigal, Zhou [20].
However, the case of pure power is still open because essentially small solitons are
delicate to deal with. Indeed, in the pure power case, a soliton Qλ can be made arbitrarily
small in H1 and not disperse. Moreover, one should keep in mind that the
asymptotic stability is false in the completely integrable case N = 1, p = 3, see [112].
3. Generic long time dynamics: In general, one expects the long time behavior
of the solution to correspond to a splitting of the solution into a non dispersive part
corresponding to a sum of decoupled solitary waves moving at different speeds and
a radiative part which disperses -ie goes to 0 in L∞ say-. Such a general behavior
2see for example[49].
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION 9
has been proved in the integrable case for the KdV system
(KdV )
ut + (uxx + u2)x = 0, (t, x) ∈ [0, T) × R,
u(0, x) = u0(x), u0 : RN → R,
but complete integrability plays a very specific role here. See Rodnianski, Soffer,
Schlag [102], Martel, Merle, Tsai [63], for the case of non integrable (NLS) systems
but with specific nonlinearities. One should think here that in general, even the
simpler question of the orbital stability of the multisolitary wave in the pure power
case for (NLS) is open.
2. The blow up problem
We focus in this section on the NLS problem (1.1) with mass critical/super critical
and energy subcritical nonlinearities, or equivalently according to (1.6):
0 ≤ sc < 1, 1 +
4
N
≤ p < 2∗
− 1.
Our aim is to collect old and new results regarding the qualitative description of
blow up solutions which involves so far many open problems.
2.1. Existence of blow up solutions: the virial law. The Cauchy theory ensures
global existence for small data in H1 but for large data, the Gagliardo Nirenberg
inequality (1.7) does not suffice anymore to ensure global existence. A well
known global obstructive argument known as the virial law allows one to very easily
prove the existence of finite time blow up solutions.
Theorem 2.1 (Virial blow up for E0 < 0). Let u0 ∈ Σ = H1 ∩ {xu ∈ L2} with
E0 < 0,
then the corresponding solution to (1.1) blows up in finite time 0 < T < +∞.
Proof of Theorem 2.1. Integrating by parts in (1.1), we find:
d2
dt2
|x|2
|u(t, x)|2
dx = 4N(p − 1)E0 −
16sc
N − 2sc
| u|2
≤ 4N(p − 1)E0 (2.1)
from sc ≥ 0. Hence from E0 < 0, the positive quantity |x|2|u(t, x)|2dx lies below
an inverted parabola and hence the solution cannot exist for all times.
This blow up argument is extraordinary because it provides a blow up criterion
based essentially on a pure Hamiltonian information E0 < 0 which applies to arbitrarily
large initial data in H1. In particular, it exhibits an open region of the
energy space -up to extra integrability condition- where blow up is proven to be
a stable phenomenon. While it may seem at first hand to be very specific to the
(NLS) problem, this kind of convexity argument is very common for parabolic or
wave type problems, see for example [30], kinetic problems [25], or even compressible
Euler equations, [104]. However, it has has two major weaknesses:
(i) It heavily relies on a very specific algebra and hence is very unstable by perturbation
of the equation. It thus is completely unable to predict blow up even in
situations where it is strongly expected. A typical case is for example (NLS) on a
domain with Dirichlet boundary conditions, [96].
(ii) More fundamentally, this argument is purely obstructive in nature and says
very little a priori on the singularity formation. In fact the blow up time formally
predicted which is the time of vanishing of the variance |x|2|u|2 is almost never
correct, solutions generically blow up before.
10 P. RAPHAËL
2.2. Scaling lower bound on blow up rate. In the setting of arbitrarily large
initial data, little is known regarding the description of the singularity formation.
This is mainly a consequence of the fact that the virial blow up argument does not
provide any insight into the blow up dynamics. More generally, the a priori control
of the blow up speed | u(t)|L2 which plays a fundamental role for the classification
of blow up dynamics for example for the heat or the wave equation, is poorly
understood. However a general lower bound on the blow up rate holds as a very
simple consequence of the scaling invariance of the problem:
Proposition 2.2 (Scaling lower bound on blow up rate). Let N ≥ 1, 0 ≤ sc < 1.
Let u0 ∈ H1 such that the corresponding solution u(t) to (1.1) blows up in finite
time 0 < T < +∞, then there holds:
∀t ∈ [0, T), | u(t)|L2 ≥
C(u0)
(T − t)
1−sc
2
. (2.2)
Proof of Proposition 2.2. We give the proof for sc = 0 which is elementary and
based on the scaling invariance of the equation and the local well posedness theory
in H1. The proof for sc > 0 is similar and requires the Cauchy theory in ˙Hsc ∩ ˙H1,
see [76]. Consider for fixed t ∈ [0, T)
vt
(τ, z) = | u(t)|
−N
2
L2 u t + | u(t)|−2
L2 τ, | u(t)|−1
L2 z .
vt is a solution to (1.1) by scaling invariance. We have | vt(0)|L2 = 1, |vt|L2 =
|u0|L2 , and thus by the resolution of the Cauchy problem locally in time by fixed
point argument, there exists τ0 > 0 independent of t such that vt is defined on
[0, τ0]. Therefore, t + | u(t)|−2
L2 τ0 ≤ T which is the desired result.
One can ask for the sharpness of the bound (2.2), or equivalently for the existence
of self similar solutions in the energy space, i.e. solutions which blow according to
the scaling law
| u(t)|L2 ∼
1
(T − t)
1−sc
2
. (2.3)
For sc = 0, it is an important open problem, [7]. It is however proved in [97], [74]
that the lower bound (2.2) is not sharp for data near the ground state in connection
with the log log law, see Theorem 4.3. On the contrary, for sc > 0, a stable self
similar blow up regime in the sense of (2.3) is observed numerically, [106], and a
rigorous derivation of these solutions is obtained in collaboration with Merle and
Szeftel in [78] for slightly super critical problems:
Theorem 2.3 (Existence and stability of self similar solutions, [78]). Let 1 ≤ N ≤ 5
and 0 < sc 1. Then there exists an open set of initial data u0 ∈ H1 such that the
corresponding solution to (1.1) blows up with in finite time T = T(u0) < +∞ with
the self similar speed:
| u(t)|L2 ∼
1
(T − t)
1−sc
2
.
The extension of this result to the full critical range sc < 1 is an important open
problem, in particular to address the physical case N = p = 3, sc = 1
2, but is
confronted to the construction and the understanding of the stationary self similar
profiles which is poorly understood, see [78] for a further discussion.
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION11
2.3. On concentration of the critical norm. A second general phenomenon of
finite blow up solutions is the concentration of the critical norm. The first result of
this type goes back to Merle, Tsutsumi, [81] in the radial case, and generalized by
Nawa, [92], for the mass critical NLS.
Theorem 2.4 (L2 concentration phenomenon for sc = 0, [81], [92]). Let sc = 0.
Let u0 ∈ H1 such that the corresponding solution u(t) to (1.1) blows up in finite
time 0 < T < +∞. Then there exists x(t) ∈ C0([0, T)RN ) such that:
∀R > 0, lim inf
t→T |x−x(t)|≤R
|u(t, x)|2
dx ≥ Q2
. (2.4)
Theorem 2.4 relies on the sharp variational characterization of the geound state
solitary wave Q and we therefore postpone the proof to section 3.1. We refer to
[108] for an extension to critical regularity u0 ∈ L2. Two natural questions following
Theorem 2.4 are still open in the general case:
(i) Does the function x(t) have a limit as t → T defining then at least one exact
blow up point in space where L2 concentration takes place?
(i) Which is the exact amount of mass focused by the blow up dynamic?
An explicit construction of blow up solutions due to Merle, [64], is the following:
let k points (xi)1≤i≤k ∈ RN , then there exists a blow up solution u(t) which blows
up in finite time 0 < T < +∞ exactly at these k points and accumulates exactly
the mass:
|u(t)|2
Σ1≤i≤k|Q|2
L2 δx=xi as t → T,
in the sense of measures. A general conjecture concerning L2 concentration is formulated
in [75] and states that a blow up solution focuses a quantized and universal
amount of mass at a finite number of points in RN , the rest of the L2 mass being
purely dispersed. The exact statement which is directly related to the soliton resolution
conjecture is the following:
Conjecture (*): Let u(t) ∈ H1 be a solution to (1.1) which blows up in finite
time 0 < T < +∞. Then there exist (xi)1≤i≤L ∈ RN with L ≤
|u0|2
Q2 , and u∗ ∈ L2
such that: ∀R > 0,
u(t) → u∗
in L2
(RN
−
1≤i≤L
B(xi, R))
and |u(t)|2
Σ1≤i≤Lmiδx=xi + |u∗
|2
with mi ∈ [ Q2
, +∞).
Let us now address the same question of the behavior of the critical norm for the
super critical NLS 0 < sc < 1. There is no simple a priori lower bound like for (2.2)
for the critical norm |u(t)| ˙Hsc which is invariant by the scaling symmetry of the
flow. Moreover, a major difference between the mass critical problem and the super
critical problem is that the critical norm is conserved by the flow for sc = 0 only,
and this leads to dramatic differences in the blow up dynamics. We for example
proved in [76] that for radial data the critical norm not only concentrates at blow
up, it explodes:
Theorem 2.5 (Blow up of the critical norm, [76]). Let 0 < sc < 1, p < 5 and
N ≥ 2. There exists a universal constant γ = γ(N, p) > 0 such that the following
holds true. Let u0 ∈ H1 with radial symmetry and assume that the corresponding
12 P. RAPHAËL
solution to (1.1) blows up in finite time T < +∞. Then there holds the lower bound
for t close enough to T:
|u(t)| ˙Hsc ≥ | log(T − t)|γ(N,p)
.
Related results were proved for the Navier Stokes equation [16], and are a first
step towards the understanding of the formation of the blow up bubble. Note that
the logarithmic lower bound can be proved to be sharp in some regimes, [78], but
there also exist regimes where the critical norm blows up polynomially, [80]. The
regimes N = 1, 2 with p ≥ 5 are still open, as well as the general non radial case.
The proof relies on the quantification of a Liouville type theorem, see [38] for recent
extensions to the wave equation.
2.4. A sharp upper bound on blow up rate. We now address the question of
upper bounds on blow up rate for general solutions. A simple observation by Merle
is that for 0 < sc < 1, the brute force time integration of the virial law (2.1) not
only implies finite time blow up for E0 < 0, it also immediately yields an upper
bound on the blow up rate for any finite time blow up solution:
Theorem 2.6 (General upper bound on blow up rate). Let 0 < sc < 1 and u0 ∈ Σ
such that the corresponding solution to (1.1) blows up in finite time 0 < T < +∞,
then:
T
0
(T − t)| u(t)|2
L2 dt < +∞. (2.5)
Note that in particular on a subsequence
| u(tn)|L2 (T − tn) → 0 as tn → T.
Interestingly enough, this bound fails for sc = 0, see (3.10), and in fact there exists
no known upper bound on blow up rate in the mass critical case which is one of
the reason why the mass critical problem is in some sense more degenerate3
. For
0 < sc < 1, we observed in collaboration with Merle and Szeftel [80] that a relatively
elementary argument based on a localization of the virial identity as initiated in [76]
implies an improved upper bound for u0 radial.
Theorem 2.7 (Sharp upper bound on blow up rate for radial data, [80]). Let
N ≥ 2, 0 < sc < 1, p < 5.
Let the interpolation number4
α =
5 − p
(p − 1)(N − 1)
. (2.6)
Let u0 ∈ H1 with radial symmetry and assume that the corresponding solution u ∈
C([0, T), H1) of (1.1) blows up in finite time T < +∞. Then there holds the space
time upper bound:
T
t
(T − τ)| u(τ)|2
L2 dτ ≤ C(u0)(T − t)
2α
1+α . (2.7)
This implies in particular
| u(tn)|L2
1
(T − tn)
1
1+α
3The example of the (gKdV) problem and Theorem 4.9 indicate that there may be no bound...
4Observe that 0 < α < 1.
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION13
on a subsequence tn → T. Note that it would be very interesting to obtain the
pointwise bound for all times.
Before proving Theorem 2.7 which relies on a sharp localization of the virial law,
let us say that we do not know if the bound (2.5) is sharp. However, we claim that
the general bound for radial data (2.7) is indeed sharp and saturated on a new class
of blow up solutions: the collapsing ring profiles.
Theorem 2.8 (Collapsing ring solutions, [80]). Let
N ≥ 2, 0 < sc < 1, p < 5,
let 0 < α < 1 be given by (2.6) and the Galilean shift:
β∞ =
5 − p
p + 3
.
Let Q be the one dimensional mass subcritical ground state (1.9). Then there exists
a time t < 0 and a solution u ∈ C([t, 0), H1) of (1.1) with radial symmetry which
blows up at time T = 0 according to the following dynamics. There exist geometrical
parameters (r(t), λ(t), γ(t)) ∈ R∗
+ × R∗
+ × R such that:
u(t, r) −
1
λ
2
p−1 (t)
Qe−iβ∞y r − r(t)
λ(t)
eiγ(t)
→ 0 in L2
(RN
). (2.8)
The blow up speed, the radius of concentration and the phase drift are given by the
asymptotic laws:
r(t) ∼ |t|
α
1+α , λ(t) ∼ |t|
1
1+α , γ(t) ∼ |t|− 1−α
1+α as t ↑ 0. (2.9)
Moreover, the blow up speed admits the equivalent:
| u(t)|L2 ∼
1
(T − t)
1
1+α
as t ↑ 0. (2.10)
Comments on the result
1. Standing and collapsing ring: The construction of ring solutions started in
[98], [100] for p = 5 in dimension N ≥ 2 where we constructed standing ring blow
up solutions which concentrate on a standing sphere r = 1 at the speed given by
the log-log law (4.14). The idea is that the geometry of the blow up set given by
a standing sphere allows one to reduce the leading order blow up dynamics to the
one dimension quintic NLS which is the mass critical one for p = 5. This has been
further extended to other geometries in higher dimensions [29], [114]. Then in the
breakthrough paper [17], Fibich, Gavish and Wang extended formally the construction
to 3 < p < 5 in dimension N = 2 and observed numerically the collapsing ring
solutions which existence is made rigorous in [80]. Note that the collapsing ring is
expected to be stable by radial perturbation of the data, but this is still an open
problem.
2. Mass concentration: The ring solutions have a quite unexpected blow up
behavior. Indeed, despite the fact that the problem is mass super critical, the
structure (2.8) coupled with the speeds (2.9) imply the concentration of the L2
mass
|u(t)|2
|Q|2
L2 δx=0 as t ↑ 0. (2.11)
A contrario the self similar blow up solutions of Theorem 2.3 constructed in [78]
have a strong limit in L2 at blow up time. In fact, by rescaling, we can let the
14 P. RAPHAËL
amount of concentrated mass in (2.11) be arbitrary, and hence the expected quantization
of Conjecture (*) for the mass critical problem does not hold here. In some
sense, the proof of Theorem 2.8 amounts showing that in the ring regime, the super
critical problem can be treated as a mass critical problem. Moreoever, this is
the first construction of blow up solutions for a large set of super critical regimes
including the physical one N = p = 3.
We now turn to the proof of the sharp upper bound (2.7) which relies on a suitably
localized virial identity in the continuation of [76].
Proof of Theorem 2.7. step 1 Localized virial identity. Let N ≥ 2, 0 < sc < 1 and
u ∈ C([0, T), H1) be a radially symmetric finite time blow up solution 0 < T < +∞.
Pick a time t0 < T and a radius 0 < R = R(t0) 1 to be chosen. Let χ ∈ D(RN )
and recall the localized virial identity5
for radial solutions:
1
2
d
dτ
χ|u|2
= Im χ · uu , (2.12)
1
2
d
dτ
Im χ · uu = χ | u|2
−
1
4
∆2
χ|u|2
−
1
2
−
1
p + 1
∆χ|u|p+1
.
Applying with χ = ψR = R2ψ( x
R ) where ψ(x) = |x|2
2 for |x| ≤ 2 and ψ(x) = 0 for
|x| ≥ 3, we get:
1
2
d
dτ
Im ψR · uu
= ψ (
x
R
)| u|2
−
1
4R2
∆2
ψ(
x
R
)|u|2
−
1
2
−
1
p + 1
∆ψ(
x
R
)|u|p+1
≤ | u|2
− N
1
2
−
1
p + 1
|u|p+1
+ C
1
R2
2R≤|x|≤3R
|v|2
+
|x|≥R
|u|p+1
.
Now from the conservation of the energy:
|u|p+1
=
p + 1
2
| u|2
− (p + 1)E(u0)
from which
| u|2
− N
1
2
−
1
p + 1
|u|p+1
=
N(p − 1)
2
E(u0) −
2sc
N − 2sc
| u|2
,
and thus:
2sc
N − 2sc
| u|2
+
1
2
d
dτ
Im ψR · uu
|E0| +
|x|≥R
|u|p+1
+
1
R2
2R≤|x|≤3R
|u|2
(2.13)
≤ C(u0) 1 +
1
R2
+
|x|≥R
|u|p+1
from the energy and L2 norm conservations.
5see [76] for further details.
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION15
step 2 Radial Gagliardo-Nirenberg interpolation estimate. In order to control
the outer nonlinear term in (2.13), we recall the radial interpolation bound:
u L∞(r≥R) ≤
u
1
2
L2 u
1
2
L2
R
N−1
2
,
which together with the L2 conservation law ensures:
|x|≥R
|u|p+1
≤ u p−1
L∞(r≥R) |u|2
≤
C(u0)
R
(N−1)(p−1)
2
u
p−1
2
L2
≤ δ
2sc
N − 2sc
| u|2
+
C
δR
2(N−1)(p−1)
(5−p)
= δ
2sc
N − 2sc
| u|2
+
C
δR
2
α
where we used Hölder for p < 5 and the definition of α (2.6). Injecting this into
(2.13) yields for δ > 0 small enough using R 1 and 0 < α < 1:
sc
N − 2sc
| u|2
+
d
dτ
Im ψR · uu ≤
C(u0, p)
R
2
α
(2.14)
step 3 Time integration. We now integrate (2.14) twice in time on [t0, t2] using
(2.12). This yields up to constants using Fubini in time:
ψR|u(t2)|2
+
t2
t0
(t2 − t) u(t) 2
L2 dt
(t2 − t0)2
R
2
α
+ (t2 − t0) Im ψR · uu (t0) + ψR|u(t0)|2
≤ C(u0)
(t2 − t0)2
R
2
α
+ R(t2 − t0) u(t0) L2 + R2
u0
2
L2
We now let t → T. We conclude that the integral in the left hand side converges
and
T
t0
(T − t)| u(t)|2
L2 dt ≤ C(u0)
(T − t0)2
R
2
α
+ R(T − t0) u(t0) L2 + R2
. (2.15)
We now optimize in R by choosing:
(T − t0)2
R
2
α
= R2
ie R(t0) = (T − t0)
α
1+α .
(2.15) now becomes:
T
t0
(T − t) u(t) 2
L2 dt ≤ C(u0) (T − t0)
2α
1+α + (T − t0)
α
1+α (T − t0) u(t0) L2
≤ C(u0)(T − t0)
2α
1+α + (T − t0)2
u(t0) 2
L2 . (2.16)
In order to integrate this differential inequality, let
g(t0) =
T
t0
(T − t) u(t) 2
L2 dt, (2.17)
then (2.16) means:
g(t) ≤ C(T − t)
2α
1+α − (T − t)g (t)
16 P. RAPHAËL
ie
g
T − t
=
1
(T − t)2
((T − t)g + g) ≤
1
(T − t)2− 2α
1+α
.
Integrating this in time yields
g(t)
T − t
≤ C(u0) +
1
(T − t)1− 2α
1+α
ie g(t) ≤ C(u0)(T − t)
2α
1+α
for t close enough to T, which together with (2.17) yields (2.7).
2.5. More blow up problems. The study of singularity formation for nonlinear
dispersive equations has experienced a substantial acceleration since the end of the
1990’s in particular in the continuation of the pioneering breakthrough works by
Merle and Zaag on the nonlinear heat equation [83], [84], [85], and Martel and
Merle on the mass critical (gKdV) problem [55], [56], [57], [58], [59]. The analysis
has spread to various other problems and led to the development of new tools. It
is not the aim of these notes to give a complete account of the existing literature,
but we would like to point out the deep unity between some of these recent works.
One particularly active direction or research is on energy critical models sc = 1
which surprisingly enough display a similar structure like the mass critical problem,
even though essential new phenomenons occur. This includes energy critical wave
or heat problems, or more geometric problems like wave and Schrödinger maps
for which the sole existence of blow up solutions in the critical regimes has been
a long standing open problem. Among the key results obtained in the past ten
years, let us mention some dynamical constructions: the first construction of blow
up solutions for the energy critical wave map problem by Krieger, Schlag, Tataru
[41], the derivation of the stable regime for the wave map jointly with Rodnianski
[99], the first construction of blow up bubble for the Schrödinger map problem and
the discovery of the rotational instability jointly with Merle and Rodnianski [77].
Moreover, a new generation of classification theorems have occurred in the direction
of the multi solitary wave resolution conjecture, see in particular Duyckaerts, Kenig,
Merle [15] for the energy critical nonlinear wave equation and the spectacular series
of works by Merle and Zaag [86], [87], [88], [89], [90] which give the first complete
classification of all blow up regimes for a nonlinear wave equation.
3. The mass critical problem
We focus in this section and for the rest of these notes onto the L2 critical case
p = 1 +
4
N
, sc = 0.
which is the smallest power nonlinearity for which blow up occurs. We will show
that a large part of the orbital stability theory developed for subcritical problems
still applies in some generalized sense and provides some essential information on
the structure of the blow up bubble.
3.1. Variational characterization of the ground state. The minimization problem
(1.12) is no longer adapted to the critical problem due to the L2 scaling invari-
ance
uλ(t, x) = λ
N
2 u(λ2
t, λx). (3.1)
Indeed, one easily proves that I(M) = 0 for M << 1 and I(M) = −∞ for M 1.
In fact, as observed by Weinstein [111], the L2 criticality of (1.1) corresponds to an
exact balance between the kinetic and potential energies which can be quantified
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION17
through the knowledge of the sharp constant in the Gagliardo-Nirenberg inequality
(1.7).
Proposition 3.1 (Sharp Gagliardo-Nirenberg estimate, [111]). Let the H1 func-
tional:
J(v) =
( | v|2)( |v|2)
2
N
|v|2+ 4
N
. (3.2)
The minimization problem
min
v∈H1, v=0
J(v)
is attained on the three parameters family:
λ
N
2
0 Q(λ0x + x0)eiγ0
, (λ0, x0, γ0) ∈ R+
∗ × RN
× R,
where Q is the unique ground state solution to:
∆Q − Q + Q1+ 4
N = 0, Q > 0, Q radial
Q(r) → 0 as r → +∞.
(3.3)
In particular, there holds the following Gagliardo-Nirenberg inequality with best con-
stant:
∀v ∈ H1
, E(v) ≥
1
2
| v|2
1 −
|v|L2
|Q|L2
4
N
. (3.4)
While E(Q) = I(M) < 0 in the subcritical case, we have in the critical case 6
E(Q) = 0.
A reformulation of (3.4) which is very useful is the following variational characterization
of Q:
Proposition 3.2 (Variational characterization of the ground state). Let v ∈ H1
such that
|v|2
= Q2
and E(v) = 0,
then
v(x) = λ
N
2
0 Q(λ0x + x0)eiγ0
,
for some parameters λ0 ∈ R∗
+, x0 ∈ RN , γ0 ∈ R.
To sum up, the situation is as follows: let v ∈ H1, then if |v|L2 < |Q|L2 , the
kinetic energy dominates the potential energy and (3.4) yields E(v) > C(v) | v|2
and the energy is in particular non negative; at the critical mass level |v|L2 = |Q|L2 ,
the only zero energy function is Q up to the symmetries of scaling, phase and
translation which generate the three dimensional manifold of minimizers of (3.2).
For |v|L2 > |Q|L2 , the sign of the energy is no longer prescribed.
Remark 3.3. Remark that on the contrary to the subcritical case, the scaling (3.1)
leaves the L2 norm invariant and hence there are no small solitary waves in the
critical case.
A simple consequence of the sharp lower bound (3.4) is the concentration of the
mass at blow up given by Theorem 2.4.
6This can be seen for example by multiplying the Q equation by N
2
Q + y · Q and integrating
by parts.
18 P. RAPHAËL
Proof of Theorem 2.4. The proof is purely variational. We prove the result in the
radial case for N ≥ 2. The general case follows from concentration compactness
techniques, see [91], [28]. Let u0 ∈ H1 radial and assume that the corresponding
solution u(t) to (1.1) blows up at time 0 < T < +∞, or equivalently:
lim
t→T
| u(t)|L2 = +∞. (3.5)
We need to prove (2.4) and argue by contradiction: assume that for some R > 0
and ε > 0, there holds on some sequence tn → T,
lim
n→+∞ |y|≤R
|u(tn, y)|2
dy ≤ Q2
− ε. (3.6)
Let us rescale the solution by its size and set:
λ(tn) =
1
| u(tn)|L2
, vn(y) = λ
N
2 (tn)u(tn, λ(tn)y),
then from explicit computation:
| vn|L2 = 1 and E(vn) = λ2
(tn)E(u). (3.7)
First observe that vn is H1 bounded and we may assume on a sequence n → +∞:
vn V in H1
.
We first claim that V is non zero. Indeed, from (3.5), (3.7) and the conservation of
the energy for u(t), E(vn) → 0 as n → +∞, and thus:
1
2 + 4
N
|vn|2+ 4
N =
1
2
| vn|2
− E(vn) =
1
2
− E(vn) →
1
2
as n → +∞.
Now from the compact embedding of H1
radial → L2+ 4
N , vn → V in L2+ 4
N up to a
subsequence, and thus 1
2+ 4
N
|V |2+ 4
N ≥ 1
2 and V is non zero. Moreover, from the
weak H1 convergence and the strong L2+ 4
N convergence,
E(V ) ≤ lim inf
n→+∞
E(vn) = 0.
Last, we have from (3.5), (3.6) and the weak H1 convergence: ∀A > 0
|y|≤A
|V (y)|2
dy ≤ lim inf
n→+∞ |y|≤A
|vn(y)|2
dy ≤ lim
n→+∞ |y|≤ R
λ(tn)
|v(tn, y)|2
dy
= lim
n→+∞ |x|≤R
|u(tn, x)|2
dx ≤ Q2
− ε.
Thus |V |2 ≤ Q2 − ε which together with V non zero and E(V ) ≤ 0 contradicts
the sharp Gagliardo-Nirenberg inequality (3.4).
The proof in the non radial case has been simplified by Hmidi, Keraani [28],
which derived the following optimal result from concentration compactness - more
precisely profile decomposition- techniques:
Lemma 3.4. Let a sequence un ∈ H1 with
lim sup
n→+∞
| un|L2 ≤ | Q|L2 , lim sup
n→+∞
|un|
L2+ 4
N
≥ |Q|
L2+ 4
N
,
then there exists xn ∈ RN and V ∈ H1 such that up to a subsequence:
vn(· + xn) V in H1
with |V |L2 ≥ |Q|L2 .
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION19
3.2. The sharp global wellposedness criterion. A generalization of Theorem
1.1 has been obtained by Weinstein [111]:
Theorem 3.5 (Global well posedness for subcritical mass, [111]). Let u0 ∈ H1 with
|u0|L2 < |Q|L2 , the corresponding solution u(t) to (1.1) is global and bounded in H1.
More precisely, the solution scatters as t ± ∞.
Proof of Theorem 3.5. From the conservation of the L2 norm, |u(t)|L2 < |Q|L2 for
all t ∈ [0, T), and thus an a priori bound on |u(t)|H1 follows from the conservation of
the energy and the sharp Gagliardo-Nirenberg inequality (3.4) applied to v = u(t).
The scattering claim is easily proved for u0 ∈ Σ = H1 ∩{xu ∈ L2} using the explicit
pseudo conformal symmetry: if u(t, x) is a solution to (1.1), then so is
v(t, x) =
1
|t|
N
2
u(
−1
t
,
x
t
)ei
|x|2
4t . (3.8)
The pseudo conformal symmetry is a well known symmetry of the linear Schrödinger
flow and a symmetry of the nonlinear problem in the mass critical case only. It is
moreover an L2 isometry and thus applying Weinstein’s criterion to v ensures that
v has a limit in Σ as t ↑ 0, and hence u scatters as t → +∞ as readily seen on (3.8).
The case when u0 ∈ L2 only is considerably more delicate and relies on the rigidity
theorem approach developed by Kenig, Merle [33], see Killip, Tao, Visan, Li, Zhang
[35], [36], [37] and references therein, Dodson [14].
A spectacular feature is that Weinstein’s criterion for global existence is sharp.
On the one hand, from (3.3),
W(t, x) = Q(x)eit
is a gobal solution to (1.1) with critical mass |W|L2 = |Q|L2 which does not disperse.
One should thus think of |Q|L2 as the minimal amount of mass required to avoid
complete dispersion of the wave, and the solitary wave is the smallest non linear
object for which dispersion and concentration exactly balance each other.
Observe now that the pseudo conformal symmetry (3.8) applied to the solitary
wave solution u(t, x) = Q(x)eit yields the explicit minimal mass blow up element:
S(t, x) =
1
|t|
N
2
Q(
x
t
)e−i
|x|2
4t
+ i
t (3.9)
which scatters as t → −∞, and blows up at the origin at the speed
| S(t)|L2 ∼
1
|t|
(3.10)
by concentrating its mass:
|S(t)|2
|Q|2
L2 δx=0 as t ↑ 0. (3.11)
Remark 3.6. For the mass critical NLS, the sharp threshold for global existence
and for scattering are therefore the same. This in fact an exceptional case induced
by the Laplace operator and the Galilean symmetry -which is again an L2 isometry-.
For a more general dispersion of the type (−∆)α, these threshold are not the same,
[39].
20 P. RAPHAËL
3.3. Orbital stability of the ground state. More can be said on the structure
of the singularity formation, and in particular on the blow up profile for initial data
with L2 mass just above the critical mass required for blow up:
u0 ∈ Bα∗ = {u0 ∈ H1
with Q2
≤ |u0|2
≤ Q2
+ α∗
} (3.12)
for some parameter α∗ > 0 small enough. This situation is moreover conjectured to
locally describe the generic blow up dynamic around one blow up point.
Let us recall that E(Q) = 0 together with the virial blow up result of Theorem
2.1 imply the instability of the solitary wave Q(x)eit. We claim however that the
orbital stability of Q may be retrieved in some sense according to the following
generalization of Theorem 1.4:
Theorem 3.7 (Orbital stability in the critical case). Let N ≥ 1. For all α∗ > 0
small enough, there exists δ(α∗) with δ(α∗) → 0 as α∗ → 0 such that the following
holds true. Let u0 ∈ H1 with
|u0|2
≤ Q2
+ α∗
, E(u) ≤ α∗
| u|2
, (3.13)
and let u(t) be the corresponding solution to (1.1) with life time 0 < T ≤ +∞, then
there exist (x(t), γ(t)) ∈ C0([0, T), RN × R) such that:
∀t ∈ [0, T), λ
N
2 (t)u(t, λ(t)x + x(t))e−iγ(t)
− Q H1 < δ(α∗
). (3.14)
Note that a finite time blow up solution with small super critical mass automatically
satisfies (3.13) near blow up time, and hence it is closed to the ground state
in H1 up to the set of H1 symmetries. This property is again purely based on
the conservation laws and the variational characterization of Q, and not on refined
properties of the flow.
Proof of Theorem 3.7. Equivalently, we need to prove the following: let a sequence
un ∈ H1 with
un L2 → Q L2 , lim sup
n→+∞
E(un)
un
2
L2
≤ 0, (3.15)
let
vn = λ
N
2
n u(λnx) with λn =
Q L2
un L2
, (3.16)
then there exist xn ∈ RN , γn ∈ R such that:
vn(· + xn)eiγn
→ Q in H1
as n → +∞. (3.17)
Indeed, observe from (3.15) and (3.16) that
vn L2 → Q L2 , vn L2 = Q L2 , lim sup
n→+∞
E(vn) ≤ 0.
We now apply Proposition 1.6 to vn. If vanishing occurs, then up to a subsequence,
we have for n large enough:
E(vn) ≥
Q 2
L2
4
which contradicts lim supn→+∞ E(vn) ≤ 0. If dichotomy occurs, then there exist
wk, zk and 0 < α < 1 such that
wk L2 → α Q L2 , zk L2 → (1 − α) Q L2 and 0 ≥ lim sup
k→+∞
(E(wk) + E(zk)).
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION21
But from the sharp Gagliardo-Nirenberg inequality (3.4) applied to wk and zk, this
implies
wk L2 + zk L2 → 0 as k → +∞
and thus
vnk
L2+ 4
N
→ 0 as k → +∞,
and we are back to the vanishing case. Hence compactness occurs and
vn(· + xn) → V strongly in L2+ 4
N , L2
up to a subsequence. But then E(v) ≤ 0 and V L2 = Q L2 imply from (3.4) and
Proposition 3.2 that V (x) = Q(x + x0)eiγ0 . This in turns implies E(V ) = 0 and
thus | vn(· + xn)|2
L2 → | Q|2
L2 which impies (3.17).
4. Dynamical construction of blow up solutions
We give in this section an overview on the known results on singularity formation
in the mass critical case which go beyond the pure variational analysis of the previous
section and rely on an explicit construction of blow up solutions for data near the
ground state. This kind of question still attracts a considerable amount of interest,
and we shall not be able to give a complete overview of the existing literature in
these notes. We shall only give some key results in connection in particular with the
question of the description of the flow near the ground state solitary wave which is
the first nonlinear object.
4.1. Minimal mass blow up. Initial data u0 ∈ H1 with subcritical mass |u0|L2 <
|Q|L2 generate global bounded solutions from Theorem 3.5. Moreoever, there exists
an explicit minimal mass blow up element S(t) induced by the pseudo conformal
symmetry (3.8) and explicitly given by (3.9). The existence of the minimal element
plays a distinguished role in the Kenig Merle approach to global existence [33]. An
essential feature of (3.9) is that S(t) is compact up to the symmetries of the flow,
meaning that all the mass is put into the singularity formation. The basic intuition
is that such a behavior is very special, and minimal elements should be classified7
.
This was proved using the pseudo conformal symmetry in a seminal work by Merle:
Theorem 4.1 (Classification of the minimal mass blow up solution, [66]). Let
u0 ∈ H1 with
|u0|L2 = |Q|L2 .
Assume that the corresponding solution to (1.1) blows up in finite time 0 < T < +∞.
Then
u(t) = S(t)
up to the symmetries.
Before giving the proof of Merle’s classification Theorem, let us say that the
question of the existence of minimal elements in various settings has been a long
standing open problem, mostly due to the fact that the existence of the minimal
element for NLS relies entirely on the exceptional pseudo conformal symmetry.
Merle in [67] considered the inhomogeneous problem
i∂tu + ∆u + k(x)u|u|2
= 0, x ∈ R2
which breaks the full symmetry group, and obtains for non smooth k non existence
results of minimal elements. A contrario and more recently, a sharp criterion for the
7This is a dispersive intuition which for example is completely false in the parabolic setting, [5].
22 P. RAPHAËL
existence and uniqueness of minimal solutions is derived in collaboration with Szeftel
in [101] which relies on a dynamical construction and new Lypapounov rigidity
functionals at the minimal mass level. A further extension to non local dispersion
can be found in [39] which shows that minimal mass blow up is in fact the generic
situation, and has little to do with the pseudo conformal symmetry, see also [2] for
an extension to curved backgrounds, and Theorem 4.6 for the case of the critical
(gKdV).
Proof of Theorem 4.1. This is the first proof of classification of minimal elements
in the Schrödinger setting. We advise the reader to compare it with the proof of
the Liouville theorem in [33] and observe the deep unity of both arguments. The
original proof by Merle [66] has been further simplified by Banica [1] and Hmidi,
Keraani [27], and it is the proof we present now.
step 1 Compactness of the flow in H1 up to scaling. Let u as in the hypothesis
of the Theorem with blow up time 0 < T < ∞. Let
λ(t) =
| Q|L2
| u(t)|L2
→ 0 as t → T.
Then
v(t, x) = λ
N
2 (t)u(t, λ(t)x + x(t))
satisfies:
| v(t)|L2 = | Q|L2 , lim
t→T
E(v) = 0, |v(t)|L2 = |Q|L2 .
Arguing as for the proof of Theorem 3.7, we conclude from standard concentration
compactness techniques and the variational characterization of the ground state
that:
v(t, x + x(t))eiγ(t)
→ Q in H1
as t → T. (4.1)
step 2 A refined Cauchy-Schwarz for critical mass functions. For |w|L2 < |Q|L2 ,
the energy controls the kinetic energy from (3.4). This controls fails for |w|L2 =
|Q|L2 but can be retrieved in some weak sense. Indeed, Banica observed the following:
let a smooth real valued ψ and w ∈ H1 with |w|L2 = |Q|L2 , then:
|Im( ψ · ww)|2
E(w) | ψ|2
|w|2
1
2
. (4.2)
Indeed, for any a > 0,
|weiaψ
|L2 = |Q|L2 and thus E(weiaψ
) ≥ 0
and the result follows by expanding in a.
step 3 L2 compactness of u and control of the concentration point. We now
claim that u is L2 compact: ∀ε > 0, ∃R > 0 such that
∀t ∈ [0, T),
|x|≥R
|u(t, x)|2
dx < ε. (4.3)
Indeed, pick ε large enough, For R > 0, let χR(x) = χ( x
R ) where χ is a smooth
radial cut off function with χ(r) = 0 for r ≤ 1
2, χ(r) = 1 for r ≥ 1. Then integrating
by parts in (1.1) and using (4.2), we get:
1
2
d
dt
χR|u|2
= |Im( χR · uu)| ≤ C E(u) | χR|2
|u|2
1
2
≤
C
R
E0|u0|L2
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION23
where we used the conservation of energy and L2 norm in the last step. Integrating
in time on [0, T] and using T < +∞ yields (4.3).
Now observe that (4.1) and (4.3) automatically imply a localization of the concentration
point:
∀t ∈ [0, T), |x(t)| ≤ C(u0). (4.4)
step 4 u ∈ Σ. From (4.4) and up to a translation in space, we may consider a
sequence of times tn → T such that
x(tn) → 0 ∈ RN
.
From (4.1), (4.3):
|u(tn, x)|2
|Q|2
δ0 as tn → T. (4.5)
This means that at time T, all the mass is at the origin. Even though there is no
finite speed of propagation for (NLS), the idea is to integrate backwards from the
singularity to conclude that this implies that there was not much mass initially at
infinity, that is
u0 ∈ Σ = H1
∩ {xu} ∈ L2
. (4.6)
This step is very important and corresponds to a non trivial gain of regularity for
the asymptotic object which is a direct consequence of its non dispersive behavior.
Let a smooth radial cut off function ψ(r) = r2 for r ≤ 1, ψ(r) = 8 for r ≥ 2 and
such that | ψ|2 ≤ Cψ. Let A > 0 and ψA(r) = A2ψ( r
A ), then:
| ψA|2
ψA. (4.7)
Then integrating by parts in (1.1), we have using (4.2) and (4.7):
1
2
d
dt
ψA|u|2
= |Im( ψA · uu)| E0 | ψA|2
|u|2
1
2
E0 ψA|u|2
1
2
or equivalently:
d
dt
ψA|u|2 E0. (4.8)
Now observe from (4.5) that
ψA|u(tn)|2
→ 0 as tn → T.
Integrating (4.8) on [t, tn] and letting tn → T, we thus get:
∀t ∈ [0, T), ψA|u(t)|2 ≤ C(E0)(T − t).
Note that the right hand side of the above expression is independent of A. We may
thus let A → ∞ and conclude to an even more precise version of (4.6):
∀t ∈ [0, T), u(t) ∈ Σ with |x|2
|u(t, x)|2
dx → 0 as t → T. (4.9)
step 5 Pseudo-conformal transformation. The conclusion of the proof is pure
magic. It relies on the following completely general fact. Let u(t) be a solution to
24 P. RAPHAËL
(1.1) leaving on [0, T), then
v(t, x) =
T
T + t
N
2
u
tT
T + t
,
Tx
T + t
e
i
|x|2
4(T +t)
is a solution to (1.1) with
|v|L2 = |u|L2 and E(v) =
1
8
lim
t→T
|x|2
|u(t, x)|2
dx.
Applying this to u and using (4.9), this implies that
|v|L2 = |u|L2 = |Q|L2 and E(v) = 0.
From Proposition 3.2, v = Q up to the symmetries of the flow, and this concludes
the proof of Theorem 4.1.
4.2. Log log blow up. The only explicit blow up solution we have encountered so
far is the minimal mass blow up bubble (3.9). This bubble is intrinsically unstable
because a mass subcritical perturbation leads to a globally defined solution. The
question of the description of stable blow up bubbles has attracted a considerable
attention which started in the 80’s with the development of sharp numerical methods
and the prediction of the "log-log law" for NLS by Landman, Papanicoalou,
Sulem, Sulem [43].
To simplify the presentation, let us restrict our attention with mass just above
the minimal required for singularity formation
u0 ∈ Bα∗ = u0 ∈ H1
with |Q|L2 < |u0|L2 < |u0|L2 + α∗
, 0 < α∗
1. (4.10)
A general and fundamental open problem is to completely describe the flow for such
initial data which in some sense corresponds according to the scattering statement of
Theorem 3.5 to the first non linear zone. The generalized orbital stability statement
of Theorem 3.7 ensures that under (4.10), if u blows up at T < +∞. then for t
close enough to T, the solution must admit a nonlinear decomposition
u(t, x) =
1
λ(t)
N
2
(Q + ε)(t,
x − x(t)
λ(t)
)eiγ(t)
, (4.11)
where
|ε(t)|H1 ≤ δ(α∗
), λ(t) ∼
1
| u(t)|L2
. (4.12)
This decomposition implies that in any blow up regime, the ground state solitary
wave Q is a good approximation of the blow up profile, and this is the starting
point for a perturbative analysis. The sharp description of the blow up bubble now
relies on the extraction of the finite dimensional and possibly universal dynamic for
the evolution of the geometrical parameters (λ(t), x(t), γ(t)) which is coupled to the
infinite dimensional dispersive dynamic driving the small excess of mass ε(t).
Remark 4.2. An illuminating computation is to reformulate (3.9) for the minimal
blow up element in terms of (4.11):
λ(t) = |t|, ε(t, y) = Q(y) e−i
b(t)|y|2
4 − 1 , b(t) = |t|.
All possible regimes of λ(t) are not known, but some progress has been done
on the understanding of stable and threshold dynamics. The following Theorem
summarizes the series of results obtained in [71], [72], [73], [74], [75], [97]:
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION25
Theorem 4.3 ([71], [72], [73], [74], [75], [97]). Let N ≤ 5. There exists a universal
constant α∗ > 0 such that the following holds true. Let u0 ∈ Bα∗ and
u ∈ C([0, T), H1), 0 < T ≤ +∞ be the corresponding solution to (1.1).
(i) Sharp L2 concentration: Assume T < +∞, then there exist parameters
(λ(t), x(t), γ(t)) ∈ C1([0, T), R∗
+ × RN × R) and an asymptotic profile u∗ ∈ L2 such
that
u(t) −
1
λ(t)
N
2
Q
x − x(t)
λ(t)
eiγ(t)
→ u∗
in L2
as t → T, (4.13)
and the blow up point is finite:
x(t) → x(T) ∈ RN
as t → T.
(ii) Classification of the speed: Under (i), the solution is either in the log-log regime
λ(t)
log | log(T − t)|
T − t
→
√
2π as t → T (4.14)
and then the asymptotic profile is not smooth:
u∗
/∈ H1
and u∗
/∈ Lp
for p > 2, (4.15)
or there holds the sharp lower bound
λ(t) C(u0)(T − t) (4.16)
and the improved regularity:
u∗
∈ H1
. (4.17)
(iii) Sufficient condition for log-log blow up: Assume E0 < 0, then the solution blows
un finite time T < +∞ in the log log regime (4.14).
(iv) H1 stability of the log log blow up: More generally, the set of initial data in Bα∗
such that the corresponding solution to (1.1) blows up in finite time with the log-log
law (4.14) is open in H1.
Comments on the result
1. The log log law. The log log law (4.14) of stable blow up was first proposed
in the pioneering formal and numerical work [43]. The first rigorous construction
of such a solution is due to Galina Perelman [95] in dimension N = 1. The proof
of Theorem 4.3 involves a mild coercivity property of the linearized operator close
to Q, see the Spectral Property 5.6, which is proved in dimension N = 1 in [71]
and checked numerically in an elementary way in [18] for N ≤ 5. Here we face the
difficulty that there is no explicit formula for the ground state in dimensions N ≥ 2.
2. Upper bound on the blow up speed: There exists no upper bound of no type on
the blow up speed | u(t)|L2 in the mass critical case, even for data u0 ∈ Bα∗ only.
The lower bound (4.16) is sharp and saturated by the minimal blow up element S(t).
The derivation of slower blow up, which through the pseudo conformal symmetry
is equivalent to the construction of infinite time grow up solutions, is linked to the
description of the flow near the ground state which is still incomplete for (NLS).
The intuition is led here by the recent classification results obtained for the mass
critical KdV problem which we present in section 4.5.
3. Quantization of the blow up mass: The strong convergence (4.13) gives a
complete description of the blow up bubble in the scaling invariance space and
26 P. RAPHAËL
implies in particular that the mass which is put into the singularity formation is
quantized
|u(t)|2
|Q|2
L2 δx=x(T) + |u∗
|2
as t → T, |u∗
|2
∈ L2
which shows the validity of the conjecture (*) for near minimal mass blow up solutions.
This kind of general asymptotic simplification theorem started in the dispersive
setting in the pioneering works by Martel and Merle [55] , and was recently
propagated to impressive classification result -without assumption of size on the
data- for energy critical wave equations [15]. Underlying the convergence (4.13) is
the asymptotic stability statement of the solitary wave as the universal attractor of
all blow up solutions which in the language (4.11) means
ε(t, x) → 0 as t → T in L2
loc.
In fact, there are steps in the proof of Theorem 4.3 and the derivation of either
upper bounds or lower bounds on the blow up rate is intimately connected to the
question of dispersion for the excess of mass ε(t, x).
4. Asymptotic profile: The regularity of the asymptotic profile u∗ sees the change
of regime because in the stable log log regime, the singular and regular parts of the
solution are very much coupled, while they are more separated in any other regimes.
4.3. Threshold dynamics. We still consider small super critical mass initial data
u0 ∈ Bα∗ . Theorem 4.3 describes the stable log log blow up. The explicit minimal
mass blow up given by (4.3) does not belong to this class and is unstable. Bourgain
and Wang [8] observed however that S(t) can be stabilized on a finite codimensional
manifold, and they do so by integrating the flow backwards from the singularity.
The excess of mass in this regime corresponds to a flat and smooth asymptotic
profile. More precisely, let N = 1, 2, fix the origin as the blow up point and let a
limiting profile u∗ ∈ H1 such that
di
dxi
u∗
(0) = 0, 1 ≤ i ≤ A, A 1, (4.18)
then one can build a solution to (1.1) which blows up at t = 0 at x = 0 and satisfies:
u(t) − S(t) → u∗
in H1
as t ↑ 0. (4.19)
We refer to [40] for a further discussion on the manifold construction. Note that this
produces blow up solutions with super critical mass |u0|L2 > |Q|L2 which saturate
the lower bound (4.16):
| u(t)|L2 ∼
1
T − t
.
Also for small L2 perturbation of S(−1), the Bourgain Wang solution blows up
at t = 0 but is global and scatters as t → −∞, simply because S(t) scatters as
t → −∞, and scattering is an L2 stable behavior8
.
We proved in collaboration with Merle and Szeftel in [79] that these solutions sit
on the border between the two open sets of solutions which scatter to the left as
t → −∞ and respectively are global to the right and scatter as t → +∞, and blow
up in finite time in the log log regime.
8This is a simple consequence of Strichartz estimates and the L2
critical Cauchy theory of
Cazenave-Weissler [13].
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION27
Theorem 4.4 (Strong instability of Bourgain Wang solutions, [79]). Let N = 1, 2.
Let u∗ be a smooth radially symmetric satisfying the degeneracy at blow up point
(4.18). Let u0
BW ∈ C((−∞, 0), H1) be the corresponding Bourgain-Wang. solution.
Then there exists a continuous map
Γ : [−1, 1] → Σ
such that the following holds true. Given η ∈ [−1, 1], let uη(t) be the solution to
(1.1) with data uη(−1) = Γ(η), then:
• Γ(0) = u0
BW (−1) ie ∀t < 0, uη=0(t) = u0
BW (t) is the Bourgain Wang solution
on (−∞, 0) with blow up profile S(t) and regular part u∗;
• ∀η ∈ (0, 1], uη ∈ C(R, Σ) is global in time and scatters forward and back-
wards;
• ∀η ∈ [−1, 0), uη ∈ C((−∞, T∗
η ), Σ) scatters to the left and blows up in finite
time T∗
η < 0 on the right in the log-log regime (4.14) with
T∗
η → 0 as η → 0. (4.20)
Note that this theorem describes the flow near the Bourgain Wang solution along
one instability solution. A major open problem in the field is to describe the flow
near the ground state Q. Theorem 4.4 is a first step towards the description of
the flow near the Bourgain Wang solutions which itself is a very interesting open
problem.
4.4. Structural instability of the log-log law. Another model with fundamental
physical relevance, [106], is the Zakharov system in dimensions N = 2, 3:
iut = −∆u + nu
1
c2
0
ntt = ∆n + ∆|u|2 (4.21)
for some fixed constant 0 < c0 < +∞. In the limit c0 → +∞, we formally recover
(1.1). In dimension N = 2, this system displays a variational structure like (1.1),
even though the scaling symmetry is destroyed by the wave coupling. In particular,
a virial law in the spirit of (2.1) holds and yields finite time blow up for radial non
positive energy initial data, see Merle [69]. Moreover, a one parameter family of
blow up solutions has been constructed as a continuation of the exact S(t) solution
for (1.1), see Glangetas, Merle, [24]. These explicit solutions have blow up speed:
| u(t)|L2 ∼
C(u0)
T − t
and appear to be stable from numerics, see Papanicolaou, Sulem, Sulem, Wang, [94].
Now from Merle, [68], all finite time blow up solutions to (4.21) satisfy
| u(t)|L2 ≥
C(u0)
T − t
.
In particular, there will be no log-log blow up solutions for (4.21). This fact suggests
that in some sense, the Zakharov system provides a much more stable and robust
blow up dynamics than its asymptotic limit (NLS). This fact enlightens the belief
that the log-log law heavily relies on the specific algebraic structure of (1.1), and
some non linear degeneracy properties will indeed be at the heart of our understanding
of the blow up dynamics. Let us insist that the fine study of the singularity
formation for the Zakharov system is mostly open, and in some sense it is the first
towards the understanding of more physical and complicated systems related to
Maxwell’s equations.
28 P. RAPHAËL
4.5. Classification of the flow near Q: the case of the generalized KdV.
We present in this section the recent series of results [62], [61], [60] which give a
complete description of the flow near the ground for an L2 critical problem: the
generalized KdV equation
(gKdV )
∂tu + (uxx + u5)x = 0
u|t=0 = u0
, (t, x) ∈ R × R. (4.22)
This problems admits the same L2 norm and energy conservation laws like (NLS),
and the same mass critical scaling. The solitary wave is here a traveling wave
solution
u(t, x) = Q(x − t)
where Q is the one dimensional ground state
Q(x) =
3
ch2
(2x)
1
4
.
This model problem has been thoroughly studied by Martel and Merle in the pioneering
breakthrough works [55], [56],[57],[58], [59]
as a toy model for which the pseudo conformal symmetry and the associated
virial algebra are lost. The long standing open problem of the existence of blow up
solutions was solved in [70], but the structure of the singularity formation was still
only poorly understood. We give in the series of works [62], [61], [60] a complete
description of the flow near the ground state and expect that the obtained picture
is canonical.
More precisely, let the set of initial data
A = u0 = Q + ε0 with ε0 H1 < α0 and
y>0
y10
ε2
0 < 1 ,
and consider the L2 tube around the family of solitary waves
Tα∗ =



u ∈ H1
with inf
λ0>0, x0∈R
u −
1
λ
1
2
0
Q
. − x0
λ0
L2 < α∗



.
We first claim the rigidity of the dynamics for data in A:
Theorem 4.5 (Rigidity of the flow in A, [62]). Let 0 < α0 α∗ 1 and u0 ∈ A.
Let u ∈ C([0, T), H1) be the corresponding solution to (4.22). Then one of the
following three scenarios occurs:
(Blow up): the solution blows up in finite time 0 < T < +∞ in the universal regime
|u(t)|H1 =
(u0) + o(1)
T − t
as t → T, (u0) > 0. (4.23)
(Soliton): the solution is global T = +∞ and converges asymptotically to a solitary
wave.
(Exit): the solution leaves the tube Tα∗ at some time 0 < t∗
u < +∞.
Moreover, the scenarios (Blow up) and (Exit) are stable by small perturbation of
the data in A.
In other words, we obtain a complete classification of solutions with data in A
which remain close in the L2 critical sense to the manifold of solitary waves. It
remains to understand the long time dynamics in the (Exit) regime. The first
step is the existence and uniqueness of a minimal blow up element which is the
generalization of the S(t) dynamics for (NLS):
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION29
Theorem 4.6 (Existence and uniqueness of the minimal mass blow up element,
[61]).
(i) Existence. There exists a solution ˜S(t) ∈ C((0, +∞), H1) to (4.22) with minimal
mass ˜S(t) L2 = Q L2 which blows up backward at the origin at the speed
| ˜S(t)|L2 ∼
1
t
as t ↓ 0,
and is globally defined on the right in time.
(ii) Uniqueness. Let u0 ∈ H1 with u0 L2 = Q L2 and assume that the corresponding
solution u(t) to (4.22) blows up in finite time. Then
u ≡ S
up to the symmetries of the flow.
In other words, we recover Merle’s result in the absence of pseudo conformal
symmetry, and the proof is here completely dynamical and deeply related to the
analysis of the inhomogeneous NLS model in [101]. We now claim that ˜S is the
universal attractor of all solutions in the (Exit) regime.
Theorem 4.7 (Description of the (Exit) scenario, [61]). Let u(t) be a solution
of (4.22) corresponding to the (Exit) scenario in Theorem 4.6 and let t∗
u 1 be
the corresponding exit time. Then there exist τ∗ = τ∗(α∗) (independent of u) and
(λ∗
u, x∗
u) such that
(λ∗
u)
1
2 u (t∗
u, λ∗
ux + x∗
u) − ˜S(τ∗
, x)
L2
≤ δI(α0),
where δI(α0) → 0 as α0 → 0.
In fact a solution at the (Exit) time acquires a specific profile with a large defocusing
spreading λ∗
u 1 -coherent with dispersion-. Understanding the flow for u
after the (Exit) is now equivalent to controlling the flow of ˜S(t) for large times. For
(NLS), we can see on the formula (3.9) that S(t) blows up at t = 0 and scatters as
t → +∞. For (gKdV), we know from Theorem 4.6 that ˜S(t) is global as t → +∞,
but scattering is not known. We however expect that this holds true, in which case
because scattering is open in L2 thanks to the Kenig, Ponce, Vega L2 critical theory
[34], we obtain the following:
Corollary 4.8. Assume that S(t) scatters as t → +∞. Then any solution in the
(Exit) scenario is global for positive time and scatters as t → +∞.
It is important to notice that the above results rely on the explicit computation
of the solution in the various regimes, and not on algebraic virial type identities.
Indeed we introduce the nonlinear decomposition of the flow
u(t, x) =
1
λ(t)
1
2
(Q + ε) t,
x − x(t)
λ(t)
and show that to leading order, λ(t) obeys the dynamical system
λtt = 0, λ(0) = 1. (4.24)
The three regimes (Exit), (Blow up), (Soliton) now correspond respectively to
λt(0) > 0, λt(0) < 0 and the threshold dynamic λt(0) = 0.
Our last result shows that the universality of the leading order ODE (4.24) is valid
under the decay assumption u0 ∈ A only, and indeed the tail of slowly decaying data
can interact with the solitary wave which for (KdV) is moving to the right, and this
may lead to new exotic singular regimes:
30 P. RAPHAËL
Theorem 4.9 (Exotic blow up regimes, [60]).
(i) Blow up in finite time: for any ν > 11
13, there exists u ∈ C((0, 1], H1) solution to
(4.22) which blows up at t = 0 with speed
ux(t) L2 ∼ t−ν
as t → 0+
. (4.25)
(ii) Blow up in infinite time: there exists u ∈ C([1, +∞), H1) solution of (4.22)
growing up at +∞ with speed
ux(t) L2 ∼ et
as t → +∞. (4.26)
For any ν > 0, there exists u ∈ C([1, +∞), H1) solution of (4.22) blowing up at
+∞ with
ux(t) L2 ∼ tν
as t → +∞. (4.27)
Such solutions can be constructed arbitrarily close in H1 to the ground state solitary
wave.
Note that this implies in particular that blow up can be arbitrarily slow.
We expect that the (KdV) picture is fairly general, and Theorem 4.4 is a first
step towards a similar description for the mass critical NLS. Let also mention that
in super critical regimes and large dimensions, Nakanishi and Schlag have obtained
a related classification of the flow near the solitary wave which in particular involves
a complete description of the scattering zone and its boundary.
5. The log log upper bound on blow up rate
Our aim in this section is to present a self contained proof of the first result
contained in Theorem 4.3 for the mass critical problem and for small super critical
mass initial data.
Theorem 5.1 ([71],[72]). Let N ≤ 4. There exist universal constants α∗, C∗ > 0
such that the following holds true. Given u0 ∈ Bα∗ with
EG(u) = E(u) −
1
2
Im( uu)
|u|L2
2
< 0, (5.1)
then the corresponding solution u(t) to (1.1) blows up in finite time 0 < T < +∞
and there holds for t close to T:
| u(t)|L2 ≤ C∗ log | log(T − t)|
T − t
1
2
. (5.2)
This theorem is the first fundamental improvement on the virial law: it not only
shows blow up in finite time of non positive energy solutions, it also gives an upper
bound on the blow up rate which in particular rules out the S(t) type of dynamic.
Moreover the steps of the proof are in some sense canonical for our study.
The heart of our analysis will be to exhibit as a consequence of dispersive properties
of (1.1) close to Q strong rigidity constraints for the dynamics of non positive energy
solutions. These will in turn imply monotonicity properties, that is the existence of
a Lyapounov function. The corresponding estimates will then allow us to prove blow
up in a dynamical way and the sharp upper bound on the blow up speed will follow.
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION31
5.1. Existence of the geometrical decomposition. Let an initial data u0 ∈
B(α∗) with EG(u0) < 0. First observe that up to a fixed Galilean transform, we
may equivalently assume
E(u0) < 0 and Im( uu0) = 0. (5.3)
Proposition 3.7 thus applies and implies for t ∈ [0, T) the existence of a geometrical
decomposition
u(t, x) =
1
λ
N
2
0 (t)
(Q + ε0)(t,
x − x0(t)
λ0(t)
)eiγ0(t)
, ε0 H1 ≤ δ(α∗
).
Let us observe that this geometrical decomposition is by no mean unique. Nevertheless,
one can freeze and regularize this decomposition by choosing a set of
orthogonality conditions on the excess of mass: this is the modulation argument
which will be examined later on. Let us so far assume that we have a smooth
decomposition of the solution: ∀t ∈ [0, T),
u(t, x) =
1
λ(t)
N
2
(Q + ε)(t,
x − x(t)
λ(t)
)eiγ(t)
(5.4)
with
λ(t) ∼
C
| u(t)|L2
and |ε(t)|H1 ≤ δ(α∗
) → 0 as α∗
→ 0.
To study the blow up dynamic is now equivalent to understanding the coupling between
the finite dimensional dynamic which governs the evolution of the geometrical
parameters (λ(t), γ(t), x(t)) and the infinite dimensional dispersive dynamic which
drives the excess of mass ε(t).
To enlighten the main issues, let us rewrite (1.1) in the so-called rescaled variables.
Let us introduce the rescaled time:
s(t) =
t
0
dτ
λ2(τ)
.
It is elementary to check that whatever is the blow up behavior of u(t), one always
has:
s([0, T)) = R+
.
Let us set:
v(s, y) = eiγ(t)
λ(t)
N
2 u(λ(t)x + x(t)).
For a given function f, we introduce the generator of L2 scaling
Λf =
N
2
f + y · f
then from direct computation, u(t, x) solves (1.1) on [0, T) iff v(s, y) solves: ∀s ≥ 0,
ivs + ∆v − v + v|v|
4
N = i
λs
λ
Λv + i
xs
λ
· v + ˜γsv, (5.5)
where ˜γ = −γ − s. Now v(s, y) = Q(y) + ε(s, y) and we linearize (5.5) close to Q.
The obtained system has the form:
iεs + Lε = i
λs
λ
ΛQ + γsQ + i
xs
λ
· Q + R(ε), (5.6)
R(ε) formally quadratic in ε, and L = (L+, L−) is the matrix linearized operator
closed to Q which has components:
L+ = −∆ + 1 − 1 +
4
N
Q
4
N , L− = −∆ + 1 − Q
4
N .
32 P. RAPHAËL
A standard approach is to think of equation (5.6) in the following way: it is essentially
a linear equation forced by terms depending on the law for the geometrical
parameters. The classical study of this kind of system relies on the understanding
of the dispersive properties of the propagator eisL of the linearized operator close to
Q. In particular, one needs to exhibit its spectral structure. This has been partially
done by Weinstein, [110], using the variational characterization of Q. The result is
the following: L is a non self adjoint operator with a generalized eigenspace at zero.
The eigenmodes are explicit and generated by the symmetries of the problem:
L+ (ΛQ) = −2Q (scaling invariance), L+( Q) = 0 (translation invariance),
L−(Q) = 0 (phase invariance), L−(yQ) = −2 Q (Galilean invariance).
An additional relation is induced by the pseudo-conformal symmetry:
L−(|y|2
Q) = −4ΛQ,
and this in turns implies the existence of an additional mode ρ solution to
L+ρ = −|y|2
Q.
These explicit directions induce “growing” solutions to the homogeneous linear equation
i∂sε + Lε = 0. More precisely, there exists a (2N+3) dimensional space S
spanned by the above directions such that H1 = M ⊕ S with |eisLε|H1 ≤ C for
ε ∈ M and |eisLε|H1 ∼ s3 for ε ∈ S. As each symmetry is at the heart of a growing
direction, a first idea is to use the symmetries from modulation theory to a priori
ensure that ε is orthogonal to S. Roughly speaking, the strategy to construct blow
up solutions is then: chose the parameters λ, γ, x so as to get good a priori dispersive
estimates on ε in order to build it from a fixed point scheme. Now the fundamental
problem is that one has (2N+2) symmetries, but (2N+3) bad modes in the set S.
Both constructions in [8] and [95] develop non trivial strategies to overcome this
intrinsic difficulty of the problem.
Our strategy will be more non linear. On the basis of the decomposition (5.4),
we will prove bounds on ε induced by the virial structure (2.1). The proof will
rely on non linear degeneracies of the structure of (1.1) around Q. Using then
the Hamiltonian information E0 < 0, we will inject these estimates into the finite
dimensional dynamic which governs λ(t) -which measures the size of the solutionand
prove rigidity properties of Lyapounov type. This will then allow us to prove
finite time blow up together with the control of the blow up speed.
5.2. Choice of the blow up profile. Before exhibiting the modulation theory
type of arguments, we present in this subsection a formal discussion regarding explicit
solutions of equation (5.5) which is inspired from a discussion in [106]. This
corresponds to a finite dimensional reduction of the problem which actually computes
the leading order terms of the solution.
First, let us observe that the key geometrical parameter is λ which measures the
size of the solution. Let us then set
−
λs
λ
= b
and look for solutions to a simpler version of (5.5):
ivs + ∆v − v + ib
N
2
v + y · v + v|v|
4
N = 0.
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION33
From the orbital stability property, we want solutions which remain close to Q in
H1. Let us look for solutions of the form v(s, y) = Qb(s)(y) where the mappings
b → Qb and the law for b(s) are the unknown. We think of b as remaining uniformly
small and Qb=0 = Q. Injecting this ansatz into the equation, we get:
i
db
ds
∂Qb
∂b
+ ∆Qb(s) − Qb(s) + ib(s)
N
2
Qb(s) + y · Qb(s) + Qb(s)|Qb(s)|
4
N = 0.
To handle the linear group, we let Pb(s) = ei
b(s)
4
|y|2
Qb(s) and solve:
i
db
ds
∂Pb
∂b
+ ∆Pb(s) − Pb(s) +
db
ds
+ b2
(s)
|y|2
4
Pb(s) + Pb(s)|Pb(s)|
4
N = 0. (5.7)
A remarkable fact related to the specific algebraic structure of (1.1) around Q is
that (5.7) admits three solutions:
• The first one is (b(s), Pb(s)) = (0, Q), that is the ground state itself. This is
just a consequence of the scaling invariance.
• The second one is (b(s), Pb(s)) = (1
s , Q). This non trivial solution is a
rewriting of the explicit critical mass blow up solution S(t) and is induced
by the pseudo-conformal symmetry.
• The third one is given by (b(s), Pb(s)) = (b, Pb) for some fixed non zero
constant b and Pb satisfies:
∆Pb − Pb +
b2
4
|y|2
Pb + Pb|Pb|
4
N = 0. (5.8)
This corresponds to self similar profiles. Indeed, recall that b = −λs
λ , so if b
is frozen, we have from ds
dt = 1
λ2 :
b = −
λs
λ
= −λλt ie λ(t) = 2b(T − t),
this is the scaling law for the blow up speed.
Now a crucial point again is -[103]- that the solutions to (5.8) never belong to L2
from a logarithmic divergence at infinity:
|Pb(y)| ∼
C(Pb)
|y|
N
2
as |y| → +∞.
This behavior is a consequence of the oscillations induced by the linear group after
the turning point |y| ≥ 2
|b|. Nevertheless, in the ball |y| < 2
|b| , the operator
−∆ + 1 − b2|y|2
4 is coercive, and no oscillations will take place in this zone.
Because we track a log-log correction to the self similar law as an upper bound
on the blow up speed, the profiles Qb = e−i b
4
|y|2
Pb with Pb solving (5.8) are natural
candidates as refinements of the Q profile in the geometrical decomposition (4.11).
Nevertheless, as they are not in L2, we need to build a smooth localized version
avoiding the non L2 tale, what according to the above discussion is doable in the
coercive zone |y| < 2
|b|.
Proposition 5.2 (Localized self similar profiles). There exist universal constants
C > 0, η∗ > 0 such that the following holds true. For all 0 < η < η∗, there exist
constants ν∗(η) > 0, b∗(η) > 0 going to zero as η → 0 such that for all |b| < b∗(η),
let
Rb =
2
|b|
1 − η, R−
b = 1 − ηRb,
34 P. RAPHAËL
BRb
= {y ∈ RN , |y| ≤ Rb}. Then there exists a unique radial solution Qb to



∆Qb − Qb + ib N
2 Qb + y · Qb + Qb|Qb|
4
N = 0,
Pb = Qbei
b|y|2
4 > 0 in BRb
,
Qb(0) ∈ (Q(0) − ν∗(η), Q(0) + ν∗(η)), Qb(Rb) = 0.
Moreover, let a smooth radially symmetric cut-off function φb(x) = 0 for |x| ≥ Rb
and φb(x) = 1 for |x| ≤ R−
b , 0 ≤ φb(x) ≤ 1 and set
˜Qb(r) = Qb(r)φb(r),
then
˜Qb → Q as b → 0
in some very strong sense, and ˜Qb satisfies
∆ ˜Qb − ˜Qb + ib( ˜Qb)1 + ˜Qb| ˜Qb|
4
N = −Ψb (5.9)
with
Supp(Ψ) ⊂ {R−
b ≤ |y| ≤ Rb} and |Ψb|C1 ≤ e
− C
|b| .
Eventually, ˜Qb has supercritical mass:
| ˜Qb|2
= Q2
+ c0b2
+ o(b2
) as b → 0 (5.10)
for some universal constant c0 > 0.
The meaning of this proposition is that one can build localized profiles ˜Qb on the
ball BRb
which are a smooth function of b and approximate Q in a very strong sense
as b → 0, and these profiles satisfy the self similar equation up to an exponentially
small term Ψb supported around the turning point 2
b . The proof of this Proposition
uses standard variational tools in the setting of non linear elliptic problems. In fact,
the implicit function theorem would do the job as well, see [95].
Now one can think of making a formal expansion of ˜Qb in terms of b, and the
first term is non zero:
∂ ˜Qb
∂b |b=0
= −
i
4
|y|2
Q.
However, the energy of ˜Qb is degenerated in b at all orders:
|E( ˜Qb)| ≤ e
− C
|b| , (5.11)
for some universal constant C > 0.
The existence of a one parameter family of profiles satisfying the self similar
equation up to an exponentially small term and having an exponentially small energy
is an algebraic property of the structure of (1.1) around Q which is at the heart of
the existence of the log-log regime.
5.3. Modulation theory. We are now in position to exhibit the sharp decomposition
needed for the proof of the log-log upper bound. From Theorem 3.7 and the
proximity of ˜Qb to Q in H1, the solution u(t) to (1.1) is for all time close to the
four dimensional manifold
M = {eiγ
λ
N
2 ˜Qb(λy + x), (λ, γ, x, b) ∈ R∗
+ × R × RN
× R}.
We now sharpen the decomposition according to the following Lemma. In the sequel,
we let
ε = ε1 + iε2
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION35
be the real and imaginary parts decomposition.
Lemma 5.3 (Non linear modulation of the solution close to M). There exist C1
functions of time (λ, γ, x, b) : [0, T) → (0, +∞) × R × RN × R such that:
∀t ∈ [0, T), ε(t, y) = eiγ(t)
λ
N
2 (t)u(t, λ(t)y + x(t)) − ˜Qb(t)(y) (5.12)
satisfies:
(i)
ε1(t), ΛΣb(t) + ε2(t), ΛΘb(t) = 0, (5.13)
ε1(t), yΣb(t) + ε2(t), yΘb(t) = 0, (5.14)
− ε1(t), Λ2
Θb(t) + ε2(t), Λ2
Σb(t) = 0, (5.15)
− ε1(t), ΛΘb(t) + ε2(t), ΛΣb(t) = 0, (5.16)
where ε = ε1 + iε2, ˜Qb = Σb + iΘb in terms of real and imaginary parts;
(ii) |1 − λ(t)
| u(t)|L2
| Q|L2
| + |ε(t)|H1 + |b(t)| ≤ δ(α∗
) with δ(α∗
) → 0 as α∗
→ 0.
Let us insist onto the fact that the reason for this precise choice of orthogonality
conditions is a fundamental issue which will be addressed in the next section.
Proof of Lemma 5.3. This Lemma follows the standard frame of modulation theory
and is obtained from Theorem 3.7 using the implicit function theorem. From
Theorem 3.7, there exist parameters γ0(t) ∈ R and x0(t) ∈ RN such that with
λ0(t) =
| Q|L2
| u(t)|L2
,
∀t ∈ [0, T), Q − eiγ0(t)
λ0(t)
N
2 u(λ0(t)x + x0(t))
H1
< δ(α∗
)
with δ(α∗) → 0 as α∗ → 0. Now we sharpen this decomposition using the fact that
˜Qb → Q in H1 as b → 0, i.e. we chose (λ(t), γ(t), x(t), b(t)) close to (λ0(t), γ0(t), x0(t), 0)
such that
ε(t, y) = eiγ(t)
λ1/2
(t)u(t, λ(t)y + x(t)) − ˜Qb(t)(y)
is small in H1 and satisfies suitable orthogonality conditions (5.13), (5.14), (5.15)
and (5.16).The existence of such a decomposition is a consequence of the implicit
function Theorem. For δ > 0, let Vδ = {v ∈ H1(C); |v − Q|H1 ≤ δ}, and for
v ∈ H1(C), λ1 > 0, γ1 ∈ R, x1 ∈ RN , b ∈ R small, define
ελ1,γ1,x1,b(y) = eiγ1
λ
N
2
1 v(λ1y + x1) − ˜Qb. (5.17)
We claim that there exists δ > 0 and a unique C1 map : Vδ → (1 − λ, 1 + λ) ×
(−γ, γ) × B(0, x) × (−b, b) such that if v ∈ Vδ, there is a unique (λ1, γ1, x1, b) such
that ελ1,γ1,x1,b = (ελ1,γ1,x1,b)1 + i(ελ1,γ1,x1,b)2 defined as in (5.17) satisfies
ρ1
(v) = ((ελ1,γ1,x1,b)1, ΛΣb) + ((ελ1,γ1,x1,b)2, ΛΘb) = 0,
ρ2
(v) = ((ελ1,γ1,x1,b)1, yΣb) + ((ελ1,γ1,x1,b)2, yΘb) = 0,
ρ3
(v) = − (ελ1,γ1,x1,b)1, Λ2
Θb + (ελ1,γ1,x1,b)2, Λ2
Σb = 0,
ρ4
(v) = ((ελ1,γ1,x1,b)1, ΛΘb) − ((ελ1,γ1,x1,b)2, ΛΣb) = 0.
36 P. RAPHAËL
Moreover, there exists a constant C1 > 0 such that if v ∈ Vδ, then |ελ1,γ1,x1 |H1 +
|λ1 − 1| + |γ1| + |x1| + |b| ≤ C1δ. Indeed, we view the above functionals ρ1, ρ2, ρ3, ρ4
as functions of (λ1, γ1, x1, b, v). We first compute at (λ1, γ1, x1, b, v) = (1, 0, 0, 0, v):
∂ελ1,γ1,x1,b
∂x1
= v,
∂ελ1,γ1,x1,b
∂λ1
=
N
2
v+x· v,
∂ελ1,γ1,x1,b
∂γ1
= iv,
∂ελ1,γ1,x1,b
∂b
= −
∂ ˜Qb
∂b
|b=0
.
Now recall that ( ˜Qb)|b=0 = Q and ∂ ˜Qb
∂b |b=0
= −i|y|2
4 Q. Therefore, we obtain at
the point (λ1, γ1, x1, b, v) = (1, 0, 0, 0, Q),
∂ρ1
∂λ1
= |ΛQ|2
2,
∂ρ1
∂γ1
= 0,
∂ρ1
∂x1
= 0,
∂ρ1
∂b
= 0,
∂ρ2
∂λ1
= 0,
∂ρ2
∂γ1
= 0,
∂ρ2
∂x1
= −
1
2
|Q|2
2,
∂ρ2
∂b
= 0,
∂ρ3
∂λ1
= 0,
∂ρ3
∂γ1
= −|ΛQ|2
2,
∂ρ3
∂x1
= 0,
∂ρ3
∂b
= 0,
∂ρ4
∂λ1
= 0,
∂ρ4
∂γ1
= 0,
∂ρ4
∂x1
= 0,
∂ρ4
∂b
=
1
4
|yQ|2
2.
The Jacobian of the above functional is non zero, thus the implicit function
Theorem applies and conclusion follows.
Let us now write down the equation satisfied by ε in rescaled variables. To
simplify notations, we note
˜Qb = Σ + Θ
in terms of real and imaginary parts. We have: ∀s ∈ R+, ∀y ∈ RN ,
bs
∂Σ
∂b
+ ∂sε1 − M−(ε) + bΛε1 =
λs
λ
+ b ΛΣ + ˜γsΘ +
xs
λ
· Σ (5.18)
+
λs
λ
+ b Λε1 + ˜γsε2 +
xs
λ
· ε1
+ Im(Ψ) − R2(ε)
bs
∂Θ
∂b
+ ∂sε2 + M+(ε) + bΛε2 =
λs
λ
+ b ΛΛΘ − ˜γsΣ +
xs
λ
· Θ (5.19)
+
λs
λ
+ b Λε2 − ˜γsε1 +
xs
λ
· ε2
− Re(Ψ) + R1(ε),
with ˜γ(s) = −s − γ(s). The linear operator close to ˜Qb is now a deformation of the
linear operator L close to Q and is M = (M+, M−) with
M+(ε) = −∆ε1 + ε1 −
4Σ2
N| ˜Qb|2
+ 1 | ˜Qb|
4
N ε1 −
4ΣΘ
N| ˜Qb|2
| ˜Qb|
4
N ε2,
M−(ε) = −∆ε2 + ε2 −
4Θ2
N| ˜Qb|2
+ 1 | ˜Qb|
4
N ε2 −
4ΣΘ
N| ˜Qb|2
| ˜Qb|
4
N ε1.
The formally quadratic in ε interaction terms are:
R1(ε) = (ε1 +Σ)|ε+ ˜Qb|
4
N −Σ| ˜Qb|
4
N −
4Σ2
N| ˜Qb|2
+ 1 | ˜Qb|
4
N ε1 −
4ΣΘ
N| ˜Qb|2
| ˜Qb|
4
N ε2,
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION37
R2(ε) = (ε2 +Θ)|ε+ ˜Qb|
4
N −Θ| ˜Qb|
4
N −
4Θ2
N| ˜Qb|2
+ 1 | ˜Qb|
4
N ε2 −
4ΣΘ
N| ˜Qb|2
| ˜Qb|
4
N ε1.
Two natural estimates may now be performed:
• First, we may rewrite the conservation laws in the rescaled variables and
linearize the obtained identities close to Q. This will give crucial degeneracy
estimates on some specific order one in ε scalar products.
• Next, we may inject the orthogonality conditions of Lemma 5.3 into the
equations (5.18), (5.19). This will compute the geometrical parameters in
their differential form λs
λ , ˜γs, xs
λ , bs in terms of ε: these are the so called modulation
equations. This step requires estimating the non linear interaction
terms. A crucial point here is to use the fact that the ground state Q is
exponentially decreasing in space.
The outcome is the following:
Lemma 5.4 (First estimates on the decomposition). We have for all s ≥ 0:
(i) Estimates induced by the conservation of the energy and the momentum:
|(ε1, Q)| ≤ δ(α∗
) | ε|2
+ |ε|2
e−|y|
1
2
+ e
− C
|b| + Cλ2
|E0|, (5.20)
|(ε2, Q)| δ(α∗
) | ε|2
+ |ε|2
e−|y|
1
2
. (5.21)
(ii) Estimate on the geometrical parameters in differential form:
λs
λ
+ b + |bs| + |˜γs| | ε|2
+ |ε|2
e−|y|
1
2
+ e
− C
|b| , (5.22)
xs
λ
δ(α∗
) | ε|2
+ |ε|2
e−|y|
1
2
+ e
− C
|b| , (5.23)
where δ(α∗) → 0 as α∗ → 0.
Remark 5.5. The exponentially small term in the degeneracy estimate (5.20) is in
fact related to the value of E( ˜Qb), so we use here in a fundamental way the non
linear degeneracy estimate (5.11).
Comments on Lemma 5.4:
1. ˙H1 norm: The norm which appears in the estimates of Lemma 5.4 is essentially
a local norm in space. The conservation of the energy indeed relates the | ε|2
norm with the local norm. These two norms will turn out to play an equivalent role
in the analysis. A key is that no global L2 norm is needed so far.
2. Degeneracy of the translation shift: Comparing estimates (5.22) and (5.23), we
see that the term induced by translation invariance is smaller than the ones induced
by scaling and phase invariances. This non trivial fact is an outcome of our use of
the Galilean transform to ensure the zero momentum condition (5.3).
5.4. The virial type dispersive estimate. We now turn to the proof of the dispersive
virial type inequality at the heart of the proof of the log-log upper bound.
This information will be obtained as a consequence of the virial structure of (1.1)
in Σ.
38 P. RAPHAËL
Let us first recall that the virial identity (2.1) corresponds to two identities:
d2
dt2
|x|2
|u|2
= 4
d
dt
Im( x · uu) = 16E0. (5.24)
We want to understand what information can be extracted from this dispersive information
in the variables of the geometrical decomposition.
To clarify the claim, let us consider an ε solution to the linear homogeneous
equation
i∂sε + Lε = 0 (5.25)
where L = (L+, L−) is the linearized operator close to Q. A dispersive information
on ε may be extracted using a similar virial law like (2.1):
1
2
d
ds
Im( y · εε) = H(ε, ε), (5.26)
where H(ε, ε) = (L1ε1, ε1) + (Lε2, ε2) is a Schrödinger type quadratic form decoupled
in the real and imaginary parts with explicit Schrödinger operators:
L1 = −∆ +
2
N
4
N
+ 1 Q
4
N
−1
y · Q , L2 = −∆ +
2
N
Q
4
N
−1
y · Q.
Note that both these operators are of the form −∆+V for some smooth well localized
time independent potential V (y), and thus from standard spectral theory, they both
have a finite number of negative eigenvalues, and then continuous spectrum on
[0, +∞). A simple outcome is then that given an ε ∈ H1 which is orthogonal to all
the bound states of L1, L2, then H(ε, ε) is coercive, that is
H(ε, ε) ≥ δ0 | ε|2
+ |ε|2
e−|y|
for some universal constant δ0 > 0. Now assume that for some reason -it will be in
our case a consequence of modulation theory and the conservation laws-, ε is indeed
for all times orthogonal to the bound states -and resonances...-, then injecting the
coercive control of H(ε, ε) into (5.26) yields:
1
2
d
ds
Im( y · εε) ≥ δ0 | ε|2
+ |ε|2
e−|y|
. (5.27)
Integrating this in time yields a standard dispersive information: a space time norm
is controlled by a norm in space.
We want to apply this strategy to the full ε equation. There are two main ob-
structions.
First, it is not reasonable to assume that ε is orthogonal to the exact bound states
of H. In particular, due to the right hand side in the ε equation, other second order
terms will appear which will need be controlled. We thus have to exhibit a set of
orthogonality conditions which ensures both the coercivity of the quadratic form H
and the control of these other second order interactions. Note that the number of
orthogonality conditions we can ensure on ε is the number of symmetries plus the
one from b. A first key is the following Spectral Property which has been proved in
dimension N = 1 in [71] using the explicit value of Q and checked numerically for
N = 2, 3, 4.
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION39
Proposition 5.6 (Spectral Property). Let N = 1, 2, 3, 4. There exists a universal
constant δ0 > 0 such that ∀ε = ε1 + iε2 ∈ H1,
H(ε, ε) ≥ δ0 | ε|2
+ |ε|2
e−|y|
−
1
δ0
(ε1, Q)2
+ (ε1, ΛQ)2
+ (ε1, yQ)2
+ (ε2, ΛQ)2
+ (ε2, Λ2
Q)2
+ (ε2, Q)2
. (5.28)
To prove this property amounts first counting exactly the number of negative
eigenvalues of each Schrödinger operator, and then prove that the specific chosen
set of orthogonality conditions, which is not exactly the set of the bound states, is
enough to ensure the coercivity of the quadratic form. Both these issues appear to
be non trivial when Q is not explicit, but obvious to check numerically through the
drawing of a small number (less than 10) explicit curves.
Then, the second major obstruction is the fact that the right hand side Im( y ·
εε) in (5.27) is an unbounded function of ε in H1. This is a priori a major obstruction
to the strategy, but an additional non linear algebra inherited from the
virial law (2.1) rules out this difficulty.
The formal computation is as follows. Given a function f ∈ Σ, we let Φ(f) =
Im( y · ff). According to (5.26), we want to compute d
ds Φ(ε). Now from (5.24)
and the conservation of the energy:
∀t ∈ [0, T), Φ(u(t)) = 4E0t + c0
for some constant c0. The key observation is that the quantity Φ(u) is scaling, phase
and also translation invariant from zero momentum assumption (5.3). Using (5.12),
we get:
∀t ∈ [0, T), Φ(ε + ˜Qb) = 4E0t + c0.
We now expand this according to:
Φ(ε + ˜Qb) = Φ( ˜Qb) − 2(ε2, ΛΣ) + 2(ε1, ΛΘ) + Φ(ε).
A simple algebra yields:
Φ( ˜Qb) = −
b
2
|y ˜Qb|2
2 ∼ −Cb
for some universal constant C > 0. Next, from the choice of orthogonality condition
(5.16),
(ε2, ΛΣ) − (ε1, ΛΘ) = 0.
We thus get using dt
ds = λ2:
(Φ(ε))s ∼ 4λ2
E0 + Cbs.
In other words, to compute the a priori unbounded quantity (Φ(ε))s for the full non
linear equation is from the virial law equivalent to computing the time derivative of
bs, what of course makes now perfectly sense in H1.
The virial dispersive structure on u(t) in Σ thus induces a dispersive structure in
L2
loc ∩ ˙H1 on ε(s) for the full non linear equation.
The key dispersive virial estimate is now the following.
40 P. RAPHAËL
Proposition 5.7 (Local viriel estimate in ε). There exist universal constants δ0 > 0,
C > 0 such that for all s ≥ 0, there holds:
bs ≥ δ0 | ε|2
+ |ε|2
e−|y|
− λ2
E0 − e
− C
|b| . (5.29)
Proof of Proposition 5.7. Using the heuristics, we can compute in a suitable way bs
using the orthogonality condition (5.16). The computation -see Lemma 5 in [72]-
yields:
1
4
|yQ|2
2bs = H(ε, ε) + 2λ2
|E0| −
xs
λ
· {(ε2, ΛΣ) − (ε1, ΛΘ)} (5.30)
−
λs
λ
+ b (ε2, Λ2
Σ) − (ε1, Λ2
Θ) − ˜γs {(ε1, ΛΣ) + (ε2, ΛΘ)}
− (ε1, ReΛΨ)) − (ε2, Im(ΛΨ)) + (l.o.t),
where the lower order terms may be estimated from the smallness of ε in H1:
|l.o.t| ≤ δ(α∗
) | ε|2
+ |ε|2
e−|y|
.
We now explain how the choice of orthogonality conditions and the conservation
laws allow us to deduce (5.29).
step 1 Modulation theory for phase and scaling. The choice of orthogonality
conditions (5.15), (5.13) has been made to cancel the two second order in ε scalar
products in (5.30):
λs
λ
+ b (ε2, Λ2
Σ) − (ε1, Λ2
Θ) + ˜γs {(ε1, ΛΣ) + (ε2, ΛΘ)} = 0.
step 2 Elliptic estimate on the quadratic form H. We now need to control
the negative directions in the quadratic form as given by Proposition 5.6. The
directions (ε1, ΛQ), (ε1, yQ), (ε2, Λ2Q) and (ε2, ΛQ) are treated thanks to the choice
of orthogonality conditions and the closeness of ˜Qb to Q for |b| small. For example,
(ε2, ΛQ)2
= | {(ε2, ΛQ − ΛΣ) + (ε1, ΛΘ)} + (ε2, ΛΣ) − (ε1, ΛΘ)|2
= |(ε2, ΛQ − ΛΣ) + (ε1, ΛΘ)|2
so that
(ε2, ΛQ)2
≤ δ(α∗
)( | ε|2
+ |ε|2
e−|y|
).
Similarly, we have:
(ε1, yQ)2
+ (ε2, Λ2
Q)2
+ (ε1, ΛQ)2
≤ δ(α∗
)( | ε|2
+ |ε|2
e−|y|
). (5.31)
The negative direction (ε1, Q)2 is treated from the conservation of the energy which
implied (5.20). The direction (ε2, Q) is treated from the zero momentum condition
which ensured (5.21). Putting this together yields:
(ε1, Q)2
+ (ε2, Q)2
≤ δ(α∗
) | ε|2
+ |ε|2
e−|y|
+ λ2
|E0| + e
− C
|b| .
step 3 Modulation theory for translation and use of Galilean invariance. The
Galilean invariance has been used to ensure the zero momentum condition (5.3)
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION41
which in turn led together with the choice of orthogonality condition (5.14) to the
degeneracy estimate (5.23):
|
xs
λ
| δ(α∗
)( | ε|2
+ |ε|2
e−|y|
)
1
2 + e
− C
|b| .
Therefore, we estimate the term induced by translation invariance in (5.30) as
xs
λ
· {(ε2, ΛΣ) − (ε1, ΛΘ)} δ(α∗
) | ε|2
+ |ε|2
e−|y|
+ e
− C
|b| .
step 4 Conclusion. Injecting these estimates into the elliptic estimate (5.28)
yields so far:
bs ≥ δ0 | ε|2
+ |ε|2
e−|y|
− 2λ2
E0 − e
− C
|b| −
1
δ0
(λ2
E0)2
.
We now use in a crucial way the sign of the energy E0 < 0 and the smallness λ2|E0| ≤
δ(α∗) which is a consequence of the conservation of the energy to conclude.
5.5. Monotonicity and control of the blow up speed. The virial dispersive
estimate (5.29) means a control of the excess of mass ε by an exponentially small
correction in b in time averaging sense. More specifically, this means that in rescaled
variables, the solution writes ˜Qb + ε where ˜Qb is the regular deformation of Q and
the rest is in a suitable norm exponentially small in b. This is thus an expansion of
the solution with respect to an internal parameter in the problem: b.
This virial control is the first dispersive estimate for the infinite dimensional dynamic
driving ε. Observe that it means little by itself if nothing is known about b(t).
We shall now inject this information into the finite dimensional dynamic driving the
geometrical parameters. The outcome will be a rigidity property for the parameter
b(t) which will in turn imply the existence of a Lyapounov functional in the problem.
This step will again heavily rely on the conservation of the energy.
We start with exhibiting the rigidity property which proof is a maximum principle
type of argument.
Proposition 5.8 (Rigidity property for b). b(s) vanishes at most once on R+.
Note that the existence of a quantity with prescribed sign in the description of
the dynamic is unexpected. Indeed, b is no more then the projection of some a
priori highly oscillatory function onto a prescribed direction. It is a very specific
feature of the blow up dynamic that this projection has a fixed sign.
Proof of Proposition 5.8. Assume that there exists some time s1 ≥ 0 such that
b(s1) = 0 and bs(s1) ≤ 0, then from (5.29), ε(s1) = 0. Thus from the conservation
of the L2 norm and ˜Qb(s1) = Q, we conclude |u0|2 = Q2 what contradicts the
strictly negative energy assumption.
The next step is to get the exact sign of b. This is done by injecting the virial
dispersive information (5.29) into the modulation equation for the scaling parameter
what will yield
−
λs
λ
∼ b. (5.32)
The key rigidity property is the following:
42 P. RAPHAËL
Proposition 5.9 (Rigidity of the flow). There exists a time s0 ≥ 0 such that
∀s > s0, b(s) > 0.
Moreover, the size of the solution is in this regime an almost Lyapounov functional
in the sense that:
∀s2 ≥ s1 ≥ s0, λ(s2) ≤ 2λ(s1). (5.33)
Proof of Proposition 5.9. step 1 Equation for the scaling parameter. The modulation
equation for the scaling parameter λ inherited from choice of orthogonality
condition (5.13) implied control (5.22):
λs
λ
+ b | ε|2
+ |ε|2
e−|y|
1
2
+ e
− C
|b| ,
which implies (5.32) in a weak sense. Nevertheless, this estimate is not good enough
to possibly use the virial estimate (5.29). We claim using extra degeneracies of the
equation that (5.22) can be improved for:
λs
λ
+ b | ε|2
+ |ε|2
e−|y|
+ e
− C
|b| (5.34)
step 2 Use of the virial dispersive relation and the rigidity property. We now
inject the virial dispersive relation (5.29) into (5.34) to get:
λs
λ
+ b bs + e
− C
|b| .
We integrate this inequality in time to get: ∀0 ≤ s1 ≤ s2,
log
λ(s2)
λ(s1)
+
s2
s1
b(s)ds ≤
1
4
+
s2
s1
e
− C
|b(s)| ds. (5.35)
The key is now to use the rigidity property of Proposition 5.8 to ensure that b(s)
has a fixed sign for s ≥ ˜s0, and thus: ∀s ≥ ˜s0,
s2
s1
e
− C
|b(s)| ds ≤
1
2
s2
s1
b(s)ds . (5.36)
step 3 b is positive for s large enough. Assume that
+∞
0 b(s)ds < +∞, then b
has a fixed sign for s ≥ ˜s0 and |bs| ≤ C, and thus: b(s) → 0 as s → +∞. Now from
(5.35) and (5.36), this implies that | log(λ(s))| ≤ C as s → +∞, and in particular
λ(s) ≥ λ0 > 0 for s large enough. Injecting this into virial control (5.29) for s large
enough yields:
bs ≥
1
2
|E0|λ2
0.
Integrating this on large time intervals contradicts the uniform boundedness of
b. Here we have used again the assumption E0 < 0. We thus have proved:
+∞
0 b(s)ds = +∞.
Now assume that b(s) < 0 for all s ≥ ˜s1, then from (5.35) and (5.36) again, we
conclude that log(λ(s)) → 0 as s → +∞. Now from λ(t) ∼ 1
| u(t)|L2
, this yields
| u(t)|L2 → 0 as t → T. But from Gagliardo-Nirenberg inequality and the conservation
of the energy and the L2 mass, this implies E0 = 0, contradicting again the
assumption E0 < 0.
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION43
step 4 Almost monotonicity of the norm. We now are in position to prove
(5.33). Indeed, injecting the sign of b into (5.35) and (5.36) yields in particular:
∀s0 ≤ s1 ≤ s2,
1
4
+
1
2
s2
s1
b(s)ds ≤ − log
λ(s2)
λ(s1)
≤
1
4
+ 2
s2
s1
b(s)ds, (5.37)
and thus:
∀s0 ≤ s1 ≤ s2, − log
λ(s2)
λ(s1)
≥
1
4
,
what yields (5.33). This concludes the proof of Proposition 5.9.
Note that from the above proof, we have obtained
+∞
0 b(s)ds = +∞, and thus
from (5.37):
λ(s) → 0 as s → ∞, (5.38)
that is finite or infinite time blow up. On the contrary to the virial argument, the
blow up proof is no longer obstructive but completely dynamical, and relies mostly
on the rigidity property of Proposition 5.8.
Let us now conclude the proof of Theorem 5.1. We need to prove finite time blow
up together with the log-log upper bound (5.2) on blow up rate.
Proof of Theorem 5.1. step 1 Lower bound on b(s). We claim: there exist some
universal constant C > 0 and some time s1 > 0 such that ∀s ≥ s1,
Cb(s) ≥
1
log | log(λ(s))|
. (5.39)
Indeed, first recall (5.29). Now that we know the sign of b(s) for s ≥ s0 from
Proposition 5.9, and we may thus view (5.29) as a differential inequality for b for
s > s0:
bs ≥ −e−C
b ≥ −b2
e− C
2b ie −
bs
b2
e
C
2b ≤ 1.
We integrate this inequality from the non vanishing property of b and get for s ≥ ˜s1
large enough:
e
C
b(s) ≤ s + e
C
b(1) s ie b(s)
1
log(s)
. (5.40)
We now recall (5.37) on the time interval [˜s1, s]:
1
2
s
˜s1
b ≤ − log(
λ(s)
λ(˜s1)
) +
1
4
≤ −2 log(λ(s))
for s ≥ ˜s2 large enough from λ(s) → 0 as s → +∞. Inject (5.40) into the above
inequality, we get for s ≥ ˜s3
s
log(s)
s
˜s2
dτ
log(τ)
≤
1
4
s
˜s2
b ≤ − log(λ(s)) ie | log(λ(s))|
s
log(s)
and thus for s large
log | log(λ(s))| ≥ log(s) − log(log(s)) ≥
1
2
log(s)
and conclusion follows from (5.40). This concludes the proof of (5.39).
44 P. RAPHAËL
step 2 Finite time blow up and control of the blow up speed. We first use
the finite or infinite time blow up result (5.38) to consider a sequence of times
tn → T ∈ [0, +∞] defined for n large such that
λ(tn) = 2−n
.
Let sn = s(tn) the corresponding sequence and t such that s(t) = s0 given by
Proposition 5.9. Note that we may assume n ≥ n such that tn ≥ t. Remark that
0 < tn < tn+1 from (5.33), and so 0 < sn < sn+1. Moreover, there holds from (5.33)
∀s ∈ [sn, sn+1], 2−n−1
≤ λ(s) ≤ 2−(n−1)
. (5.41)
We now claim that (5.2) follows from a control from above of the size of the intervals
[tn, tn+1] for n ≥ n.
Let n ≥ n. (5.39) implies
sn+1
sn
ds
log | log(λ(s))|
sn+1
sn
b(s)ds.
(5.37) with s1 = sn and s2 = sn+1 yields:
1
2
sn+1
sn
b(s) ≤
1
4
− |yQ|2
L2 log(
λ(sn+1)
λ(sn)
) 1.
Therefore,
∀n ≥ n,
sn+1
sn
ds
log | log(λ(s))|
1.
Now we change variables in the integral at the left of the above inequality according
to ds
dt = 1
λ2(s)
and estimate with (5.41):
1
sn+1
sn
ds
log | log(λ(s))|
=
tn+1
tn
dt
λ2(t) log | log(λ(t))|
≥
1
10λ2(tn) log | log(λ(tn))|
tn+1
tn
dt
so that
tn+1 − tn λ2
(tn) log | log(λ(tn))|.
From λ(tn) = 2−n and summing the above inequality in n, we first get
T < +∞
and
T − tn
k≥n
2−2k
log(k) =
n≤k≤2n
2−2k
log(k) +
k≥2n
2−2k
log(k)
2−2n
log(n) + 2−4n
log(2n)
k≥0
2−2k log(2n + k)
log(2n)
2−2n
log(n) + 2−4n
log(n) 2−2n
log(n) λ2
(tn) log | log(λ(tn))|.
From the monotonicity of λ (5.33), we extend the above control to the whole sequence
t ≥ t. Let t ≥ t, then t ∈ [tn, tn+1] for some n ≥ n, and from 1
2λ(tn) ≤
λ(t) ≤ 2λ(tn), we conclude
λ2
(t) log | log(λ(t))| λ2
(tn) log | log(λ(tn))| T − tn T − t.
Now remark that the function f(x) = x2 log | log(x)| is non decreasing in a neighborhood
at the right of x = 0, and moreover
f
C
2
T − t
log | log(T − t)|
=
C2
4
(T − t)
log | log(T − t)|
log log C
T − t
log | log(T − t)|
≤ C(T−t)
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION45
for t close enough to T, so that we get for some universal constant C∗:
f(λ(t)) ≥ f C∗ T − t
log | log(T − t)|
ie λ(t) ≥ C∗ T − t
log | log(T − t)|
and (5.2) is proved.
Appendix
This Appendix is devoted to the proof of the concentration compactness Lemma,
i.e. Proposition 1.6. We follow Cazenave [11].
Proof of Proposition 1.6. . Let un ∈ H1 be as in the hypothesis of Proposition 1.6.
step 1 Concentration function. Let the sequence of concentration functions:
ρn(R) = sup
y∈RN B(y,R)
|un(x)|2
dx.
The following facts are elementary and left to the reader:
• Monotonicity: ∀n ≥ 0, ρn(R) is a nondecreasing function of R.
• The concentration point is attained:
∀R > 0, ∀n ≥ 0, ∃yn(R) ∈ RN
such that ρn(R) =
B(yn(R),R)
|un(x)|2
dx.
• Uniform Hölder continuity: ∃C, α > 0 independent of n such that
∀R1, R2 > 0, ∀n ≥ 0, |ρn(R1) − ρn(R2)| ≤ C|RN
1 − RN
2 |α
. (5.42)
This last fact is a simple consequence of the H1 bound (1.13).
step 2 Limit of concentration functions. From (5.42) and Ascoli’s theorem, there
exists a subsequence nk → +∞ and a nondecreasing limit ρ such that
∀R > 0, lim
k→+∞
ρnk
(R) = ρ(R). (5.43)
Let now
µ = lim
R→+∞
lim inf
n→+∞
ρn(R).
By definition, there exists Rk → +∞ such that
lim
k→+∞
ρnk
(Rk) = µ.
We now claim some stability of the sequence Rk which is a very general and simple
fact but crucial for the rest of the argument:
µ = lim
k→+∞
ρnk
(Rk) = lim
k→+∞
ρnk
(
Rk
2
) = lim
R→+∞
ρ(R). (5.44)
Proof of (5.44): First oberve from the monotonicity of ρnk
that
lim sup
k→+∞
ρnk
(
Rk
2
) ≤ lim sup
k→+∞
ρnk
(Rk) = µ. (5.45)
For the other sense, we argue as follows. For every R > 0, there holds:
ρ(R) = lim inf
k→+∞
ρnk
(R) ≥ lim inf
n→+∞
ρn(R)
and thus:
lim
R→+∞
ρ(R) ≥ µ. (5.46)
46 P. RAPHAËL
Eventually, for any R > 0, we have Rk
2 ≥ R for k large enough and thus:
ρnk
(
Rk
2
) ≥ ρnk
(R).
Letting k → +∞ implies:
∀R > 0, lim
k→+∞
ρnk
(
Rk
2
) ≥ ρ(R).
Letting now R > 0 yields:
lim
k→+∞
ρnk
(
Rk
2
) ≥ lim
R→+∞
ρ(R) ≥ µ
where we used (5.46) in the last step. This together with (5.45) concludes the proof
of (5.44).
The proof now proceed by making an hypothesis on µ.
Step 3: µ = 0 is vanishing. Assume µ = 0. Then from (5.44), limR→+∞ ρ(R) =
0. But ρ is nondecreasing positive so: ∀R > 0, ρ(R) = 0. In particular, ρ(1) = 0
and thus
lim
k→+∞
ρnk
(1) = lim
k→+∞
sup
y∈RN B(y,1)
|unk
|2
= 0. (5.47)
We claim that this together with the H1 bound on unk
implies (1.15). There is a
slight difficulty here which is that we need to pass from a local information -vanishing
on every ball- to a global information -vanishing of the global Lq norm-. This relies
on a refinement of the Gagliardo Nirenberg interpolation inequality. Indeed, we
claim that
∀u ∈ H1
, |u|2+ 4
N ≤ C|u|2
H1 |u|
4
N
L2 (5.48)
can be refined for:
∀u ∈ H1
, |u|2+ 4
N ≤ C|u|2
H1 sup
y∈RN B(y,1)
|u|2
2
N
. (5.49)
This together with (5.47) implies
unk
→ 0 in L2+ 4
N as k → +∞
and (1.15) follows by interpolation using the global H1 bound.
Proof of (5.49): Let a partition of Rd with disjoint rectangles Qj of side 1
2. Assume
N ≥ 3 and write Hölder noticing:
1
2 + 4
N
=
α
2
+
1 − α
2N
N−2
with α =
2
N + 2
so that
|u|
L2+ 4
N (Qi)
≤ |u|α
L2(Qj)|u|1−α
L2∗
(Qj)
and hence using Sobolev in Qj:
|u|
2+ 4
d
L2+ 4
d
≤ C|u|
4
d
L2(Qj)
|u|2
H1(Qj)
ON THE SINGULARITY FORMATION FOR THE NON LINEAR SCHRÖDINGER EQUATION47
where the Sobolev constant does not dŐpend on j thanks to the translation invariance
of Lebesgue’s mesure. We may now sum on the disjoint cubes:
|u|2+ 4
N dx = Σj≥1
Qj
|u|2+ 4
N dx sup
j≥1
|u|2
L2(Qj)
2
d
Σj≥1|u|2
H1(Qj)
= sup
j≥1
|u|2
L2(Qj)
2
N
|u|2
H1
and (5.49) is proved. The cases N = 1, 2 is similar and left to the reader.
Step 4: µ = M is compactness. Let nk be the sequence satisfying (5.43). For
R > 0, let yk(R) such that
ρnk
(R) =
B(yk(R),R)
|unk
(x)|2
dx. (5.50)
Pick ε > 0. Then from (5.44), there exist R0, R(ε) such that
ρ(R0) >
M
2
, ρ(R(ε)) > M − ε.
Hence there exists k0(ε) such that ∀k ≥ k0(ε),
ρnk
(R0) =
B(yk(R0),R0)
|unk
|2
>
M
2
, ρnk
(R(ε)) =
B(yk(R(ε)),R(ε))
|unk
|2
> M − ε.
But the total L2 mass being M, this implies that the balls B(yk(R0), R0) and
B(yk(R(ε)), R(ε)) cannot be disjoint. Hence -draw a picture- we can find R1(ε)
such that:
∀ε > 0, ∀k ≥ k0(ε),
B(yk(R0),R1(ε))
|unk
|2
≥ M − ε.
By possibly raising the value of R1(ε) for the values k ∈ [1, k0(ε)], this implies that
the sequence vk = unk
(· + yk(R0)) is L2 compact:
∀ε > 0, ∃R2(ε) > 0 such that ∀k ≥ 1,
|y|≥R2(ε)
|vk(y)|2
dy < ε.
The compactness of the embedding H1 → L2(B(0, R(ε))) then implies that vk a
Cauchy sequence in L2, and the H1 boundedness now implies (1.14) by interpolation.
Step 5: 0 < µ < M is dichotomy. Let again (nk, Rk) satisfying (5.43), (5.44).
Then we can write:
unk
= vk + wk + zk
with
vk = unk
1|y−yk(
Rk
2
)|≤
Rk
2
, wk = unk
1|y−yk(
Rk
2
)|≥Rk
, zk = unk
1Rk
2
<|y−yk(
Rk
2
)|<Rk
.
The key is to observe from (5.50) and (5.44) that:
|zk|2
=
B(yk(
Rk
2
),Rk)
|unk
|2
−
B(yk(
Rk
2
),
Rk
2
)
|unk
|2
≤ ρnk
(Rk) −
B(yk(
Rk
2
),
Rk
2
)
|unk
|2
= ρnk
(Rk) − ρnk
(
Rk
2
)
→ 0 as k → +∞.
48 P. RAPHAËL
The claim dichotomy now follows by taking smooth cut off in the localization. The
Lp norm of zk will go to zero using the vanishing of the L2 norm and the global H1
bound, and the error introduced by localization will be treated using Rk → +∞.
This is left to the reader.
This concludes the proof of Proposition 1.6.
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Université de Nice Sophia Antipolis & Institut Universitaire de France
E-mail address: praphael@math.unice.fr