Annals of Mathematics
The blow-up dynamic and
upper bound on the blow-up
rate for critical nonlinear
Schr¨odinger equation
By Frank Merle and Pierre Raphael
Annals of Mathematics,161 (2005), 157–222
The blow-up dynamic and upper bound
on the blow-up rate for critical
nonlinear Schr¨odinger equation
By Frank Merle and Pierre Raphael
Abstract
We consider the critical nonlinear Schr¨odinger equation iut=−∆u−|u|4
Nu
with initial condition u(0,x)=u0in dimension N=1. Foru0∈H1, local
existence in the time of solutions on an interval [0,T) is known, and there exist
finite time blow-up solutions, that is, u0such that limt↑T<+∞|ux(t)|L2=+∞.
This is the smallest power in the nonlinearity for which blow-up occurs, and
is critical in this sense. The question we address is to understand the blow-up
dynamic. Even though there exists an explicit example of blow-up solution and
a class of initial data known to lead to blow-up, no general understanding of the
blow-up dynamic is known. At first, we propose in this paper a general setting
to study and understand small, in a certain sense, blow-up solutions. Blow-up
in finite time follows for the whole class of initial data in H1with strictly
negative energy, and one is able to prove a control from above of the blow-up
rate below the one of the known explicit explosive solution which has strictly
positive energy. Under some positivity condition on an explicit quadratic form,
the proof of these results adapts in dimension N>1.
1. Introduction
1.1. Setting of the problem. In this paper, we consider the critical nonlin-
ear Schr¨odinger equation
(NLS) iut=−∆u−|u|4
Nu, (t, x)∈[0,T)×RN
u(0,x)=u0(x),u
0:RN→C
(1)
with u0∈H1=H1(RN) in dimension N≥1. The problem we address is the
one of formation of singularities for solutions to (1). Note that this equation
is Hamiltonian and in this context few results are known.
It is a special case of the following equation
iut=−∆u−|u|p−1u(2)
where 1 <p< N+2
N−2and the initial condition u0∈H1. From a result of
Ginibre and Velo [8], (2) is locally well-posed in H1. In addition, (1) is locally
158 FRANK MERLE AND PIERRE RAPHAEL
well-posed in L2=L2(RN) from Cazenave and Weissler [5]. See also [3], [2]
for the periodic case and global well posedness results. Thus, for u0∈H1,
there exists 0 <T ≤+∞such that u(t)∈C([0,T),H1) and either T=+∞,
where the solution is global, or T<+∞and then lim supt↑T|∇u(t)|L2=+∞.
We first recall the main known facts about (1), (2). For 1 <p<N+2
N−2,(2)
admits a number of symmetries in the energy space H1, explicitly:
•Space-time translation invariance: If u(t, x) solves (2), then so does
u(t+t0,x+x0), t0,x
0∈R.
•Phase invariance: If u(t, x) solves (2), then so does u(t, x)eiγ ,γ∈R.
•Scaling invariance: If u(t, x) solves (2), then so does λ2
p−1u(λ2t, λx),
λ>0.
•Galilean invariance: If u(t, x) solves (2), then so does u(t, x−βt)eiβ
2(x−β
2t),
β∈R.
From Ehrenfest’s law or direct computation, these symmetries induce invari-
ances in the energy space H1, respectively:
•L2-norm:
|u(t, x)|2=|u0(x)|2;(3)
•Energy:
E(u(t, x)) = 1
2|∇u(t, x)|2−1
p+1|u(t, x)|p+1 =E(u0);(4)
•Momentum:
Im ∇uu(t, x)=Im∇u0u0(x).(5)
The conservation of energy expresses the Hamiltonian structure of (2) in H1.
For p<1+ 4
N, (3), (4) and the Gagliardo-Nirenberg inequality imply
|∇u(t)|2
L2≤C(u0)|∇u(t)|2α
L2+1
for some α<1,
so that (2) is globally well posed in H1:
∀t∈[0,T[,|∇u(t)|L2≤C(u0) and T=+∞.
The situation is quite different for p≥1+ 4
N. Let an initial condition u0∈
Σ=H1∩{xu ∈L2}and assume E(u0)<0, then T<+∞follows from
the so-called virial Identity. Indeed, the quantity y(t)=|x|2|u|2(t, x) is well
defined for t∈[0,T) and satisfies
y′′(t)≤C(p)E(u0)
with C(p)>0. The positivity of y(t) yields the conclusion.
THE BLOW-UP DYNAMIC 159
The critical power in this problem is p=1+ 4
N. From now on, we focus
on it. First, note that the scaling invariance now can be written:
•Scaling invariance: If u(t, x) solves (1), then so does
uλ(t, x)=λN
2u(λx, λ2t),λ>0,
and by direct computation
|uλ|L2=|u|L2.
Moreover, (1) admits another symmetry which is not in the energy space H1,
the so-called pseudoconformal transformation:
•Pseudoconformal transformation: If u(t, x) solves (1), then so does
v(t, x)= 1
|t|N
2
u1
t,x
tei|x|2
4t.
This additional symmetry yields the conservation of the pseudoconformal en-
ergy for initial datum u0∈Σ which is most frequently expressed as (see [30]):
d2
dt2|x|2|u(t, x)|2=4d
dtIm x∇uu(t, x)=16E(u0).(6)
At the critical power, special regular solutions play an important role. They
are the so-called solitary waves and are of the form u(t, x)=eiωtWω(x), ω>0,
where Wωsolves
∆Wω+Wω|Wω|4
N=ωWω.(7)
Equation (7) is a standard nonlinear elliptic equation. In dimension N=1,
there exists a unique solution up to translation to (7) and infinitely many with
growing L2-norm for N≥2. Nevertheless, from [1], [7] and [11], there is a
unique positive solution up to translation Qω(x). In addition Qωis radially
symmetric. When Q=Qω=1, then Qω(x)=ωN
4Q(ω1
2x) from the scaling
property. Therefore, one computes
|Qω|L2=|Q|L2.
Moreover, the Pohozaev identity yields E(Q) = 0, so that
E(Qω)=ωE(Q)=0.
In particular, none of the three conservation laws in H1(3), (4), (5) of (1) sees
the variation of size of the Wωstationary solutions. These two facts are deeply
related to the criticality of the problem, that is the value p=1+ 4
N. Note that
in dimension N=1,Qcan be written explicitly
Q(x)=3
ch2(2x)1
4
.(8)
160 FRANK MERLE AND PIERRE RAPHAEL
Weinstein in [29] used the variational characterization of the ground state
solution Qto (7) to derive the explicit constant in the Gagliardo-Nirenberg
inequality
∀u∈H1,1
2+ 4
N|u|4
N+2 ≤1
2|∇u|2|u|2
Q2
2
N
,(9)
so that for |u0|L2<|Q|L2, for all t≥0, |∇u(t)|L2≤C(u0) and T=+∞,
the solution is global in H1. In addition, blow-up in H1has been proved to
be equivalent to “blow-up” for the L2theory from the following concentration
result: If a solution blows up at T<+∞in H1, then there exists x(t) such
that
∀R>0,lim inf
t↑T|x−x(t)|≤R|u(t, x)|2≥|Q|2
L2.
See for example [18].
On the other hand, for |u0|L2≥|Q|L2, blow-up may occur. Indeed, since
E(Q)=0and∇E(Q)=−Q, there exists u0ε∈Σ with |u0ε|L2=|Q|L2+ε
and E(u0ε)<0, and the corresponding solution must blow-up from the virial
identity (6).
The case of critical mass |u0|L2=|Q|L2has been studied in [19]. The pseu-
doconformal transformation applied to the stationary solution eitQ(x) yields
an explicit solution
S(t, x)= 1
|t|N
2
Q(x
t)ei|x|2
4t−i
t
(10)
which blows up at T= 0. Note that
|S(t)|L2=|Q|L2and |∇S(t)|L2∼1
|t|.
It turns out that S(t) is the unique minimal mass blow-up solution in H1in
the following sense: Let u(−1) ∈H1with |u(−1)|L2=|Q|L2and assume that
u(t) blows up at T= 0; then u(t)=S(t) up to the symmetries of the equation.
In the case of super critical mass |u0|2>Q2, the situation is more
complicated:
- There still exist in dimension N= 2 from a result by Bourgain and Wang,
[4], solutions of type S(t), that is, with blow-up rate |∇u(t)|L2∼1
T−t.
- Another fact suggested by numerical simulations, see Landman, Papan-
icolaou, Sulem, Sulem, [12], is the existence of solutions blowing up as
|∇u(t)|L2∼ln(|ln|t||)
|t|.(11)
