Blow-up solutions for mass
supercritical NLS equations
Jérémie Szeftel
Département de Mathématiques et Applications,
Ecole Normale Supérieure
(Joint work with Frank Merle and Pierre Raphaël)
1
The focusing NLS

 iu = −∆u − |u|p−1 u, (t, x) ∈ [0, T ) × RN
t
 u(0, x) = u0 (x), u0 : RN → C
1 < p < +∞ for N = 1, 2 and 1 < p <
N +2
for N ≥ 3
N −2
Well-posed in H 1 (RN ) (Ginibre and Velo)
For u0 ∈ H 1 : 0 < T ≤ +∞ and u(t) ∈ C([0, T ), H 1 )
Either T = +∞ or T < +∞ and then:
lim |∇u(t)|L2 = +∞
t→T
2
L − norm :
1
2
R
|u(t, x)|2 dx
2
1
p+1
|∇u(t, x)| dx −
|u(t, x)|p+1 dx
R
Momentum : Im( ∇u(t, x)u(t, x)dx)
Energy :
R
2
R
The scaling of NLS and the virial law
uλ (t, x) = λ
2
p−1
u(λ2 t, λx) solution for all λ > 0
Ḣ σc (RN ) invariant with σc =
N
2
−
2
p−1
σc < 0 (mass subcritical): T = +∞
2
d
dt2
Z
|x|2 |u(t, x)|2 dx = 4N (p−1)E0 −
16σc
N − 2σc
Z
|∇u|2
mass critical and supercritical cases (σc ≥ 0):
Z
d2
2
2
|x|
|u(t,
x)|
dx ≤ 4N (p − 1)E0
2
dt
E0 < 0 ⇒
d2
dt2
R
|x|2 |u(t, x)|2 dx < 0 ⇒ blow up
Goal: description of the blow up (profile, speed, . . . )
3
4
)
N
The critical case (σc = 0, pc = 1 +

 iu = −∆u − |u| N4 u, (t, x) ∈ [0, T ) × RN
t
 u(0, x) = u0 (x), u0 : RN → C
Q unique positive radial solution to:
4
∆Q − Q + Q1+ N = 0
If |u0 |L2 < |Q|L2 , then T = +∞
Conformal transform: v(t, x) =
1
N
|t| 2
u
1 x
t, t
2
i |x|
4t
e
Explicit finite time blow-up solution S(t, x):
x |x|2 i
1
ei 4t − t , |S(t)|L2 = |Q|L2
S(t, x) = N Q
t
|t| 2
Blow up speed: |∇S(t)|L2 ∼
4
1
|t|
The case σc = 0 and |u0 |L2 > |Q|L2
Blow up solutions with conformal blow up speed:
|∇u|L2
1
Bα∗ = {u0 ∈ H with
R
Solutions with u0 in Bα∗
1
∼
T −t
2
|u0 | < Q2 +α∗ }
2
R
Im(
∇u
u
)
0 0
and E0 − 21
<0
|u0 | 2
Q <
R
2
R
L
blow up with the blow up speed:
12
log|log(T − t)|
|∇u|L2 ∼
T −t
Question: what happens in the supercritical case?
5
Blow up on a sphere for the quintic
NLS in dimension N ≥ 2

 iu = −∂ 2 u − N −1 ∂ u − |u|4 u, (t, r) ∈ [0, T ) × R+
r
t
r
r
 u(0, r) = u0 (r), u0 : R+ → C
Theorem [N = 2 (Raphaël), N ≥ 3 (Raphaël, S.)]:
N
(RN ) blowing up
There is an open subset of Hrad
in finite time on a sphere of strictly positive
radius according to the log-log law
Singularity formation happens around r ∼ 1
∂r u ∼ |∂r u| << ∂r2 u
r Near singularity: blow up dynamic is 1D (critical)
Goal: prove |u|
H
N −1
2
(r≤ 12 )
≪1
Propagation of regularity outside of the blow-up sphere
6
Stable self similar blow up for
slightly supercritical NLS
Theorem [Merle, Raphaël, S.] Let 1 ≤ N ≤ 5.
There exists p∗ > pc and for all p ∈ (pc , p∗ ), there
exists δ(p) > 0 with δ(p) → 0 as p → pc and an
open set O in H 1 such that: for any u0 ∈ O, there
exists (λ(t), x(t), γ(t)) and ε(t) ∈ H 1 such that:
1
x − x(t) iγ(t)
u(t, x) =
[Q
+
ε(t)]
e
p
2
λ(t)
λ p−1 (t)
with:
|∇ε(t)|L2 ≤ δ(p) (orbital stability)
x(t) → x(T ) ∈ RN (one blow-up point solution)
λ(t)
√
∼ C > 0 (selfsimilar blow-up speed)
T −t
∀σ ∈ [0, σc ), u(t) → u∗ in H σ as t → T and u∗ ∈
/ Ḣ σc
7
Approximate self-similar solutions
u(t, x) =
1
λ
2
p−1
(t)
v t,
x − x(t)
λ(t)
eiγ(t) ,
1
ds
= 2
dt
λ (t)
xs
λs
p−1
= (γs −1)v+i ·∇v
i∂s v+∆v−v−i Λv+v|v|
λ
λ
λs
γs = 1, xs = 0, −
= b, bs = 0, v(s, y) = Qb(s) (y)
λ
∆Qb − Qb + ibΛQb + Qb |Qb |p−1 = 0
Conformal change of variables Pb = Qb e
ib|y|2
4
1
∆Pb − Pb − iσc bPb + b2 |y|2 Pb + Pb |Pb |p−1 = 0
4
σc small : σc bPb treated as error term
|y| <
2
b:
linear operator −∆ + 1 −
N
b2 |y|2
4
coercive
|y| ≥ 2/b: |Pb (y)| ∼ |y|− 2 (not in L2 (RN ))
8
Modulation theory
∆Qb − Qb + ibΛQb + Qb |Qb |
p−1
π
− 2b
=O e
x − x(t)
(Qb(t) + ε) t,
u(t, x) =
2
λ(t)
p−1
λ
(t)
1
+ bσc
eiγ(t)
Orthogonality conditions: ∀t ∈ [0, T ∗ ],
2
ℜ ε(t), |y| Qb = 0
ℜ (ε(t), yQb ) = 0
2
ℑ ε(t), Λ Qb = 0
ℑ (ε(t), ΛQb ) = 0
Goal: show that T = T ∗ , 0 < b1 ≤ b ≤ b2 ≪ 1,
√
λ ∼ T − t, |∇ε|L2 ≤ C ≪ 1 and x(t) → x(T )
9
The modulation equations
λs
π
+ b . |∇ε|L2 + e− 2b
+ bσc
λ
x π
s
− 2b
+ bσc
. |∇ε|L2 + e
λ
π
− 2b
:
If 0 < b1 ≤ b ≤ b2 ≪ 1 and |∇ε|L2 . e
√
λs
Then: λλt = λ ∼ −b ⇒ λ ∼ T − t
x and: |xt | = λs2 ≪ λ1 ∼ √T1−t ⇒ x(t) → x(T )
Bootstrap method: assume 0 < b1 ≤ b ≤ b2 ≪ 1
π
and |∇ε|L2 ≤ Ce− 2b on [0, T ∗ ) for some possibly
small T ∗ > 0 and improve on the constants to
show that in fact T = T ∗
10
First monotonicity formula
Computation of the modulation equation of bs
• Imaginary part of scalar product of the
equation of ε with ΛQb
• Conservation of energy
• Quadratic form is coercive up to a finite
number of directions
• Negative directions controlled with
orthogonality conditions and conservation of
energy and momentum
Z
1 −π
2
bs ≥ c1 σc + |∇ε| − e b
c1
11
Second monotonicity formula
Truncate further to take oscillations at infinity
better into account: |y| ≤ A with A ≫ 2b
⇒ New approximate self-similar solution Q̂b
• Rerun the first monotonicity formula with
Q̂b and ε̂ = ε + Qb − Q̂b
R 2A 2
• Control of A |ε|
• Multiplication by b and injection of the
conservation of mass
Z
b
2
−π
b
+ |∇ε̂| − σc
− {J }s ≥ c2 b e
c2
with J ∼ b2
12
Control of b and ε
Using the first and second monotinicity formulae:
c1 σc −
1 −π
π
π
1
e b ≤ bs ≤ −c2 e− b + σc ⇒ σc ∼ e− b
c1
c2
and
Z
2
−π
b
|∇ε| . e
b and ε may oscillate
13
−π
b
+ σc . e