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On blow up for the energy super critical defocusing nonlinear Schrödinger equations
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On blow up for the energy super critical defocusing nonlinear Schrödinger equations
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Journal Article
On blow up for the energy super critical defocusing nonlinear Schrödinger equations
2022
Overview
We consider the energy supercritical
defocusing
nonlinear Schrödinger equation
i
∂
t
u
+
Δ
u
-
u
|
u
|
p
-
1
=
0
in dimension
d
≥
5
. In a suitable range of energy supercritical parameters (
d
,
p
), we prove the existence of
C
∞
well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a
front mechanism
. Blow up is achieved by
compression
for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of
C
∞
spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V,Springer Verlag
