TY - JOUR
T1 - On blow up for the energy super critical defocusing nonlinear Schrödinger equations
AU - Merle, Frank
AU - Raphaël, Pierre
AU - Rodnianski, Igor
AU - Szeftel, Jeremie
N1 - Funding Information:
P.R. is supported by the ERC-2014-CoG 646650 SingWave. P.R. would like to thank the Université de la Côte d’Azur where part of this work was done for its kind hospitality, and N. Burq (Orsay) and L. Vega (Bilbao) for stimulating discussions at very early stages of this work. I.R. is partially supported by the NSF Grant DMS #1709270 and a Simons Investigator Award. J.S. is supported by the ERC Grant ERC-2016 CoG 725589 EPGR. The authors would also like to thank the anonymous referees for their careful reading and numerous valuable suggestions and comments.
Publisher Copyright:
© 2021, The Author(s).
PY - 2022/1
Y1 - 2022/1
N2 - We consider the energy supercritical defocusing nonlinear Schrödinger equation i∂tu+Δu-u|u|p-1=0in 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.
AB - We consider the energy supercritical defocusing nonlinear Schrödinger equation i∂tu+Δu-u|u|p-1=0in 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.
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U2 - 10.1007/s00222-021-01067-9
DO - 10.1007/s00222-021-01067-9
M3 - Article
AN - SCOPUS:85114955444
VL - 227
SP - 247
EP - 413
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
SN - 0020-9910
IS - 1
ER -