Nonlinear Schrödinger equation on four-dimensional compact manifolds

Patrick Gérard; Vittoria Pierfelice

Bulletin de la Société Mathématique de France (2010)

  • Volume: 138, Issue: 1, page 119-151
  • ISSN: 0037-9484

Abstract

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We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness for small energy data in the Hamiltonian case. Both results are based on new multilinear Strichartz-type estimates for the Schrödinger group.

How to cite

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Gérard, Patrick, and Pierfelice, Vittoria. "Nonlinear Schrödinger equation on four-dimensional compact manifolds." Bulletin de la Société Mathématique de France 138.1 (2010): 119-151. <http://eudml.org/doc/272391>.

@article{Gérard2010,
abstract = {We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness for small energy data in the Hamiltonian case. Both results are based on new multilinear Strichartz-type estimates for the Schrödinger group.},
author = {Gérard, Patrick, Pierfelice, Vittoria},
journal = {Bulletin de la Société Mathématique de France},
keywords = {nonlinear Schrödinger; eigenfunction estimates; dispersive equations},
language = {eng},
number = {1},
pages = {119-151},
publisher = {Société mathématique de France},
title = {Nonlinear Schrödinger equation on four-dimensional compact manifolds},
url = {http://eudml.org/doc/272391},
volume = {138},
year = {2010},
}

TY - JOUR
AU - Gérard, Patrick
AU - Pierfelice, Vittoria
TI - Nonlinear Schrödinger equation on four-dimensional compact manifolds
JO - Bulletin de la Société Mathématique de France
PY - 2010
PB - Société mathématique de France
VL - 138
IS - 1
SP - 119
EP - 151
AB - We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness for small energy data in the Hamiltonian case. Both results are based on new multilinear Strichartz-type estimates for the Schrödinger group.
LA - eng
KW - nonlinear Schrödinger; eigenfunction estimates; dispersive equations
UR - http://eudml.org/doc/272391
ER -

References

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