Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics

Sebastian Herr[1]

  • [1] Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Journées Équations aux dérivées partielles (2010)

  • page 1-10
  • ISSN: 0752-0360

Abstract

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This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds M with small initial data in H 1 ( M ) . The results include small data GWP for the quintic NLS in the case of the 3 d flat rational torus M = 𝕋 3 and small data GWP for the corresponding cubic NLS in the cases M = ℝ 2 × 𝕋 2 and M = ℝ 3 × 𝕋 . The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling, as well as critical function space theory. All results mentioned above have been obtained in collaboration with D. Tataru and N. Tzvetkov.

How to cite

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Herr, Sebastian. "Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics." Journées Équations aux dérivées partielles (2010): 1-10. <http://eudml.org/doc/116376>.

@article{Herr2010,
abstract = {This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds $M$ with small initial data in $H^1(M)$. The results include small data GWP for the quintic NLS in the case of the $3d$ flat rational torus $M=\mathbb\{T\}^3$ and small data GWP for the corresponding cubic NLS in the cases $M=\{\mathbb\{R\}\}^2\times \mathbb\{T\}^2$ and $M=\{\mathbb\{R\}\}^3\times \mathbb\{T\}$. The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling, as well as critical function space theory. All results mentioned above have been obtained in collaboration with D. Tataru and N. Tzvetkov.},
affiliation = {Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany},
author = {Herr, Sebastian},
journal = {Journées Équations aux dérivées partielles},
keywords = {energy critical nonlinear Schrödinger equations; global well-posedness; critical function spaces; Strichartz estimates},
language = {eng},
month = {6},
pages = {1-10},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics},
url = {http://eudml.org/doc/116376},
year = {2010},
}

TY - JOUR
AU - Herr, Sebastian
TI - Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics
JO - Journées Équations aux dérivées partielles
DA - 2010/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 10
AB - This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds $M$ with small initial data in $H^1(M)$. The results include small data GWP for the quintic NLS in the case of the $3d$ flat rational torus $M=\mathbb{T}^3$ and small data GWP for the corresponding cubic NLS in the cases $M={\mathbb{R}}^2\times \mathbb{T}^2$ and $M={\mathbb{R}}^3\times \mathbb{T}$. The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling, as well as critical function space theory. All results mentioned above have been obtained in collaboration with D. Tataru and N. Tzvetkov.
LA - eng
KW - energy critical nonlinear Schrödinger equations; global well-posedness; critical function spaces; Strichartz estimates
UR - http://eudml.org/doc/116376
ER -

References

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