Centre de diffusion de revues académiques mathématiques


Séminaire Laurent Schwartz — EDP et applications

Table des matières de ce volume | Article précédent | Article suivant
Zaher Hani; Benoit Pausader; Nikolay Tzvetkov; Nicola Visciglia
Growing Sobolev norms for the cubic defocusing Schrödinger equation
Séminaire Laurent Schwartz — EDP et applications (2013-2014), Exp. No. 16, 11 p., doi: 10.5802/slsedp.60
Article PDF

Résumé - Abstract

This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.


[1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3, (1993), 107–156.  MR 1209299 |  Zbl 0787.35097
[2] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices 1996, no. 6, 277–304.  MR 1386079 |  Zbl 0934.35166
[3] J. Bourgain, Problems in Hamiltonian PDE’s, Geom. Funct. Anal., 2000. (Special volume, Part I), 32–56.  MR 1826248 |  Zbl 1050.35016
[4] J. Bourgain, Refinements of Strichartz inequality and applications to $2$D-NLS with critical nonlinearity, Int. Math. Res. Not., (1998), 253-283.  MR 1616917 |  Zbl 0917.35126
[5] J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. Israel J. Math., 193 (2013), no. 1, 441–458.  MR 3038558 |  Zbl 1271.42039
[6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649–669.  MR 1871414 |  Zbl 1002.35113
[7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math., 181 (2010), no. 1, 39–113.  MR 2651381 |  Zbl 1197.35265
[8] Z. Hani, Long-time strong instability and unbounded orbits for some periodic nonlinear Schödinger equations, Arch. Rat. Mech. Anal., to appear (DOI:10.1007/s00205-013-0689-6).  MR 3158811
[9] Z. Hani, B. Pausader. N. Tzvetkov. N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Preprint 2013.
[10] S. Herr, D. Tataru, and N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in $4d$ and applications, J. Ang. Math., to appear, DOI:10.1515/crelle-2012-0013.
[11] M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc, to appear.
[12] J. Kato and F. Pusateri, A new proof of long range scattering for critical nonlinear Schrödinger equations, J. Diff. Int. Equ., Vol. 24, no. 9–10 (2011).  MR 2850346 |  Zbl 1249.35307
[13] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys., 139 (1991), pp. 479–493.  MR 1121130 |  Zbl 0742.35043
Copyright Cellule MathDoc 2019 | Crédit | Plan du site