Comptes Rendus
no. 7-8
Partial Differential Equations
A blowup result for the periodic NLS without gauge invariance

Presented by: the Editorial Board

Tadahiro Oh1
1 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA
Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 389-392.

In this Note, we prove a finite-time blowup result for the periodic nonlinear Schrödinger equation on Td with nonlinearity |u|p for p>1. In particular, our blowup result holds above the Strauss exponent. This is in contrast with the non-periodic setting, where global existence for small data is known above the Strauss exponent.

Dans cette Note, nous démontrons un résultat dʼexplosion en temps fini pour lʼéquation de Schrödinger non linéaire sur le tore Td avec une non linéarité du type |u|p, p>1. En particulier, notre résultat dʼexplosion est vrai pour des puissances p plus grandes que lʼexposant de Strauss. Cette situation est contraire au cas non périodique où lʼon connaît que pour p supérieur à lʼexposant de Strauss, le problème de Cauchy est globalement bien posé.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.04.009

Tadahiro Oh 1

1 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA 
@article{CRMATH_2012__350_7-8_389_0,
     author = {Tadahiro Oh},
     title = {A blowup result for the periodic {NLS} without gauge invariance},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {389--392},
     publisher = {Elsevier},
     volume = {350},
     number = {7-8},
     year = {2012},
     doi = {10.1016/j.crma.2012.04.009},
     language = {en},
}
TY  - JOUR
AU  - Tadahiro Oh
TI  - A blowup result for the periodic NLS without gauge invariance
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 389
EP  - 392
VL  - 350
IS  - 7-8
PB  - Elsevier
DO  - 10.1016/j.crma.2012.04.009
LA  - en
ID  - CRMATH_2012__350_7-8_389_0
ER  - 
%0 Journal Article
%A Tadahiro Oh
%T A blowup result for the periodic NLS without gauge invariance
%J Comptes Rendus. Mathématique
%D 2012
%P 389-392
%V 350
%N 7-8
%I Elsevier
%R 10.1016/j.crma.2012.04.009
%G en
%F CRMATH_2012__350_7-8_389_0
Tadahiro Oh. A blowup result for the periodic NLS without gauge invariance. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 389-392. doi : 10.1016/j.crma.2012.04.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.04.009/

[1] J. Bourgain Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrödinger equations, Geom. Funct. Anal., Volume 3 (1993) no. 2, pp. 107-156

[2] J. Bourgain Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal., Volume 3 (1993) no. 2, pp. 157-178

[3] J. Bourgain Nonlinear Schrödinger equations, Park City, UT, 1995 (IAS/Park City Math. Ser.), Volume vol. 5, Amer. Math. Soc., Providence, RI (1999), pp. 3-157

[4] T. Cazenave Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, New York University, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI, 2003 (xiv+323 pp)

[5] S. Herr; D. Tataru; N. Tzvetkov Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in H1(T3), Duke Math. J., Volume 159 (2011) no. 2, pp. 329-349

[6] M. Ikeda; Y. Wakasugi Nonexistence of a non-trivial global weak solution for the nonlinear Schrödinger equation with a nongauge invariant power nonlinearity | arXiv

[7] Q.S. Zhang Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., Volume 97 (1999) no. 3, pp. 515-539

[8] Q.S. Zhang A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris, Ser. I, Volume 333 (2001) no. 2, pp. 109-114

Cited by Sources:

Comments - Policy