Comptes Rendus
no. 11-12
Ordinary Differential Equations
Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball
[Mesures de Gibbs et équations non-linéaires des ondes et de Schrödinger sur la boule]

Présenté par : Jean Bourgain

Jean Bourgain1 ; Aynur Bulut1
1 School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 571-575.

On démontre des résultats nouveaux sur les solutions radiales de lʼéquation des ondes et lʼéquation de Schrödinger sur la boule B dans R2 et R3 pour des conditions initiales aléatoires. Plus exactement, on établit une dynamique bien définie et unique sur le support de la mesure de Gibbs. Ceci complète des résultats de Burq et Tzvetkov (2008) [8,9] et Tzvetkov (2006, 2008) [10,11].

We establish new results for the radial nonlinear wave and Schrödinger equations on the ball in R2 and R3, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from Burq and Tzvetkov (2008) [8,9] and Tzvetkov (2006, 2008) [10,11].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.05.006
Jean Bourgain 1 ; Aynur Bulut 1

1 School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA 
@article{CRMATH_2012__350_11-12_571_0,
     author = {Jean Bourgain and Aynur Bulut},
     title = {Gibbs measure evolution in radial nonlinear wave and {Schr\"odinger} equations on the ball},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {571--575},
     publisher = {Elsevier},
     volume = {350},
     number = {11-12},
     year = {2012},
     doi = {10.1016/j.crma.2012.05.006},
     language = {en},
}
TY  - JOUR
AU  - Jean Bourgain
AU  - Aynur Bulut
TI  - Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 571
EP  - 575
VL  - 350
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2012.05.006
LA  - en
ID  - CRMATH_2012__350_11-12_571_0
ER  - 
%0 Journal Article
%A Jean Bourgain
%A Aynur Bulut
%T Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball
%J Comptes Rendus. Mathématique
%D 2012
%P 571-575
%V 350
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2012.05.006
%G en
%F CRMATH_2012__350_11-12_571_0
Jean Bourgain; Aynur Bulut. Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 571-575. doi : 10.1016/j.crma.2012.05.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.006/

[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 Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV equation, Geom. Funct. Anal., Volume 3 (1993) no. 3, pp. 209-262

[3] J. Bourgain Periodic nonlinear Schrödinger equation in invariant measures, CMP, Volume 166 (1994), pp. 1-26

[4] J. Bourgain Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, CMP, Volume 176 (1996), pp. 421-445

[5] J. Bourgain Invariant measures for the Gross–Piatevskii equation, J. Math. Pures Appl. (9), Volume 76 (1997) no. 8, pp. 649-702

[6] J. Bourgain Nonlinear Schrödinger Equations, IAS/Park City Math. Ser., vol. 5, American Mathematical Society, Providence, RI, 1999

[7] J. Bourgain A remark on normal forms and the ‘I-method’ for periodic NLS, J. Anal. Math., Volume 94 (2004), pp. 125-157

[8] N. Burq; N. Tzvetkov Random data Cauchy theory for supercritical wave equations, I (local theory), Invent. Math., Volume 173 (2008) no. 3, pp. 449-475

[9] N. Burq; N. Tzvetkov Random data Cauchy theory for supercritical wave equations, II (a global existence result), Invent. Math., Volume 173 (2008) no. 3, pp. 977-996

[10] N. Tzvetkov Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ., Volume 3 (2006), pp. 111-160

[11] N. Tzvetkov Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 7, pp. 2543-2609

Cité par Sources :

☆ The research of J.B. was partially supported by NSF grants DMS-0808042 and DMS-0835373 and the research of A.B. was supported by NSF under agreement Nos. DMS-0635607 and DMS-0808042.

Commentaires - Politique

Ces articles pourraient vous intéresser

Concerning the pathological set in the context of probabilistic well-posedness

Chenmin Sun; Nikolay Tzvetkov

C. R. Math (2020)