Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball

Jean Bourgain; Aynur Bulut

doi:10.1016/j.crma.2012.05.006

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 Bourgain ¹ ; Aynur Bulut ¹

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.

Résumé
Abstract

On démontre des résultats nouveaux sur les solutions radiales de lʼéquation des ondes et lʼéquation de Schrödinger sur la boule B dans $R^{2}$ et $R^{3}$ 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 $R^{2}$ and $R^{3}$ , 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 : 2012-05-06
Accepté le : 2012-05-14
Publié le : 2012-06-01

DOI : 10.1016/j.crma.2012.05.006

Affiliations des auteurs :

Jean Bourgain ¹ ; Aynur Bulut ¹

¹ School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

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     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},
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     doi = {10.1016/j.crma.2012.05.006},
     language = {en},
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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/

Bibliographie
Cité par

[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.

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