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 
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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/

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[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

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  • Vedran Sohinger A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials, International Mathematics Research Notices, Volume 2022 (2022) no. 19, p. 14964 | DOI:10.1093/imrn/rnab132
  • Vedran Sohinger Gibbs Measures of Nonlinear Schrödinger Equations as Limits of Quantum Many-Body States in Dimension d ≤ 3, Frontiers in Analysis and Probability (2020), p. 371 | DOI:10.1007/978-3-030-56409-4_9
  • Jürg Fröhlich; Antti Knowles; Benjamin Schlein; Vedran Sohinger Gibbs Measures of Nonlinear Schrödinger Equations as Limits of Many-Body Quantum States in Dimensions d⩽3 d ⩽ 3, Communications in Mathematical Physics, Volume 356 (2017) no. 3, p. 883 | DOI:10.1007/s00220-017-2994-7
  • Vedran Sohinger; Gigliola Staffilani Randomization and the Gross–Pitaevskii Hierarchy, Archive for Rational Mechanics and Analysis, Volume 218 (2015) no. 1, p. 417 | DOI:10.1007/s00205-015-0863-0
  • Seckin Demirbas Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus, Canadian Mathematical Bulletin, Volume 58 (2015) no. 3, p. 471 | DOI:10.4153/cmb-2015-025-7
  • Vedran Sohinger Local existence of solutions to randomized Gross-Pitaevskii hierarchies, Transactions of the American Mathematical Society, Volume 368 (2015) no. 3, p. 1759 | DOI:10.1090/tran/6479
  • Jean Bourgain; Aynur Bulut Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 31 (2014) no. 6, p. 1267 | DOI:10.1016/j.anihpc.2013.09.002
  • Jean Bourgain; Aynur Bulut Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball, Journal of Functional Analysis, Volume 266 (2014) no. 4, p. 2319 | DOI:10.1016/j.jfa.2013.06.002

Cité par 8 documents. Sources : Crossref

☆ 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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