Remarks on the Gibbs measures for nonlinear dispersive equations
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 3, pp. 527-597.

On montre, grâce à différents exemples, comment on peut utiliser des mesures de Gibbs pour construire des solutions globales, à basse régularité, pour des équations dispersives. La construction repose sur le théorème de compacité de Prokhorov, combiné avec le théorème de convergence de Skorokhod. D’abord, on considère l’équation de Schrödinger non-linéaire (NLS) sur la sphère de dimension 3. Ensuite, on étudie l’équation de Benjamin–Ono et l’équation de Schrödinger avec dérivée sur le cercle. Puis, on construit une mesure de Gibbs et une solution globales aux équations des demi-ondes et de Szegő avec conditions périodiques. Enfin, on considère NLS cubique défocalisante, en dimension deux, sur un domaine quelconque et on construit des solutions globales sur le support de la mesure de Gibbs correspondante.

We show, by the means of several examples, how we can use Gibbs measures to construct global solutions to dispersive equations at low regularity. The construction relies on the Prokhorov compactness theorem combined with the Skorokhod convergence theorem. To begin with, we consider the nonlinear Schrödinger equation (NLS) on the tri-dimensional sphere. Then we focus on the Benjamin–Ono equation and on the derivative nonlinear Schrödinger equation on the circle. Next, we construct a Gibbs measure and global solutions to the so-called periodic half-wave equation and of the Szegő equation. Finally, we consider the cubic $2d$ defocusing NLS on an arbitrary spatial domain and we construct global solutions on the support of the associated Gibbs measure.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1578
Classification : 35BXX,  37K05,  37L50,  35Q55
Mots clés : nonlinear Schrödinger equation, Benjamin–Ono equation, derivative nonlinear Schrödinger equation, half-wave equation, Szegő equation, random data, Gibbs measure, weak solutions, global solutions
@article{AFST_2018_6_27_3_527_0,
author = {Burq, Nicolas and Thomann, Laurent and Tzvetkov, Nikolay},
title = {Remarks on the {Gibbs} measures for nonlinear dispersive equations},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {527--597},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 27},
number = {3},
year = {2018},
doi = {10.5802/afst.1578},
mrnumber = {3869074},
zbl = {1405.35193},
language = {en},
url = {http://www.numdam.org/articles/10.5802/afst.1578/}
}
TY  - JOUR
AU  - Burq, Nicolas
AU  - Thomann, Laurent
AU  - Tzvetkov, Nikolay
TI  - Remarks on the Gibbs measures for nonlinear dispersive equations
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2018
DA  - 2018///
SP  - 527
EP  - 597
VL  - Ser. 6, 27
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1578/
UR  - https://www.ams.org/mathscinet-getitem?mr=3869074
UR  - https://zbmath.org/?q=an%3A1405.35193
UR  - https://doi.org/10.5802/afst.1578
DO  - 10.5802/afst.1578
LA  - en
ID  - AFST_2018_6_27_3_527_0
ER  - 
Burq, Nicolas; Thomann, Laurent; Tzvetkov, Nikolay. Remarks on the Gibbs measures for nonlinear dispersive equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 3, pp. 527-597. doi : 10.5802/afst.1578. http://www.numdam.org/articles/10.5802/afst.1578/

 Albeverio, Sergio; Cruzeiro, Ana-Bela Global flows with invariant (Gibbs) measures for Euler and Navier–Stokes two dimensional fluids, Commun. Math. Phys., Volume 129 (1990) no. 3, pp. 431-444 | Article | MR 1051499 | Zbl 0702.76041

 Ayache, Antoine; Tzvetkov, Nikolay ${L}^{p}$ properties for Gaussian random series, Trans. Am. Math. Soc., Volume 360 (2008) no. 8, pp. 4425-4439 | Article | MR 2395179 | Zbl 1145.60019

 Blair, Matthew D.; Smith, Hart F.; Sogge, Christopher D. On multilinear spectral cluster estimates for manifolds with boundary, Math. Res. Lett., Volume 15 (2008) no. 2-3, pp. 419-426 | Article | MR 2407219 | Zbl 1146.58019

 Bourgain, Jean 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 | Article | Zbl 0787.35097

 Bourgain, Jean Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., Volume 166 (1994) no. 1, pp. 1-26 | Article | Zbl 0822.35126

 Bourgain, Jean Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., Volume 176 (1996) no. 2, pp. 421-445 | Article | Zbl 0852.35131

 Bourgain, Jean; Bulut, Aynur Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case, J. Eur. Math. Soc., Volume 16 (2014) no. 6, pp. 1289-1325 | Article | Zbl 1301.35145

 Brydges, David C.; Slade, Gordon Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation, Commun. Math. Phys., Volume 182 (1996) no. 2, pp. 485-504 | Article | Zbl 0867.35090

 Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Am. J. Math., Volume 126 (2004) no. 3, pp. 569-605 | Article | Zbl 1067.58027

 Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., Volume 159 (2005) no. 1, pp. 187-223 | Article | Zbl 1092.35099

 Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. Éc. Norm. Supér., Volume 38 (2005) no. 2, pp. 255-301 | Article | Numdam | Zbl 1116.35109

 Burq, Nicolas; Lebeau, Gilles Injections de Sobolev probabilistes et applications, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 6, pp. 917-962 | Article | Numdam | MR 3134684 | Zbl 1296.46031

 Burq, Nicolas; Thomann, Laurent; Tzvetkov, Nikolay Long time dynamics for the one dimensional nonlinear Schrödinger equation, Ann. Inst. Fourier, Volume 63 (2013) no. 6, pp. 2137-2198 | Article | Zbl 1317.35226

 Burq, Nicolas; Thomann, Laurent; Tzvetkov, Nikolay Global infinite energy solutions for the cubic wave equation, Bull. Soc. Math. Fr., Volume 143 (2015) no. 2, pp. 301-313 | Article | MR 3351181 | Zbl 1320.35217

 Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations I: Local theory, Invent. Math., Volume 173 (2008) no. 3, pp. 449-475 | Article | MR 2425133 | Zbl 1156.35062

 Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations. II. A global existence result, Invent. Math., Volume 173 (2008) no. 3, pp. 477-496 | Article | MR 2425134 | Zbl 1187.35233

 Cazenave, Thierry Semilinear Schrödinger Equations, Courant Lecture Notes in Math., 10, American Mathematical Society, 2003, xiii+323 pages | Zbl 1055.35003

 Christ, Michael Power series solution of a nonlinear Schrödinger equation, Mathematical aspects of nonlinear dispersive equations (Annals of Mathematics Studies), Volume 163, Princeton University Press, 2007, pp. 131-155 | Zbl 1142.35084

 Colliander, James; Oh, Tadahiro Almost sure well-posedness of the cubic nonlinear Schrödinger equation below ${L}^{2}\left(𝕋\right)$, Duke Math. J., Volume 161 (2012) no. 3, pp. 367-414 | Article | Zbl 1260.35199

 Da Prato, Giuseppe; Debussche, Arnaud Two-dimensional Navier-Stokes equations driven by a space-time white noise, J. Funct. Anal., Volume 196 (2002) no. 1, pp. 180-210 | Article | MR 1941997 | Zbl 1013.60051

 Deng, Yu Invariance of the Gibbs measure for the Benjamin-Ono equation, J. Eur. Math. Soc., Volume 17 (2015) no. 5, pp. 1107-1198 | Article | MR 3346690 | Zbl 1379.37135

 Deng, Yu; Tzvetkov, Nikolay; Visciglia, Nicola Invariant measures and long time behaviour for the Benjamin-Ono equation III, Commun. Math. Phys., Volume 339 (2015) no. 3, pp. 815-857 | Article | MR 3385985 | Zbl 1379.37136

 Gérard, Patrick; Grellier, Sandrine The cubic Szegő equation, Ann. Sci. Éc. Norm. Supér., Volume 43 (2010) no. 5, pp. 761-810 | Article | Numdam | Zbl 1228.35225

 Gérard, Patrick; Grellier, Sandrine The cubic Szegő equation, Sémin. Équ. Dériv. Partielles, Volume 2008-2009 (2010), 2, 19 pages (Exp. No. 2, 19 p.) | Numdam | Zbl 1213.35397

 Gérard, Patrick; Grellier, Sandrine Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, Volume 5 (2012) no. 5, pp. 1139-1155 | Article | MR 3022852 | Zbl 1268.35013

 Gérard, Patrick; Grellier, Sandrine Invariant tori for the cubic Szegö equation, Invent. Math., Volume 187 (2012) no. 3, pp. 707-754 | Article | Zbl 1252.35026

 Grünrock, Axel; Herr, Sebastian Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., Volume 39 (2008) no. 6, pp. 1890-1920 | Article | Zbl 1156.35471

 Guo, Zihua; Kwon, Soonsik; Oh, Tadahiro Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS, Commun. Math. Phys., Volume 322 (2013) no. 1, pp. 19-48 | Zbl 1308.35270

 Hörmander, Lars The analysis of linear partial differential operators. III. Pseudo-differential operators, Classics in Mathematics, Springer, 2007, xii+525 pages (reprint of the 1994 ed.) | Zbl 1115.35005

 Kallenberg, Olav Foundations of modern probability, Probability and Its Applications, Springer, 2002, xvii+638 pages | Article | Zbl 0996.60001

 Koralov, Leonid B.; Sinai, Yakov G. Theory of probability and random processes, Universitext, Springer, 2007, xi+353 pages | Zbl 1181.60004

 Krieger, Joachim; Lenzmann, Enno; Raphaël, Pierre Nondispersive solutions to the ${L}^{2}$-critical half-wave equation, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 1, pp. 61-129 | Article | MR 3054599 | Zbl 1288.35433

 Molinet, Luc Global well-posedness in ${L}^{2}$ for the periodic Benjamin-Ono equation, Am. J. Math., Volume 130 (2008) no. 3, pp. 635-683 | Article | MR 2418924 | Zbl 1157.35001

 Molinet, Luc Sharp ill-posedness result for the periodic Benjamin-Ono equation, J. Funct. Anal., Volume 257 (2009) no. 11, pp. 3488-3516 | Article | MR 2571435 | Zbl 1181.35246

 Nahmod, Andrea R.; Oh, Tadahiro; Rey-Bellet, Luc; Staffilani, Gigliola Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc., Volume 14 (2012) no. 4, pp. 1275-1330 | Article | MR 2928851 | Zbl 1251.35151

 Nahmod, Andrea R.; Rey-Bellet, Luc; Sheffield, Scott; Staffilani, Gigliola Absolute continuity of Brownian bridges under certain gauge transformations, Math. Res. Lett., Volume 18 (2011) no. 5, pp. 875-887 | Article | MR 2875861 | Zbl 1250.60018

 Oh, Tadahiro Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system, SIAM J. Math. Anal., Volume 41 (2009) no. 6, pp. 2207-2225 | Zbl 1205.35268

 Oh, Tadahiro Invariant Gibbs measures and a.s. global well-posedness for coupled KdV systems, Differ. Integral Equ., Volume 22 (2009) no. 7-8, pp. 637-668 | MR 2532115 | Zbl 1240.35477

 Oh, Tadahiro Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegő equation, Funkc. Ekvacioj, Volume 54 (2011) no. 3, pp. 335-365 | MR 2918143 | Zbl 05991997

 Oh, Tadahiro; Sulem, Catherine On the one-dimensional cubic nonlinear Schrödinger equation below ${L}^{2}$, Kyoto J. Math., Volume 52 (2012) no. 1, pp. 99-115 | Zbl 1258.35184

 Pocovnicu, Oana First and second order approximations for a nonlinear wave equation, J. Dyn. Differ. Equations, Volume 25 (2013) no. 2, pp. 305-333 | Article | MR 3054639 | Zbl 1270.65060

 Seeley, Robert T. An estimate near the boundary for the spectral function of the Laplace operator, Am. J. Math., Volume 102 (1980), pp. 869-902 | Article | MR 590638 | Zbl 0447.35029

 Simon, Barry The $P{\left(\phi \right)}_{2}$ Euclidean (Quantum) Field Theory, Princeton Series in Physics, Princeton University Press, 1974, xx+392 pages | Zbl 1175.81146

 Smith, Hart F.; Sogge, Christopher D. On the ${L}^{p}$ norm of spectral clusters for compact manifolds with boundary, Acta Math., Volume 198 (2007) no. 1, pp. 107-153 | Article | MR 2316270 | Zbl 1189.58017

 Sogge, Christopher D. Eigenfunction and Bochner Riesz estimates on manifolds with boundary, Math. Res. Lett., Volume 9 (2002) no. 2-3, pp. 205-216 | Article | MR 1903059 | Zbl 1017.58016

 Thomann, Laurent; Tzvetkov, Nikolay Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, Volume 23 (2010) no. 11, pp. 2771-2791 | Article | Zbl 1204.35154

 Tzvetkov, Nikolay Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ., Volume 3 (2006) no. 2, pp. 111-160 | Article | MR 2227040 | Zbl 1142.35090

 Tzvetkov, Nikolay Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier, Volume 58 (2008) no. 7, pp. 2543-2604 | Article | Numdam | Zbl 1171.35116

 Tzvetkov, Nikolay Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Relat. Fields, Volume 146 (2010) no. 3-4, pp. 481-514 | Article | MR 2574736 | Zbl 1188.35183

 Tzvetkov, Nikolay; Visciglia, Nicola Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 2, pp. 249-299 | Article | Numdam | MR 3112200 | Zbl 1317.35208

 Tzvetkov, Nikolay; Visciglia, Nicola Invariant measures and long time behaviour for the Benjamin-Ono equation. I, Int. Math. Res. Not., Volume 2014 (2014) no. 17, pp. 4679-4714 | Article | Zbl 1301.35141

 Tzvetkov, Nikolay; Visciglia, Nicola Invariant measures and long time behaviour for the Benjamin-Ono equation. II, J. Math. Pures Appl., Volume 103 (2015) no. 1, pp. 102-141 | Article | MR 3281949 | Zbl 1315.37051

 Zhidkov, Peter E. Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory, Lecture Notes in Math., 1756, Springer, 2001, 147 pages | MR 1831831 | Zbl 0987.35001

 Zhong, Sijia Global existence of solutions to Schrödinger equations on compact Riemannian manifolds below ${H}^{1}$, Bull. Soc. Math. Fr., Volume 138 (2010) no. 4, pp. 583-613 | Article | Numdam | MR 2794885 | Zbl 1236.35002

Cité par Sources :