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- Série 6 : Tome 27 (2018)

- no. 3

- p. 527-597

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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.

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.

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Nicolas Burq 1 ; Laurent Thomann 2 ; Nikolay Tzvetkov 3

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[1] Sergio Albeverio; Ana-Bela Cruzeiro 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 | MR | DOI | Zbl

[2] Antoine Ayache; Nikolay Tzvetkov L p properties for Gaussian random series, Trans. Am. Math. Soc., Volume 360 (2008) no. 8, pp. 4425-4439 | MR | DOI | Zbl

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

[4] Jean 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 | DOI | Zbl

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

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

[7] Jean Bourgain; Aynur Bulut 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 | DOI | Zbl

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

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

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

[11] Nicolas Burq; Patrick Gérard; Nikolay Tzvetkov 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 | Numdam | DOI | Zbl

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

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

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

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

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

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

[18] Michael Christ 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

[19] James Colliander; Tadahiro Oh Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L 2 (𝕋), Duke Math. J., Volume 161 (2012) no. 3, pp. 367-414 | DOI | Zbl

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

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

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

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

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

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

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

[27] Axel Grünrock; Sebastian Herr 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 | DOI | Zbl

[28] Zihua Guo; Soonsik Kwon; Tadahiro Oh 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

[29] Lars Hörmander 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

[30] Olav Kallenberg Foundations of modern probability, Probability and Its Applications, Springer, 2002, xvii+638 pages | DOI | Zbl

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

[32] Joachim Krieger; Enno Lenzmann; Pierre Raphaël Nondispersive solutions to the L 2 -critical half-wave equation, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 1, pp. 61-129 | MR | DOI | Zbl

[33] Luc Molinet Global well-posedness in L 2 for the periodic Benjamin-Ono equation, Am. J. Math., Volume 130 (2008) no. 3, pp. 635-683 | MR | DOI | Zbl

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

[35] Andrea R. Nahmod; Tadahiro Oh; Luc Rey-Bellet; Gigliola Staffilani 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 | MR | DOI | Zbl

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

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

[38] Tadahiro Oh 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 | Zbl

[39] Tadahiro Oh 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 | Zbl

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

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

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

[43] Barry Simon The P(φ) 2 Euclidean (Quantum) Field Theory, Princeton Series in Physics, Princeton University Press, 1974, xx+392 pages | Zbl

[44] Hart F. Smith; Christopher D. Sogge On the L p norm of spectral clusters for compact manifolds with boundary, Acta Math., Volume 198 (2007) no. 1, pp. 107-153 | MR | DOI | Zbl

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

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

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

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

[49] Nikolay Tzvetkov 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 | MR | DOI | Zbl

[50] Nikolay Tzvetkov; Nicola Visciglia 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 | MR | Numdam | DOI | Zbl

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

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

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

[54] Sijia Zhong 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 | MR | Numdam | DOI | Zbl

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