Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 2, p. 249-299

Inspired by the work of Zhidkov on the KdV equation, we perform a construction of weighted Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. The resulting measures are supported by Sobolev spaces of increasing regularity. We also prove a property on the support of these measures leading to the conjecture that they are indeed invariant by the flow of the Benjamin-Ono equation.

Inspirés par le travail de Zhidkov sur l'équation KdV, nous construisons des mesures gaussiennes à poids associées à une loi de conservation arbitraire de l'équation de Benjamin-Ono. Les supports de ces mesures sont constitués de fonctions de régularité de Sobolev croissantes. On démontre aussi une propriété-clé des mesures qui nous conduit à conjecturer leur invariance par le flot de l'équation.

DOI : https://doi.org/10.24033/asens.2189
Classification:  35Q35,  37L40,  28C20
Keywords: dispersive equations, Wiener chaos, invariant measures
@article{ASENS_2013_4_46_2_249_0,
     author = {Tzvetkov, Nikolay and Visciglia, Nicola},
     title = {Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {2},
     year = {2013},
     pages = {249-299},
     doi = {10.24033/asens.2189},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2013_4_46_2_249_0}
}
Tzvetkov, Nikolay; Visciglia, Nicola. Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 2, pp. 249-299. doi : 10.24033/asens.2189. http://www.numdam.org/item/ASENS_2013_4_46_2_249_0/

[1] L. Abdelouhab, J. L. Bona, M. Felland & J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D 40 (1989), 360-392. | MR 1044731

[2] J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), 1-26. | MR 1309539

[3] J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), 421-445. | MR 1374420

[4] D. C. Brydges & G. Slade, Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation, Comm. Math. Phys. 182 (1996), 485-504. | MR 1447302

[5] N. Burq & F. Planchon, On well-posedness for the Benjamin-Ono equation, Math. Ann. 340 (2008), 497-542. | MR 2357995

[6] N. Burq, L. Thomann & N. Tzvetkov, Long time dynamics for the one dimensional non linear Schrödinger equation, to appear in Ann. Inst. Fourier.

[7] A. D. Ionescu & C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc. 20 (2007), 753-798. | MR 2291918

[8] J. L. Lebowitz, H. A. Rose & E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys. 50 (1988), 657-687. | MR 939505

[9] M. Ledoux & M. Talagrand, Probability in Banach spaces, Ergebn. Math. Grenzg. 23, Springer, 1991. | MR 1102015

[10] Y. Matsuno, Bilinear transformation method, Mathematics in Science and Engineering 174, Academic Press Inc., 1984. | MR 759718

[11] L. Molinet, Global well-posedness in L 2 for the periodic Benjamin-Ono equation, Amer. J. Math. 130 (2008), 635-683. | MR 2418924

[12] A. R. Nahmod, T. Oh, L. Rey-Bellet & G. Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. (JEMS) 14 (2012), 1275-1330. | MR 2928851

[13] T. Tao, Global well-posedness of the Benjamin-Ono equation in H 1 (𝐑), J. Hyperbolic Differ. Equ. 1 (2004), 27-49. | MR 2052470

[14] N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Related Fields 146 (2010), 481-514. | MR 2574736

[15] P. E. Zhidkov, Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory, Lecture Notes in Math. 1756, Springer, 2001. | MR 1831831