Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation
[Mesures gaussiennes associées à une loi de conservation arbitraire de l'équation de Benjamin-Ono]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 2, pp. 249-299.

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.

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.

DOI : 10.24033/asens.2189
Classification : 35Q35, 37L40, 28C20
Keywords: dispersive equations, Wiener chaos, invariant measures
Mot clés : Équations dispersives, chaos de Wiener, mesures invariantes 
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     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},
     pages = {249--299},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
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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, Série 4, Tome 46 (2013) no. 2, pp. 249-299. doi : 10.24033/asens.2189. http://www.numdam.org/articles/10.24033/asens.2189/

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

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

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

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

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

[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

[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

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

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

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

[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

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

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

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

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