Invariant measures and long-time behavior for the Benjamin-Ono equation
Journées équations aux dérivées partielles (2014), article no. 11, 14 p.

We summarize the main ideas in a series of papers ([20], [21], [22], [5]) devoted to the construction of invariant measures and to the long-time behavior of solutions of the periodic Benjamin-Ono equation.

DOI : 10.5802/jedp.114
Deng, Yu 1 ; Tzvetkov, Nikolay 2 ; Visciglia, Nicola 3

1 Mathematics Department Princeton University Fine Hall Washington road, Princeton NJ 08544-4200, USA
2 Institut Universitaire de France and Département de Mathématiques Université de Cergy-Pontoise 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France
3 Dipartimento di Matematica Università Degli Studi di Pisa Largo Bruno Pontecorvo 5 56127 Pisa, Italy
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Deng, Yu; Tzvetkov, Nikolay; Visciglia, Nicola. Invariant measures and long-time behavior for the Benjamin-Ono equation. Journées équations aux dérivées partielles (2014), article  no. 11, 14 p. doi : 10.5802/jedp.114. http://www.numdam.org/articles/10.5802/jedp.114/

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

[2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), 209-262 | MR | Zbl

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

[4] Y. Deng, Invariance of the Gibbs measure for the Benjamin-Ono equation. arxiv:1210.1542

[5] Y. Deng, N. Tzvetkov, N. Visciglia, Invariant Measures and long time behavior for the Benjamin-Ono equation III, arXiv:1405.4954

[6] J. L. Lebowitz, H. A. Rose, E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys. 50 (1988), 657Ð687. | MR | Zbl

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

[8] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988) 891-907. | MR | Zbl

[9] C. Kenig, K. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math. Res. Lett., 10 (2003) 879-895. | MR | Zbl

[10] H. Koch, N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in H s (), Int. Math. Res. Not. 2003, n.26, 1449-1464. | MR | Zbl

[11] Y. Matsuno, Bilinear transformation method, Academic Press, 1984. | MR | Zbl

[12] H.P. McKean, E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143-226. | MR | Zbl

[13] L. Molinet, Global well-posendess in L 2 for the periodic Benjamin-Ono equation, Amer. J. Math. 130 (2008) 635-685. | MR | Zbl

[14] L. Molinet, D. Pilod, The Cauchy problem of the Benjamin-Ono equation in L 2 revisited, arXiv:1007.1545v1 | MR

[15] L. Molinet, J.-C. Saut, N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33 (2001), 982-988. | MR | Zbl

[16] A. 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. 14 (2012), 1275-1330. | MR | Zbl

[17] G. Ponce, On the global well-posedness of the Benjamin-Ono equation, Diff. Int. Eq. 4 (1991) 527-542. | MR | Zbl

[18] T. Tao, Global well-posedness of the Benjamin-Ono equation in H 1 , J. Hyperbolic Diff. Equations, 1 (2004) 27-49. | MR | Zbl

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

[20] N. Tzvetkov, N. Visciglia, Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation , Ann. Scient. Ec. Norm. Sup. 46 (2013) 249-299. | MR

[21] N. Tzvetkov, N. Visciglia, Invariant measures and long time behaviour for the Benjamin-Ono equation, Int. Math. Res. Not. 2013; doi: 10.1093/imrn/rnt094. | MR

[22] N. Tzvetkov, N. Visciglia, Invariant measures and long time behaviour for the Benjamin-Ono equation II, J. Math. Pures Appl. 2014; doi:10.1016/j.matpur.2014.03.009 | MR

[23] P. Zhidkov, KdV and Nonlinear Schrödinger equations : qualitative theory, Lecture notes in Mathematics 1756, Springer, 2001. | Zbl

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