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July  2015, 35(7): 2905-2920. doi: 10.3934/dcds.2015.35.2905

## Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures

 1 Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS ( UMR 7539), 99, avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France

Received  July 2014 Revised  September 2014 Published  January 2015

We use Wasserstein metrics adapted to study the action of the flow of the BBM equation on probability measures. We prove the continuity of this flow and the stability of invariant measures for finite times.
Citation: Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905
##### References:
 [1] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4. [2] ________, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory, Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010. [3] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252. doi: 10.3934/dcds.2009.23.1241. [4] J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26. doi: 10.1007/BF02099299. [5] R. Brout and I. Prigogine, Statistical mechanics of irreversible processes part viii: general theory of weakly coupled systems, Physica, 22 (1956), 621-636. doi: 10.1016/S0031-8914(56)90009-X. [6] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), 449-475. doi: 10.1007/s00222-008-0124-z. [7] F. Cacciafesta and A.-S. de Suzzoni, Continuity of the flow of KdV with regard to the Wasserstein metrics and application to an invariant measure, ArXiv e-prints, 2013. [8] A.-S. de Suzzoni, Wave Turbulence for the BBM Equation: Stability of a Gaussian Statistics Under the Flow of BBM, Comm. Math. Phys., 326 (2014), 773-813. doi: 10.1007/s00220-014-1897-0. [9] J. L. Lebowitz, H. A. Rose and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys., 50 (1988), 657-687. doi: 10.1007/BF01026495. [10] R. Peierls, Zur kinetischen theorie der wärmeleitung in kristallen, Annalen der Physik, 395 (1929), 1055-1101. doi: 10.1002/andp.19293950803. [11] V. E. Zakharov and N. N. Filonenko, Weak turbulence of capillary waves, Journal of Applied Mechanics and Technical Physics, 8 (1967), 37-40. doi: 10.1007/BF00915178. [12] P. E. Zhidkov, On invariant measures for some infinite-dimensional dynamical systems, Ann. Inst. H. Poincaré Phys. Théor., 62 (1995), 267-287.

show all references

##### References:
 [1] J. L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4. [2] ________, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory, Nonlinearity, 17 (2004), 925-952. doi: 10.1088/0951-7715/17/3/010. [3] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252. doi: 10.3934/dcds.2009.23.1241. [4] J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26. doi: 10.1007/BF02099299. [5] R. Brout and I. Prigogine, Statistical mechanics of irreversible processes part viii: general theory of weakly coupled systems, Physica, 22 (1956), 621-636. doi: 10.1016/S0031-8914(56)90009-X. [6] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), 449-475. doi: 10.1007/s00222-008-0124-z. [7] F. Cacciafesta and A.-S. de Suzzoni, Continuity of the flow of KdV with regard to the Wasserstein metrics and application to an invariant measure, ArXiv e-prints, 2013. [8] A.-S. de Suzzoni, Wave Turbulence for the BBM Equation: Stability of a Gaussian Statistics Under the Flow of BBM, Comm. Math. Phys., 326 (2014), 773-813. doi: 10.1007/s00220-014-1897-0. [9] J. L. Lebowitz, H. A. Rose and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys., 50 (1988), 657-687. doi: 10.1007/BF01026495. [10] R. Peierls, Zur kinetischen theorie der wärmeleitung in kristallen, Annalen der Physik, 395 (1929), 1055-1101. doi: 10.1002/andp.19293950803. [11] V. E. Zakharov and N. N. Filonenko, Weak turbulence of capillary waves, Journal of Applied Mechanics and Technical Physics, 8 (1967), 37-40. doi: 10.1007/BF00915178. [12] P. E. Zhidkov, On invariant measures for some infinite-dimensional dynamical systems, Ann. Inst. H. Poincaré Phys. Théor., 62 (1995), 267-287.
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