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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 & Continuous Dynamical Systems - A, 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.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

[2]

________, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory,, Nonlinearity, 17 (2004), 925.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[3]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Syst., 23 (2009), 1241.  doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[4]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures,, Comm. Math. Phys., 166 (1994), 1.  doi: 10.1007/BF02099299.  Google Scholar

[5]

R. Brout and I. Prigogine, Statistical mechanics of irreversible processes part viii: general theory of weakly coupled systems,, Physica, 22 (1956), 621.  doi: 10.1016/S0031-8914(56)90009-X.  Google Scholar

[6]

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory,, Invent. Math., 173 (2008), 449.  doi: 10.1007/s00222-008-0124-z.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s00220-014-1897-0.  Google Scholar

[9]

J. L. Lebowitz, H. A. Rose and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation,, J. Statist. Phys., 50 (1988), 657.  doi: 10.1007/BF01026495.  Google Scholar

[10]

R. Peierls, Zur kinetischen theorie der wärmeleitung in kristallen,, Annalen der Physik, 395 (1929), 1055.  doi: 10.1002/andp.19293950803.  Google Scholar

[11]

V. E. Zakharov and N. N. Filonenko, Weak turbulence of capillary waves,, Journal of Applied Mechanics and Technical Physics, 8 (1967), 37.  doi: 10.1007/BF00915178.  Google Scholar

[12]

P. E. Zhidkov, On invariant measures for some infinite-dimensional dynamical systems,, Ann. Inst. H. Poincaré Phys. Théor., 62 (1995), 267.   Google Scholar

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.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

[2]

________, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory,, Nonlinearity, 17 (2004), 925.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[3]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Syst., 23 (2009), 1241.  doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[4]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures,, Comm. Math. Phys., 166 (1994), 1.  doi: 10.1007/BF02099299.  Google Scholar

[5]

R. Brout and I. Prigogine, Statistical mechanics of irreversible processes part viii: general theory of weakly coupled systems,, Physica, 22 (1956), 621.  doi: 10.1016/S0031-8914(56)90009-X.  Google Scholar

[6]

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory,, Invent. Math., 173 (2008), 449.  doi: 10.1007/s00222-008-0124-z.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s00220-014-1897-0.  Google Scholar

[9]

J. L. Lebowitz, H. A. Rose and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation,, J. Statist. Phys., 50 (1988), 657.  doi: 10.1007/BF01026495.  Google Scholar

[10]

R. Peierls, Zur kinetischen theorie der wärmeleitung in kristallen,, Annalen der Physik, 395 (1929), 1055.  doi: 10.1002/andp.19293950803.  Google Scholar

[11]

V. E. Zakharov and N. N. Filonenko, Weak turbulence of capillary waves,, Journal of Applied Mechanics and Technical Physics, 8 (1967), 37.  doi: 10.1007/BF00915178.  Google Scholar

[12]

P. E. Zhidkov, On invariant measures for some infinite-dimensional dynamical systems,, Ann. Inst. H. Poincaré Phys. Théor., 62 (1995), 267.   Google Scholar

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