# American Institute of Mathematical Sciences

December  2012, 5(4): 673-695. doi: 10.3934/krm.2012.5.673

## Exponential stability of the solutions to the Boltzmann equation for the Benard problem

 1 Mathematical Sciences Chalmers 41296 Gothenburg, Sweden 2 MEMOCS, Università dell'Aquila, Cisterna di Latina (LT), 04012, Italy 3 Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma 4 LATP, CMI, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France

Received  May 2012 Revised  July 2012 Published  November 2012

We complete the result in [2] by showing the exponential decay of the perturbation of the laminar solution below the critical Rayleigh number and of the convective solutions above the critical Rayleigh number, in the kinetic framework.
Citation: Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic and Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673
##### References:
 [1] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability of the laminar solution of the Boltzmann equation for the Benard problem, Bull. Academia Sinica, 3 (2008), 51-97. [2] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Archive for Rational Mechanics, 198 (2010), 125-187. doi: 10.1007/s00205-010-0292-z. [3] L. Arkeryd and A. Nouri, Asymptotic techniques for kinetic problems of Boltzmann type, Proceedings of the 3rd Edition of the Summer School in "Methods and Models of Kinetic Theory," Riv. Mat. Univ. Parma, 7 (2007), 1-74. [4] P. G. Drazin and W. H. Reid, "Hydrodynamic Instability," Cambridge Univ. Press, Cambridge, 1981. [5] R. Esposito, R. Marra and J. L. Lebowitz, Solutions to the Boltzmann equation in the Boussinesq Regime, J. Stat. Phys., 90 (1998), 1129-1178. [6] C. Foias, O. P. Manley and R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Non-Linear Analysis, 11 (1987), 939-967. [7] J. M. Ghidaglia, "Etude d'Écoulements Fluides Visqueux Incompressibles: Comportement pour les Grands Temps et Applications aux Attracteurs," Ph. D thesis, Orsay, 1984. [8] V. I. Iudovich, On the origin of convection, J. Appl. Math. Mech., 30 (1966), 1193-1199. [9] V. I. Iudovich, Free convection and bifurcation, J. Appl. Math. Mech., 31 (1967), 103-114. [10] V. I. Iudovich, Stability of convection flows, J. Appl. Math. Mech., 31 (1967), 272-281. [11] D. D. Joseph, On the stability of the Boussinesq equation, Arch. Rat. Mech. Anal., 20 (1965), 59-71. [12] N. B. Maslova, "Nonlinear Evolution Equations: Kinetic Approach," World Scientific, 1993. [13] T. Ma and S. Wang, Dynamic bifurcation and stability in the Rayleigh- Bénard convection, Comm. Math. Sci., 2 (2004), 159-183. [14] Y. Sone, "Kinetic Theory and Fluid Dynamics," Birkhäuser Boston, 2002; Molecular gas dynamics, theory, techniques, and applications, World Scientific, Birkhäuser Boston, 2007.

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##### References:
 [1] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability of the laminar solution of the Boltzmann equation for the Benard problem, Bull. Academia Sinica, 3 (2008), 51-97. [2] L. Arkeryd, R. Esposito, R. Marra and A. Nouri, Stability for Rayleigh-Benard convective solutions of the Boltzmann equation, Archive for Rational Mechanics, 198 (2010), 125-187. doi: 10.1007/s00205-010-0292-z. [3] L. Arkeryd and A. Nouri, Asymptotic techniques for kinetic problems of Boltzmann type, Proceedings of the 3rd Edition of the Summer School in "Methods and Models of Kinetic Theory," Riv. Mat. Univ. Parma, 7 (2007), 1-74. [4] P. G. Drazin and W. H. Reid, "Hydrodynamic Instability," Cambridge Univ. Press, Cambridge, 1981. [5] R. Esposito, R. Marra and J. L. Lebowitz, Solutions to the Boltzmann equation in the Boussinesq Regime, J. Stat. Phys., 90 (1998), 1129-1178. [6] C. Foias, O. P. Manley and R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Non-Linear Analysis, 11 (1987), 939-967. [7] J. M. Ghidaglia, "Etude d'Écoulements Fluides Visqueux Incompressibles: Comportement pour les Grands Temps et Applications aux Attracteurs," Ph. D thesis, Orsay, 1984. [8] V. I. Iudovich, On the origin of convection, J. Appl. Math. Mech., 30 (1966), 1193-1199. [9] V. I. Iudovich, Free convection and bifurcation, J. Appl. Math. Mech., 31 (1967), 103-114. [10] V. I. Iudovich, Stability of convection flows, J. Appl. Math. Mech., 31 (1967), 272-281. [11] D. D. Joseph, On the stability of the Boussinesq equation, Arch. Rat. Mech. Anal., 20 (1965), 59-71. [12] N. B. Maslova, "Nonlinear Evolution Equations: Kinetic Approach," World Scientific, 1993. [13] T. Ma and S. Wang, Dynamic bifurcation and stability in the Rayleigh- Bénard convection, Comm. Math. Sci., 2 (2004), 159-183. [14] Y. Sone, "Kinetic Theory and Fluid Dynamics," Birkhäuser Boston, 2002; Molecular gas dynamics, theory, techniques, and applications, World Scientific, Birkhäuser Boston, 2007.
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