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 & 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.

[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. 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, 7 (2007), 1.

[4]

P. G. Drazin and W. H. Reid, "Hydrodynamic Instability,", Cambridge Univ. Press, (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.

[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.

[7]

J. M. Ghidaglia, "Etude d'Écoulements Fluides Visqueux Incompressibles: Comportement pour les Grands Temps et Applications aux Attracteurs,", Ph. D thesis, (1984).

[8]

V. I. Iudovich, On the origin of convection,, J. Appl. Math. Mech., 30 (1966), 1193.

[9]

V. I. Iudovich, Free convection and bifurcation,, J. Appl. Math. Mech., 31 (1967), 103.

[10]

V. I. Iudovich, Stability of convection flows,, J. Appl. Math. Mech., 31 (1967), 272.

[11]

D. D. Joseph, On the stability of the Boussinesq equation,, Arch. Rat. Mech. Anal., 20 (1965), 59.

[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.

[14]

Y. Sone, "Kinetic Theory and Fluid Dynamics,", Birkhäuser Boston, (2002).

show all references

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.

[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. 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, 7 (2007), 1.

[4]

P. G. Drazin and W. H. Reid, "Hydrodynamic Instability,", Cambridge Univ. Press, (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.

[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.

[7]

J. M. Ghidaglia, "Etude d'Écoulements Fluides Visqueux Incompressibles: Comportement pour les Grands Temps et Applications aux Attracteurs,", Ph. D thesis, (1984).

[8]

V. I. Iudovich, On the origin of convection,, J. Appl. Math. Mech., 30 (1966), 1193.

[9]

V. I. Iudovich, Free convection and bifurcation,, J. Appl. Math. Mech., 31 (1967), 103.

[10]

V. I. Iudovich, Stability of convection flows,, J. Appl. Math. Mech., 31 (1967), 272.

[11]

D. D. Joseph, On the stability of the Boussinesq equation,, Arch. Rat. Mech. Anal., 20 (1965), 59.

[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.

[14]

Y. Sone, "Kinetic Theory and Fluid Dynamics,", Birkhäuser Boston, (2002).

[1]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869

[2]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[3]

Seiji Ukai, Tong Yang, Huijiang Zhao. Exterior Problem of Boltzmann Equation with Temperature Difference. Communications on Pure & Applied Analysis, 2009, 8 (1) : 473-491. doi: 10.3934/cpaa.2009.8.473

[4]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557

[5]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for the Ibragimov-Shabat equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 661-673. doi: 10.3934/dcdss.2016020

[6]

Hao Tang, Zhengrong Liu. On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2229-2256. doi: 10.3934/dcds.2016.36.2229

[7]

Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic & Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443

[8]

Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159

[9]

Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 55-75. doi: 10.3934/era.2015.22.55

[10]

Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145

[11]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205

[12]

Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic & Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499

[13]

El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401

[14]

Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic & Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237

[15]

Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic & Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551

[16]

Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039

[17]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation. Networks & Heterogeneous Media, 2016, 11 (2) : 281-300. doi: 10.3934/nhm.2016.11.281

[18]

Marzia Bisi, Giampiero Spiga. A Boltzmann-type model for market economy and its continuous trading limit. Kinetic & Related Models, 2010, 3 (2) : 223-239. doi: 10.3934/krm.2010.3.223

[19]

Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741

[20]

Arnaud Debussche, Julien Vovelle. Diffusion limit for a stochastic kinetic problem. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2305-2326. doi: 10.3934/cpaa.2012.11.2305

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (0)

[Back to Top]