ESAIM: Mathematical Modelling and Numerical Analysis
- ESAIM: Mathematical Modelling and Numerical Analysis / Volume 42 / Issue 02 / March 2008, pp 263-275
- © EDP Sciences, SMAI, 2008
- Published online by Cambridge University Press: 02 March 2011
- DOI: http://dx.doi.org/10.1051/m2an:2008007 (About DOI)
Research Article
On the motion of a body in thermal equilibrium immersed in a perfect gas
Aoki, Kazuoa1, Cavallaro, Guidoa2, Marchioro, Carloa2 and Pulvirenti, Marioa2
a1 Department of Mechanical Engineering and Science and Advanced Research Institute of Fluid Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan. aoki@aero.mbox.media.kyoto-u.ac.jp
a2 Dipartimento di Matematica, Università di Roma “La Sapienza", Piazzale A. Moro 2, 00185, Roma, Italy. cavallar@mat.uniroma1.it; marchior@mat.uniroma1.it; pulvirenti@mat.uniroma1.it
Abstract
We consider a body immersed in a perfect gas and moving under the action of a constant force.
Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body,
it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction.
We study the approach of the body velocity V(t) to the limiting velocity
and prove that, under suitable smallness
assumptions, the approach to equilibrium is
where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in
friction problems, and even slower than for the same problem with elastic collisions.
(Received March 31 2007)
(Revised October 16 2007)
(Online publication March 27 2008)
Key Words:
- Kinetic theory of gases;
- Boltzmann equation;
- free molecular gas;
- friction problem;
- approach to equilibrium.
Mathematics Subject Classification:
- 76P05;
- 82B40;
- 82C40;
- 35L45;
- 35L50
- [1]
W.
Braun
and
K.
Hepp
, The Vlasov dynamics and its fluctuations in the
limit of interacting classical particles.
Comm. Math. Phys.
56 (1977) 101–113.
[OpenURL Query Data]
[CrossRef] [Google Scholar] - [2] P. Buttà , E. Caglioti and C. Marchioro , On the long time behavior of infinitely extended systems of particles interacting via Kac Potentials. J. Stat. Phys. 108 (2002) 317–339. [OpenURL Query Data] [CrossRef] [MathSciNet] [Google Scholar]
- [3] S. Caprino , C. Marchioro and M. Pulvirenti , Approach to equilibrium in a microscopic model of friction. Comm. Math. Phys. 264 (2006) 167–189. [OpenURL Query Data] [CrossRef] [MathSciNet] [Google Scholar]
- [4] S. Caprino , G. Cavallaro and C. Marchioro , On a microscopic model of viscous friction. Math. Models Methods Appl. Sci. 17 (2007) 1369–1403. [OpenURL Query Data] [CrossRef] [MathSciNet] [Google Scholar]
- [5] G. Cavallaro , On the motion of a convex body interacting with a perfect gas in the mean-field approximation. Rend. Mat. Appl. 27 (2007) 123–145. [OpenURL Query Data] [Google Scholar]
- [6] R.L. Dobrushin , Vlasov equations. Sov. J. Funct. Anal. 13 (1979) 115–123. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [7] C. Gruber and J. Piasecki , Stationary motion of the adiabatic piston. Physica A 268 (1999) 412–423. [OpenURL Query Data] [CrossRef] [Google Scholar]
- [8] J.L. Lebowitz, J. Piasecki and Y. Sinai, Scaling dynamics of a massive piston in a ideal gas, in Hard Ball Systems and the Lorentz Gas, Encycl. Math. Sci. 101, Springer, Berlin (2000) 217–227.
- [9] H. Neunzert, An Introduction to the Nonlinear Boltzmann-Vlasov Equation, in Kinetic Theories and the Boltzmann Equation, Montecatini (1981), Lecture Notes in Math. 1048, Springer, Berlin (1984) 60–110.
- [10] H. Spohn , On the Vlasov hierarchy. Math. Meth. Appl. Sci. 3 (1981) 445–455. [OpenURL Query Data] [CrossRef] [MathSciNet] [Google Scholar]
