# The Boltzmann Equation: Mathematics and Applications

Bollettino dell'Unione Matematica Italiana (2007)

- Volume: 10-B, Issue: 2, page 293-315
- ISSN: 0392-4033

## Access Full Article

top## Abstract

top## How to cite

topCercignani, Carlo. "The Boltzmann Equation: Mathematics and Applications." Bollettino dell'Unione Matematica Italiana 10-B.2 (2007): 293-315. <http://eudml.org/doc/290450>.

@article{Cercignani2007,

abstract = {The paper is subdivided into two parts. The first presents a recent result by the author concerning the existence of the solution of the Boltzmann equation for Maxwell molecules, without any cutoff in the collision kernel, when the solution depends on just one variable. At variance with the well-known theorem of DiPerna- Lions, conservation of energy is also shown to hold. The second part will concern rarefied gas dynamics problems, governed by the Boltzmann equation and con- cerning the theory of micromachines (MEMS) and nanomachines (NENS).},

author = {Cercignani, Carlo},

journal = {Bollettino dell'Unione Matematica Italiana},

language = {eng},

month = {6},

number = {2},

pages = {293-315},

publisher = {Unione Matematica Italiana},

title = {The Boltzmann Equation: Mathematics and Applications},

url = {http://eudml.org/doc/290450},

volume = {10-B},

year = {2007},

}

TY - JOUR

AU - Cercignani, Carlo

TI - The Boltzmann Equation: Mathematics and Applications

JO - Bollettino dell'Unione Matematica Italiana

DA - 2007/6//

PB - Unione Matematica Italiana

VL - 10-B

IS - 2

SP - 293

EP - 315

AB - The paper is subdivided into two parts. The first presents a recent result by the author concerning the existence of the solution of the Boltzmann equation for Maxwell molecules, without any cutoff in the collision kernel, when the solution depends on just one variable. At variance with the well-known theorem of DiPerna- Lions, conservation of energy is also shown to hold. The second part will concern rarefied gas dynamics problems, governed by the Boltzmann equation and con- cerning the theory of micromachines (MEMS) and nanomachines (NENS).

LA - eng

UR - http://eudml.org/doc/290450

ER -

## References

top- BONY, M., Existence globale et diffusion en théorie cinétique discrète, In Advances in Kinetic Theory and Continuum Mechanics, R. Gatignol and Soubbarameyer, Eds., 81-90, Springer-Verlag, Berlin (1991)
- CERCIGNANI, C., Weak solutions of the Boltzmann equation and energy conservation, Appl. Math. Lett., 8 (1995), 53-59. Zbl0830.35107MR1357252DOI10.1016/0893-9659(95)00011-E
- CERCIGNANI, C., Theory and Application of the Boltzmann equation. Springer Verlag, New York (1988) Zbl0646.76001MR1007992DOI10.1007/978-1-4612-1039-9
- CERCIGNANI, C., Global Weak Solutions of the Boltzmann Equation, Journal of Statistical Physics118 (2005), 333-342. Zbl1097.82022MR2122558DOI10.1007/s10955-004-8786-4
- CERCIGNANI, C., Estimating the solutions of the Boltzmann equation, submitted to Jour. Stat. Phys. (2005). MR2266453DOI10.1007/s10955-006-9192-x
- CERCIGNANI, C. - ILLNER, R., Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions, Arch. Rational Mech. Anal.134 (1996), 1-16. Zbl0937.45007MR1392307DOI10.1007/BF00376253
- CERCIGNANI, C. - ILLNER, R. - PULVIRENTI, M., The Mathematical Theory of Dilute Gases. Springer Verlag, New York (1994). Zbl0813.76001MR1307620DOI10.1007/978-1-4419-8524-8
- DIPERNA, R. - LIONS, P. L., On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math.130 (1989), 321-366. Zbl0698.45010MR1014927DOI10.2307/1971423
- IKENBERRY, E. - TRUESDELL, C., On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, I, Jour. Rat. Mech. Anal.5 (1956), 1-54. Zbl0070.23504MR75725
- MAXWELL, J. C., On the dynamical theory of gases, Phil. Trans. Roy Soc. (London) 157 (1866), 49-88.
- CERCIGNANI, C., Slow Rarefied Flows. Theory and Application to Micro-Electro-Mechanical Systems, Birkhauser, Basel, (2006). Zbl1097.82001MR2227952DOI10.1007/3-7643-7537-X
- REYNOLDS, O., On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments including an experimental determination of the viscosity of the olive oil, Philos. Trans. R. Soc. London, A177 (1886), 157-234. Zbl18.0946.04
- FAN, J. - SHEN, C., Statistical simulation of low-speed unidirectional flows in transitional regime, in Rarefied Gas Dynamics, edited by R. Brun, R. Campargue, R. Gatignol, J.C. Lengrand, Cepadues Editions, Vol. 2 (1999), 245-252.
- SHEN, C. - FAN, J. - XIE C, C., Statistical simulation of rarefied gas flows in micro-channels, J. Comp. Physics189 (2003), 512-526. Zbl1061.76515
- CERCIGNANI, C. - LAMPIS, M. - LORENZANI, S., Variational approach to gas flows in microchannels, Phys. Fluids, 14 (2004),3426-3437. Zbl1187.76086MR2082039DOI10.1063/1.1764700
- FUKUI, S. - KANEKO, R., Analysis of ultra-thin gas film lubrication based on linearized Boltzmann equation: first report-derivation of a generalized lubrication equation including thermal creep flow, Journal of Tribology, 110 (1988), 253-262.
- FUKUI, S. - KANEKO, R., Analysis of ultra-thin gas film lubrication based on the linearized Boltzmann equation, JSME International Journal, 30 (1987), 1660-1666.
- CERCIGNANI, C. - LAMPIS, M. - LORENZANI, S., Flow of a Rarefied Gas between Parallel and Almost Parallel Plates, in Rarefied Gas Dynamics, 24th Int. Symp., M. Capitelli Ed., AIP Conf. Proc.762, pp. 719-724, New York (2005).
- CERCIGNANI, C. - DANERI, A., Flow of a rarefied gas between two parallel plates, Journal of Applied Physics, 34 (1963), 3509-3513. MR160603
- ALEXANDER, F. J. - GARCIA, A.L. - ALDER, B. J., Direct simulation Monte Carlo for thin film bearings, Phys. Fluids, 6 (1994), 3854-3860. Zbl0832.76064

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.