The Boltzmann Equation: Mathematics and Applications

Carlo Cercignani

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 2, page 293-315
  • ISSN: 0392-4041

Abstract

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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).

How to cite

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Cercignani, 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

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  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
  9. 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
  10. MAXWELL, J. C., On the dynamical theory of gases, Phil. Trans. Roy Soc. (London) 157 (1866), 49-88. 
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  12. 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
  13. 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. 
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  15. CERCIGNANI, C. - LAMPIS, M. - LORENZANI, S., Variational approach to gas flows in microchannels, Phys. Fluids, 14 (2004),3426-3437. Zbl1187.76086MR2082039DOI10.1063/1.1764700
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  20. 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

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