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The discrete model of the Boltzmann equation. (English) Zbl 0624.76102

We study the general model of a gas with a discrete distribution of velocities. The medium is composed of identical particles which can only have velocity vectors belonging to a finite set of p vectors. When the medium is sufficiently rarefied, only binary collisions are considered. The original Boltzmann equation is replaced by a system of p first order partial differential equations. Each equation in the system is linear with respect to the derivatives of the unknown functions and quadratic with respect to the functions themselves.
As in classical kinetic theory, by introducing a properly defined H- Boltzmann function, we can prove that for a gas in a uniform state, the distribution of velocities tends to a distribution, called Maxwellian, in which each collision brings no contribution to the evolution of densities. Among all distributions of velocities which correspond to given state variables, one and only one is Maxwellian, and the corresponding H-function is minimal. When the collision cross-section is infinite, the velocity distribution is Maxwellian, and the evolution of the gas is governed by the Euler equations (which form a hyperbolic system) and by the associated shock equations.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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[1] Beale J. T., Communications on Mathematical Physics
[2] Bellomo N., Comptes-Rendus Ac. Sci. Paris 299 pp 835– (1984)
[3] Bellomo N., Proc. of German-Italian Seminar on Applications of Mathematics in Technology (1984)
[4] DOI: 10.1063/1.1711368 · Zbl 0123.21102 · doi:10.1063/1.1711368
[5] DOI: 10.1017/S0022112064000817 · Zbl 0151.41001 · doi:10.1017/S0022112064000817
[6] DOI: 10.1017/S0022112076000633 · Zbl 0351.76079 · doi:10.1017/S0022112076000633
[7] Cabannes H., Comptes-rendus Ac. Sci. Paris 284 pp 269– (1977)
[8] Cabannes H., J. de Méc 17 pp 1– (1978)
[9] Cabannes H., Mathematical Problems in Kinetic Theory pp 25– (1979)
[10] DOI: 10.1016/0093-6413(83)90058-7 · Zbl 0578.76076 · doi:10.1016/0093-6413(83)90058-7
[11] Cabannes H., Mechanics Research and Communications
[12] Carleman T., Problèmes mathématiques dans la théorie cinétique des gaz (1957)
[13] Cercignani C., Comptes-rendus Ac. Sci. Paris 301 pp 89– (1985)
[14] Gatignol R., Zeitschrift für Flugwissenschaften 18 pp 93– (1970)
[15] DOI: 10.1063/1.861121 · Zbl 0303.76042 · doi:10.1063/1.861121
[16] Gatignol R., Lecture Notes in Physics 36 (1975) · Zbl 0325.76108
[17] DOI: 10.1063/1.861834 · Zbl 0373.76065 · doi:10.1063/1.861834
[18] Gatignol R., Eleventh Rarefied Gas Dynamics Symposium pp 195– (1979)
[19] Goudunov S. K., Russian Math. Surveys 26 (1971)
[20] Hamdache K., Comptes-rendus Ac. Sci. Paris 299 pp 431– (1984)
[21] Hamdache K., J. de Mécanique théorique et appliquée 3 pp 761– (1984)
[22] DOI: 10.1063/1.1761848 · doi:10.1063/1.1761848
[23] DOI: 10.1063/1.1705172 · Zbl 0155.32604 · doi:10.1063/1.1705172
[24] Illner R., Habilitationnsschrift (1981)
[25] Illner R., J. de Mécanique théorique et appliquee 1 pp 611– (1982)
[26] DOI: 10.1002/mma.1670030110 · Zbl 0563.76073 · doi:10.1002/mma.1670030110
[27] DOI: 10.1007/BF02106189 · Zbl 0361.76081 · doi:10.1007/BF02106189
[28] Kaniel S., J. de Mécanique théorique et appliquée Journal de Mécanique 19 pp 581– (1980)
[29] DOI: 10.3792/pjaa.57.19 · Zbl 0476.76071 · doi:10.3792/pjaa.57.19
[30] Kawashima S., Lecture Notes in Num. Appl. Anal. 6 pp 59– (1983)
[31] Maxwell J. C., Scientific Papers II (1890)
[32] DOI: 10.3792/pja/1195518755 · Zbl 0326.35051 · doi:10.3792/pja/1195518755
[33] Shizuta Y., Hokkaido Math. J.
[34] Tartar L., Ecole polytechnique, Seminaire Goulaouic-Schwartz (1975)
[35] Tartar L., Some existence theorems for semilinear hyperbolic systems in one space variable (1980)
[36] Toscani G., Annali di Matematica pura ed applicata 138 pp 297– (1984)
[37] Toscani G., J. Math. Physics
[38] DOI: 10.1007/BF02412176 · Zbl 0158.11201 · doi:10.1007/BF02412176
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