On discrete models of the quantum Boltzmann equation

V. V. Vedenyapin   ; I. V. Mingalev   ; O. V. Mingalev

Sbornik. Mathematics, Tome 80 (1995) no. 2, pp. 271-285

Résumé

A certain class of discrete models for the quantum Boltzmann equation is proposed. In the case of fermions a theorem of existence and uniqueness in the large is proved. In the case of bosons such a theorem is proved for models with the maximum principle. For one simple model analytic solution of the traveling wave problem is given.

Bibliographie

[1] Karleman T., Matematicheskie zadachi kineticheskoi teorii gazov, IL, M., 1960

[2] Broadwell J. E., “Shock Structure in a Simple Discrete Velocity Gas”, The Phys. of Fluids, 7:8 (1964), 1243 | DOI | Zbl

[3] Godunov S. K., Sultangazin U. M., “O diskretnykh modelyakh kineticheskogo uravneniya Boltsmana”, UMN, 36:3 (1971), 1–51 | MR

[4] Vedenyapin V. V., O teoreme suschestvovaniya v tselom reshenii zadachi Koshi nekotorykh giperbolicheskikh nelineinykh sistem uravnenii, Preprint No 42, IPM im. Keldysha AN SSSR, 1973

[5] Vedenyapin V. V., “O razreshimosti v tselom zadachi Koshi dlya diskretnykh modelei uravneniya Boltsmana”, DAN SSSR, 215:1 (1974), 21–23 | MR

[6] Gatignol R., Theoric cinetique des gaz a repartion descrete de Vitesses, Lecture Notes in Physics, 36, 1975 | MR

[7] Gatignol R., “Kinetic theoric for a discrete velocity gas and application to the shock structure”, Phys. of fluids, 18 (1975), 153–161 | DOI | Zbl

[8] Cabannes H., “Global solution for the Discrete Boltzmann Equation”, XIII International Symposium on rarefied gas dynamics, V. 1 (Novosibirsk, July 5–9, 1982), Plenum Press, N.-Y.–London, 1985, 3–17

[9] Illner R., “On discrete models for the Boltzmann equation”, Math. Meth. in the Appl. Sci., 1 (1979), 187–203 | DOI | MR

[10] Kaniel S., Shinbrot M., “The Boltzmann Equation. II: Some Discrete Velocity Models”, J. Mech., 19 (1978), 581–593 | MR

[11] Bellomo N., de Socio L., “On the Discrete Boltzmann Equation for Binary Gas Mixtures”, XIII International Symp. on rarepied gas dynamics, V. II (Novosibirsk, July 5–9, 1982), Plenum Press, N.-Y.–London, 1985, 1269–1276

[12] Shinbrot M., Cercignani C., “Global existence for some discrete velocity models”, Comm. Partial Diff. Equat., 12 (1987), 307–310 | DOI | MR

[13] Di Perna R., Lions P.-L., “On the Cauchy problem for the Boltzmann: global existence and weak stability”, Ann. Math., 130 (1989), 321–366 | DOI | MR

[14] Bobylev A. V., Vedenyapin V. V., “O printsipe maksimuma dlya diskretnykh modelei uravneniya Boltsmana”, DAN SSSR, 233:4 (1977), 519–522 | MR | Zbl

[15] Vedenyapin V. V., “Differentsialnye formy v prostranstvakh bez normy. Teorema o edinstvennosti $H$-funktsii Boltsmana”, UMN, 43:1 (1986), 159–179 | MR

[16] Suslin V. M., Teorema suschestvovaniya i edinstvennosti dlya neodnorodnogo kineticheskogo uravneniya Yulinga–Ulenbeka (sluchai statistiki Fermi–Diraka), Preprint No 56, IPM im. Keldysha AN SSSR, 1989 | MR

[17] Dobrushin R. L., Maslova N. B., Sinai Ya. G., Sukhov Yu. M., “Dinamicheskie sistemy statisticheskoi mekhaniki i kineticheskie uravneniya, III”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 2, VINITI, M., 1985, 233–307 | MR

[18] Illner R., “Global existence results for discrete velocity models of the B.E. in several dimension”, J. Mecan. Th. Appl., 4 (1982), 611–622 | MR

[19] Arkerid L., Esposito R., Pulvirenti M., “The Boltzmann Equation for Weakly Inhomogeneous Data”, Comm. Math. Phys., 111 (1987), 393–407 | DOI | MR

[20] Beale J. T., “Large-time behavior of discrete velocity B.E.”, Comm. Math. Phys., 102 (1985), 217–235 | DOI | MR | Zbl

[21] Dedonne, Osnovy sovremennogo analiza, Mir, M., 1964