Cercignani: Recent results on the Boltzmann equation

Cercignani, Carlo:
Recent results on the Boltzmann equation (Recenti risultati sull'equazione di Boltzmann)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 7 (1996), fasc. n.2, p. 83-112, (English)
pdf (2.83 MB), djvu (579 Kb). | MR1438608 | Zbl 0870.76072

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Negli ultimi anni la teoria dell'equazione di Boltzmann nonlineare ha registrato una vera folla di contributi, stimolati dal risultato fondamentale di DiPerna e Lions. In questa Memoria vogliamo passare in rassegna questi risultati dedicando un'attenzione particolare ad alcuni tra gli sviluppi più recenti.

Referenze Bibliografiche

[1] C. CERCIGNANI, Mathematical Methods in Kinetic Theory. 2nd edition, Plenum Press, New York 1990. | MR 1069558 | Zbl 0191.25103

[2] C. CERCIGNANI, The Boltzmann Equation and its Applications. Springer, New York 1987. | fulltext (doi) | MR 1313028 | Zbl 0646.76001

[3] C. CERCIGNANI - R. ILLNER - M. PULVIRENTI, The Mathematical Theory of Dilute Gases. Springer-Verlag, New York 1994. | MR 1307620 | Zbl 0813.76001

[4] T. KATO, Perturbation Theory of Linear Operators. Springer, New York 1966. | MR 203473 | Zbl 0435.47001

[5] J. P. GUIRAUD, An H theorem for a gas of rigid spheres in a bounded domain. In: G. PICHON (ed.), Théories cinétiques classiques et rélativistes. CNRS, Paris 1975, 29-58. | MR 449410 | Zbl 0364.76067

[6] Y. SHIZUTA - K. ASANO, Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Japan Acad., vol. 53A, 1977, 3-5. | fulltext mini-dml | MR 466988 | Zbl 0382.35047

[7] S. UKAI, Solutions of the Boltzmann equation. Pattern and Waves-Qualitative Analysis of Nonlinear Differential Equations, 1986, 37-96. | fulltext (doi) | MR 882376 | Zbl 0633.76078

[8] S. UKAI - K. ASANO, On the initial boundary value problem of the linearized Boltzmann equation in an exterior domain. Proc. Japan Acad., vol. 56, 1980, 12-17. | fulltext mini-dml | MR 562862 | Zbl 0462.76077

[9] R. DIPERNA - P. L. LIONS, On the Cauchy problem for Boltzmann equations. Ann. of Math., vol. 130, 1989, 321-366. | fulltext (doi) | MR 1014927 | Zbl 0698.45010

[10] F. GOLSE - B. PERTHAME - P. L. LIONS - R. SENTIS, Regularity of the moments of the solution of a transport equation. J. Funct. Anal., vol. 76, 1988, 110-125. | fulltext (doi) | MR 923047 | Zbl 0652.47031

[11] P. GÉRARD, Solutions globales du problème de Cauchy pour l'équation de Boltzmann. Séminaire Bourbaki, nr. 699, 1987-88. | fulltext EuDML | fulltext mini-dml | Zbl 0704.76041

[12] K. HAMDACHE, Initial boundary value problems for Boltzmann equation. Global existence of weak solutions. Arch. Rat. Mech. Analysis, vol. 119, 1992, 309-353. | fulltext (doi) | MR 1179690 | Zbl 0777.76084

[13] L. ARKERYD - C. CERCIGNANI - R. ILLNER, Measure solutions of the steady Boltzmann equation in a slab. Commun. Math. Phys., vol. 142, 1991, 285-296. | fulltext mini-dml | MR 1137065 | Zbl 0733.76063

[14] L. DESVILLETTES, Convergence to equilibrium in large time for Boltzmann and BGK equations. Arch. Rat. Mech. Analysis, vol. 110, 1990, 73-91. | fulltext (doi) | MR 1031086 | Zbl 0705.76070

[15] C. CERCIGNANI, Equilibrium states and trend to equilibrium in a gas according to the Boltzmann equation. Rend. Mat. Appl., vol. 10, 1990, 77-95. | MR 1060473 | Zbl 0723.76079

[16] L. ARKERYD, On the strong \( L^{1} \) trend to equilibrium for the Boltzmann equation. Studies in Appl. Math., vol. 87, 1992, 283-288. | MR 1180486 | Zbl 0762.35089

[17] P. L. LIONS, Compactness in Boltzmann's equation via Fourier integral operators and applications. I. Journal of Mathematics of Kyoto University, vol. 34, 1994, 391-427. | fulltext mini-dml | MR 1284432 | Zbl 0831.35139

[18] L. ARKERYD - C. CERCIGNANI, A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rat. Mech. Analysis, vol. 125, 1993, 271-288. | fulltext (doi) | MR 1245073 | Zbl 0789.76075

[19] J. DARROZÉS - J.-P. GUIRAUD, Généralisation formelle du théorème H en présence de parois. Applications. C.R.A.S. Paris, vol. A262, 1966, 1368-1371.

[20] C. CERCIGNANI - M. LAMPIS, Kinetic models for gas-surface interactions. Transp. Th. Stat. Phys., vol. 1, 1971, 101-114. | MR 432101 | Zbl 0288.76041

[21] C. CERCIGNANI, Scattering kernels for gas-surface interactions. Transp. Th. Stat. Phys., vol. 2, 1972, 27-53. | Zbl 0892.76076

[22] C. TRUESDELL - R. G. MUNCASTER, Fundamentals of Maxwell's Kinetic Theory of a Simple Monoatomic Gas. Academic Press, New York 1980. | MR 554086

[23] L. ARKERYD - N. MASLOVA, On diffuse reflection at the boundary for the Boltzmann equation and related equations. Preprint ISSN 03407-2809, University of Göteborg, January 1994. | fulltext (doi) | MR 1301931 | Zbl 0839.76073

[24] C. CERCIGNANI, On global existence theorems for the initial value problem for the Boltzmann Equation. In: P. A. Bois et al. (eds.), Asymptotic Modelling in Fluid Mechanics. LNP 442, 1995, 165-177. | fulltext (doi) | MR 1359296 | Zbl 0938.76095

[25] P. L. LIONS, Conditions at infinity for Boltzmann's equation. Cahiers de Mathématiques de la décision no. 9334, CEREMADE 1993. | Zbl 0799.35210

[26] C. CERCIGNANI - R. ILLNER, Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions. Arch. Rat. Mech. Analysis, vol. 134, 1996, 1-16. | fulltext (doi) | MR 1392307 | Zbl 0937.45007

[27] C. CERCIGNANI, Weak solutions of the Boltzmann equation and energy conservation. Appl. Math. Lett., vol. 8, 1995, 53-59. | fulltext (doi) | MR 1357252 | Zbl 0830.35107

[28] C. CERCIGNANI, A remarkable estimate for the solutions of the Boltzmann equation. Appl. Math. Lett., vol. 5, 1992, 59-62. | fulltext (doi) | MR 1345903 | Zbl 0762.35090

[29] C. CERCIGNANI, Errata: Weak solutions of the Boltzmann equation and energy conservation. Appl. Math. Lett., vol. 8, 1995, 95-99. | fulltext (doi) | MR 1356304 | Zbl 0841.35093

[30] L. ARKERYD, Existence theorems for certain kinetic equations and large data. Arch. Rat. Mech. Analysis, vol. 103, 1988, 139-149. | fulltext (doi) | MR 946972 | Zbl 0654.76073

[31] M. BONY, Existence globale et diffusion en théorie cinétique discrète. In: R. GATIGNOL - SOUBBARAMEYER (eds.), Advances in Kinetic Theory and Continuum Mechanics. Springer-Verlag, Berlin 1991, 81-90.

[32] R. DIPERNA - P. L. LIONS, Global solutions of Boltzmann's equation and the entropy inequality. Arch. Rat. Mech. Analysis, vol. 114, 1991, 47-55. | fulltext (doi) | MR 1088276 | Zbl 0724.45011

[33] C. CERCIGNANI, Trend to equilibrium of weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions. J. Stat. Phys., 1996, to appear. | fulltext (doi) | MR 1400189 | Zbl 1081.82614

[34] L. ARKERYD - A. NOURI, Asymptotics of the Boltzmann equation with diffuse reflection boundary conditions. Submitted to Monatshefte für Mathematik, 1995. | Zbl 0877.76063

[35] C. CERCIGNANI, Are there more than linearly independent collision invariants for the Boltzmann equation?. J. Stat. Phys., vol. 58, 1990, 817-823. | fulltext (doi) | MR 1049048 | Zbl 0718.60113

[36] L. BOLTZMANN, Über das Wärmegleichgewicht von Gasen, auf welche äussere Kräfte wirken. Sitzungsberichte der Akademie der Wissenschaften Wien, vol. 72, 1875, 427-457. | Jbk 07.0683.03

[37] L. BOLTZMANN, Über die Aufstellung und Integration der Gleichungen, welche die Molekularbewegungen in Gasen bestimmen. Sitzungsberichte der Akademie der Wissenschaften Wien, vol. 74, 1876, 503-552.

[38] B. WENNBERG, On an entropy dissipation inequality for the Boltzmann equation. C.R.A.S. Paris, vol. 315, I, 1992, 1441-1446. | MR 1199019 | Zbl 0770.76059

[39] G. A. BIRD, Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford 1994. | MR 1352466

[40] C. CERCIGNANI, Aerodynamical Applications of the Boltzmann Equation. La Rivista del Nuovo Cimento, vol. 18, 7, 1995, 1-40. | Zbl 1178.82042

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Referenza completa

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Referenze Bibliografiche