Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation

Archive for Rational Mechanics and Analysis

Volumn 194, Issue 1, 2009, Pages 253-282

  Gamba, I M ^a   Panferov, V ^b   Villani, C ^c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b MCMASTER UNIVERSITY   (Canada)

^c UNIVERSITÉ DE LYON   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70349628429     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-009-0250-9     Document Type: Article

References (35)

1. Arkeryd L.: On the Boltzmann equation. I. Existence II. The full initial value problem. Arch. Ration. Mech. Anal. 45, 1-34 (1972)
2. Arkeryd L.: L∞ estimates for the space-homogeneous Boltzmann equation. J. Stat. Phys. 31(2), 347-361 (1983)
3. Arkeryd L., Esposito R., Pulvirenti M.: The Boltzmann equation for weakly inhomogeneous data. Comm. Math. Phys. 111(3), 393-407 (1987)
4. Bellomo N., Toscani G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic stability. J. Math. Phys. 26(2), 334-338 (1985)
5. Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Mathematical Physics Reviews, Vol. 7, vol. 7 of Soviet Sci. Rev. Sect. C Math. Phys. Rev. Harwood Academic, Churchill, 1988, pp. 111-233
6. Bobylev A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Stat. Phys. 88(5-6), 1183-1214 (1997)
7. Bobylev A.V., Gamba I. M., Panferov V.: Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Stat. Phys. 116(5-6), 1651-1682 (2004)
8. Carleman T.: Sur la théorie de l'équation intégrodifférentielle de Boltzmann. Acta Math. 60, 91-146 (1933)
9. Carleman, T.: Problèmes mathématiques dans la théoriecinétique des gaz. Publ. Sci. Inst. Mittag-Leffler, 2. Almqvist and Wiksell, Uppsala, 1957
10. Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences. Springer, New York (1994)
11. Crandall M.G., Tartar L.: Some relations between nonexpansive and order preserving mappings. Proc. Am. Math. Soc. 78(3), 385-390 (1980)
12. Desvillettes L.: Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Ration. Mech. Anal. 123(4), 387-404 (1993)
13. Di Blasio G.: Differentiability of spatially homogeneous solutions of the Boltzmann equation in the non Maxwellian case. Comm. Math. Phys. 38, 331-340 (1974)
14. DiPerna R., Lions P.-L.: On the Cauchy problem for the Boltzmann equation: Global existence and weak stability. Ann. Math. 130, 321-366 (1989)
15. Elmroth T.: Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Ration. Mech. Anal. 82(1), 1-12 (1983)
16. Evans L.C.: Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)
17. Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
18. Gamba I.M., Panferov V., Villani C.: On the Boltzmann equation for diffusively excited granular media. Comm. Math. Phys. 246(3), 503-541 (2004)
19. Gel'fand, I.M., Shilov, G.E.: Generalized Functions, Vol. 1. Academic Press, New York, London, 1964. Translation of the second Russian edition, Moscow, 1958.
20. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd ed., Vol. 224 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 1983
21. Goudon T.: Generalized invariant sets for the Boltzmann equation. Math. Models Methods Appl. Sci. 7(4), 457-476 (1997)
22. Grad, H.: Principles of the kinetic theory of gases. In: Handbuch der Physik, Vol.12 (Ed. S. Flügge). Springer, Berlin, 1958, 205-294
23. Hamdache K.: Existence in the large and asymptotic behaviour for the Boltzmann equation. Jpn. J. Appl. Math. 2(1), 1-15 (1985)
24. Illner R., Shinbrot M.: The Boltzmann equation: global existence for a rare gas in an infinite vacuum. Comm. Math. Phys. 95(2), 217-226 (1984)
25. Kaniel S., Shinbrot M.: The Boltzmann equation. I. Uniqueness and local existence. Comm. Math. Phys. 58(1), 65-84 (1978)
26. Landau, L.D., Lifshitz, E. M.: Mechanics, 3rd edn. Course of theoretical physics, Vol. 1. Pergamon Press, Oxford, 1976. Translation of the third Russian edition, Moscow, 1973
27. Lanford, III, O.E.: Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), 1-111. Springer, Berlin, 1975. Lecture Notes in Phys., Vol. 38
28. Lions P.-L.: Compactness in Boltzmann's equation via Fourier integral operators and applications. II. J. Math. Kyoto Univ. 34(2), 429-461 (1994)
29. Mischler S., Perthame B.: Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to a Maxwellian. SIAM J. Math. Anal. 28(5), 1015-1027 (1997)
30. Mischler S., Wennberg B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(4), 467-501 (1999)
31. Povzner A.J.: On the Boltzmann equation in the kinetic theory of gases. Mat. Sb. (N.S.) 58(100), 65-86 (1962)
32. Pulvirenti A., Wennberg B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Comm. Math. Phys. 183(1), 145-160 (1997)
33. Vedenjapin, V.V.: On an inequality for convex functions, and on an estimate of the collision integral of the Boltzmann equation for a gas of elastic spheres. Dokl. Akad. Nauk SSSR 226(5) 997-1000, (1976). English translation in Soviet Math. Dokl. 17(1) (1976) 218-222
34. Villani, C.: A Review of Mathematical Topics in Collisional Kinetic Theory. Handbook of mathematical fluid dynamics, Vol. I. North-Holland, Amsterdam, 2002, 71-305
35. Wennberg B.: An example of nonuniqueness for solutions to the homogeneous Boltzmann equation. J. Stat. Phys. 95(1-2), 469-477 (1999)