Abstract
We prove in this work that under suitable assumptions, the solution of the spatially homogeneous non-cut-off Kac equation (or of the spatially homogeneous non cut-off 2D Boltzmann equation with Maxwellian molecules in the radial case) becomes very regular with respect to the velocity variable as soon as the time is strictly positive.
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[A 1] Arkeryd, L.: On the Boltzmann equation, I and II. Arch. Rat. Mech. Anal.45, 1–34 (1972)
[A 2] Arkeryd, L.: Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rat. Mech. Anal.77, 11–21 (1981)
[A 3] Arkeryd, L.: Stability inL 1 for the spatially homogeneous Boltzmann equation. Arch. Rat. Mech. Anal.103, 151–167 (1988)
[B] Brezis, H.: Analyse fonctionnelle, théorie et applications. Paris: Masson, (1983)
[Ce] Cercignani, C.: The Boltzmann equation and its applications. Berlin, Heidelberg, New York: Springer, (1988)
[Ch, Co] Chapman, S., Cowling, T.G.: The mathematical theory of non-uniform gases. London: Cambridge Univ. Press., (1952)
[Dg, Lu] Degond, P., Lucquin-Desreux, B.: The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Mod. Meth. Appl. Sc.2 (1992)
[De 1] Desvillettes, L.: Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Rat. Mech. Anal.123, 387–404 (1993)
[De 2] Desvillettes, L.: On asymptotics of the Boltzmann equation when the collisions become grazing. Tr. Th. and Stat. Phys.21, 259–276 (1992)
[De 3] Desvillettes, L.: Convergence to equilibrium in large time for Boltzmann and B.G.K equations. Arch. Rat. Mech. Anal.110, 73–91 (1990)
[DP, L] DiPerna, R.J., Lions, P-L.: On the Cauchy problem for Boltzmann equations, Global existence and weak stability. Ann. Math.130, 321–366 (1989)
[El] Elmroth, T.: Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Rat. Mech. Anal.82, 1–12 (1983)
[G] Gabetta, E.: On a conjecture of McKean with application to Kac's model. To appear in Tr. Th. and Stat. Phys.
[Gr] Grad, H.: Principles of the kinetic theory of gases. Flügge's Handbuch der Physik.12, 205–294 (1958)
[K] Kac, M.: Probability and related topics in the Physical Sciences. New York: (needs publisher) (1959)
[L 1] Lions, P-L.: Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II and III. Preprint
[L 2] Lions, P-L.: On Boltzmann and Landau equations. Cahiers de mathématiques de la décision, n. 9220, and to appear in Phil. Trans. Roy. Soc.
[MK] McKean, H.P.: Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Rat. Mech. Anal.21, 347–367 (1966)
[Tr] Truesdell, C.: On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory II. J. Rat. Mech. Anal.5, 55 (1955)
[Tr, Mu] Truesdell, C., Muncaster, R.: Fundamentals of Maxwell kinetic theory of a simple monoatomic gas. New York: Acad. Press., (1980)
[We] Wennberg, B.: Preprint
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Communicated by J.L. Lebowitz
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Desvillettes, L. About the regularizing properties of the non-cut-off Kac equation. Commun.Math. Phys. 168, 417–440 (1995). https://doi.org/10.1007/BF02101556
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DOI: https://doi.org/10.1007/BF02101556