Global existence and stability of mild solutions to the inelastic Boltzmann system for gas mixtures

Ha, Seung-Yeal; Noh, Se

doi:10.1090/S0033-569X-2010-01183-5

Authors: Seung-Yeal Ha and Se Eun Noh
Journal: Quart. Appl. Math. 68 (2010), 671-699
MSC (2000): Primary 35A05, 35B65, 78A35
DOI: https://doi.org/10.1090/S0033-569X-2010-01183-5
Published electronically: September 15, 2010
MathSciNet review: 2761510
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Abstract: We study the global existence and uniform stability estimate of mild solutions to the inelastic Boltzmann system for gas mixtures, when initial data are small and decay exponentially in phase space, and we also provide a general multi-dimensional Bony-type potential which yields a priori weighted two-point correlation estimates in phase space to the mild solutions with finite mass and energy without any smallness restriction.

References

Alonso, R.: Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data. To appear in Indiana Univ. Math. J.

Pierre Andries, Kazuo Aoki, and Benoit Perthame, A consistent BGK-type model for gas mixtures, J. Statist. Phys. 106 (2002), no. 5-6, 993–1018. MR 1889599, DOI https://doi.org/10.1023/A%3A1014033703134

Kazuo Aoki, Claude Bardos, and Shigeru Takata, Knudsen layer for gas mixtures, J. Statist. Phys. 112 (2003), no. 3-4, 629–655. MR 1997264, DOI https://doi.org/10.1023/A%3A1023876025363

N. Bellomo and G. Toscani, On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic stability, J. Math. Phys. 26 (1985), no. 2, 334–338. MR 776503, DOI https://doi.org/10.1063/1.526664

Dario Benedetto and Mario Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, M2AN Math. Model. Numer. Anal. 35 (2001), no. 5, 899–905. MR 1866273, DOI https://doi.org/10.1051/m2an%3A2001141

A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys. 110 (2003), no. 1-2, 333–375. MR 1966332, DOI https://doi.org/10.1023/A%3A1021031031038

A. V. Bobylev, C. Cercignani, and G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, J. Statist. Phys. 111 (2003), no. 1-2, 403–417. MR 1964277, DOI https://doi.org/10.1023/A%3A1022273528296

A. V. Bobylev, J. A. Carrillo, and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys. 98 (2000), no. 3-4, 743–773. MR 1749231, DOI https://doi.org/10.1023/A%3A1018627625800

A. V. Bobylev, I. M. Gamba, and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Statist. Phys. 116 (2004), no. 5-6, 1651–1682. MR 2096050, DOI https://doi.org/10.1023/B%3AJOSS.0000041751.11664.ea

Bony, J. M.: Solutions globales bornées pour les modèles discrete de l’équation de Boltzmann en dimension 1 d’espace. Acte Journees E. D. P. St. Jean de Monts, $n^o$ XVI (1987).

Carlo Cercignani, Reinhard Illner, and Cristina Stoica, On diffusive equilibria in generalized kinetic theory, J. Statist. Phys. 105 (2001), no. 1-2, 337–352. MR 1861207, DOI https://doi.org/10.1023/A%3A1012246513712

Chapman, S. and Cowling, T. G.: The mathematical theory of non-uniform gases. Cambridge University Press, London (1952).

Myeongju Chae and Seung-Yeal Ha, Stability estimates of the Boltzmann equation with quantum effects, Contin. Mech. Thermodyn. 17 (2006), no. 7, 511–524. MR 2240606, DOI https://doi.org/10.1007/s00161-006-0012-y

I. M. Gamba, V. Panferov, and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys. 246 (2004), no. 3, 503–541. MR 2053942, DOI https://doi.org/10.1007/s00220-004-1051-5

Garzo, V., Santos, A. and Brey, J. J.: A kinetic model for a multi-component gas. Phys. Fluids A 1, 380-383 (1989).

Shi Jin and Marshall Slemrod, Regularization of the Burnett equations for rapid granular flows via relaxation, Phys. D 150 (2001), no. 3-4, 207–218. MR 1820735, DOI https://doi.org/10.1016/S0167-2789%2800%2900224-4

Seung-Yeal Ha, $L^1$ stability estimate for a one-dimensional Boltzmann equation with inelastic collisions, J. Differential Equations 190 (2003), no. 2, 621–642. MR 1971148, DOI https://doi.org/10.1016/S0022-0396%2802%2900113-4

Seung-Yeal Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations 215 (2005), no. 1, 178–205. MR 2146347, DOI https://doi.org/10.1016/j.jde.2004.07.022

Seung-Yeal Ha and Seok-Bae Yun, Uniform $L^1$-stability estimate of the Boltzmann equation near a local Maxwellian, Phys. D 220 (2006), no. 1, 79–97. MR 2252152, DOI https://doi.org/10.1016/j.physd.2006.06.011

Seung-Yeal Ha and Se Eun Noh, New a priori estimate for the Boltzmann-Enskog equation, Nonlinearity 19 (2006), no. 6, 1219–1232. MR 2229996, DOI https://doi.org/10.1088/0951-7715/19/6/001

Hamel, B.: Kinetic model for binary gas mixtures. Phys. Fluids 8, 418-425 (1965).

Reinhard Illner and Marvin Shinbrot, The Boltzmann equation: global existence for a rare gas in an infinite vacuum, Comm. Math. Phys. 95 (1984), no. 2, 217–226. MR 760333

Shmuel Kaniel and Marvin Shinbrot, The Boltzmann equation. I. Uniqueness and local existence, Comm. Math. Phys. 58 (1978), no. 1, 65–84. MR 475532

Jacek Polewczak, Classical solution of the nonlinear Boltzmann equation in all ${\bf R}^3$: asymptotic behavior of solutions, J. Statist. Phys. 50 (1988), no. 3-4, 611–632. MR 939503, DOI https://doi.org/10.1007/BF01026493

Yoshio Sone, Kinetic theory and fluid dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1919070

Yoshio Sone, Molecular gas dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2007. Theory, techniques, and applications. MR 2274674

Cédric Villani, Mathematics of granular materials, J. Stat. Phys. 124 (2006), no. 2-4, 781–822. MR 2264625, DOI https://doi.org/10.1007/s10955-006-9038-6

Cédric Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 71–305. MR 1942465, DOI https://doi.org/10.1016/S1874-5792%2802%2980004-0

Yasuda, S., Takata, S. and Aoki, K.: Evaporation and condensation of a binary mixture of vapors on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation. Phys. Fluids 17, 047105 (2005).

Yasuda, S., Takata S. and Aoki, K.: Numerical analysis of the shear flow of a binary mixture of hard-sphere gases over a plane wall. Phys. Fluids 16, 1989-2003 (2004).