Global existence and stability of mild solutions to the inelastic Boltzmann system for gas mixtures
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
Full-text PDF Free Access
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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
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- 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
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- 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).
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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.
- Andries, P., Aoki, K. and Perthame, B.: A consistent BGK-type model for gas mixtures. J. Stat. Phys. 106, 993-1018 (2002). MR 1889599 (2002k:82062)
- Aoki, K., Bardos, C. and Takata, S.: Knudsen layer for gas mixtures. J. Stat. Phys. 112, 629-655 (2003). MR 1997264 (2004f:82065)
- Bellomo, N. and Toscani, G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic behavior. J. Math. Phys. 26, 334-338 (1985). MR 776503 (86h:82038)
- Benedetto, D. and Pulvirenti, M.: On the one-dimensional Boltzmann equation for granular flow. Math. Model. Numer. Anal. 35, 899-905 (2001). MR 1866273 (2002h:82066)
- Bobylev, A. V. and Cercignani, C.: Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions. J. Stat. Phys. 110, 333-375 (2003). MR 1966332 (2004g:82110)
- Bobylev, A. V., Cercignani, C. and Toscani, G.: Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials. J. Stat. Phys. 111, 403-417 (2003). MR 1964277 (2004d:82032)
- Bobylev, A. V., Carrillo, J. A. and Gamba, I. M.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys. 98, 743-773 (2000). MR 1749231 (2001c:82063)
- Bobylev, A. V., Gamba, I. M. and Panferov, V. A.: Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Stat. Phys. 116, 1651-1682 (2004). MR 2096050 (2005g:82111)
- 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).
- Cercignani, C., Illner, R. and Stoica, C.: On diffusive equilibria in generalized kinetic theory. J. Stat. Phys. 105, 337-352 (2001). MR 1861207 (2002j:82093)
- Chapman, S. and Cowling, T. G.: The mathematical theory of non-uniform gases. Cambridge University Press, London (1952).
- Chae, M. and Ha, S.-Y.: Stability estimates of the Boltzmann equation with quantum effects. Contin. Mech. Thermodyn. 17, 511-524 (2006). MR 2240606 (2007j:82095)
- Gamba, I. M., Panferov, V. and Villani, C.: On the Boltzmann equation for diffusively excited granular media. Comm. Math. Phys. 246, 503-541 (2004). MR 2053942 (2005b:82076)
- Garzo, V., Santos, A. and Brey, J. J.: A kinetic model for a multi-component gas. Phys. Fluids A 1, 380-383 (1989).
- Jin, S. and Slemrod, M.: Regularization of the Burnett equations for rapid granular flows via relaxation. Physica D 150, 207-218 (2001). MR 1820735 (2001k:76094)
- Ha, S.-Y.: $L^1$ stability estimate for a one-dimensional Boltzmann equation with inelastic collisions. J. Differential Equations 190, 621-642 (2003). MR 1971148 (2004h:35035)
- Ha, S.-Y.: Nonlinear functionals of the Boltzmann equation and uniform stability estimates. J. Differential Equations 215, 178-205 (2005). MR 2146347 (2006b:35033)
- Ha, S.-Y. and Yun, S.-B.: Uniform $L^1$-stability estimate of the Boltzmann equation near a local Maxwellian. Physica D 220, 79-97 (2006). MR 2252152 (2007f:82082)
- Ha, S.-Y., and Noh, S. E.: New a priori estimate for the Boltzmann-Enskog equation. Nonlinearity 19, 1219-1232 (2006). MR 2229996 (2009c:35363)
- Hamel, B.: Kinetic model for binary gas mixtures. Phys. Fluids 8, 418-425 (1965).
- Illner, R. and Shinbrot, M.: Global existence for a rare gas in an infinite vacuum. Comm. Math. Phys. 95, 217-226 (1984). MR 760333 (86a:82019)
- Kaniel, S. and Shinbrot, M.: The Boltzmann equation 1: Uniqueness and local existence. Comm. Math. Phys. 58, 65-84 (1978). MR 0475532 (57:15133)
- Polewczak, J.: Classical solution of the nonlinear Boltzmann equation in all $R^3$: asymptotic behavior of solutions. J. Stat. Phys. 50, 611-632 (1988). MR 939503 (89h:82021)
- Sone, Y.: Kinetic theory and fluid dynamics. Birkhauser, Boston 2002. MR 1919070 (2003h:76113)
- Sone, Y.: Molecular gas dynamics. Birkhauser, Boston 2006. MR 2274674 (2007m:82081)
- Villani, C.: Mathematics of granular materials. J. Stat. Phys. 124, 781-822 (2006). MR 2264625 (2007j:82116)
- Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of Fluid Mechanics, S. Friedlander and D. Serre, Eds. (2001). MR 1942465 (2003k:82087)
- 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).
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Se Eun Noh
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Email:
senoh@math.snu.ac.kr
Keywords:
Boltzmann system,
gas mixture,
Lyapunov functional,
multi-dimensional Bony functional,
nonlinear functional approach.
Received by editor(s):
February 23, 2009
Published electronically:
September 15, 2010
Additional Notes:
The work of S.-Y. Ha is partially supported by KRF-2008-C00023 and research grant of CNS-SNU, and the work of S. Noh is supported by the BK21-Mathematical Division of SNU
Article copyright:
© Copyright 2010
Brown University
The copyright for this article reverts to public domain 28 years after publication.