# American Institute of Mathematical Sciences

March  2013, 6(1): 137-157. doi: 10.3934/krm.2013.6.137

## Diffusion asymptotics of a kinetic model for gaseous mixtures

 1 UPMC Univ Paris 06, UMR 7598 LJLL, Paris, F-75005 2 MAP5, CNRS UMR 8145, Université Paris Descartes, Sorbonne Paris Cité, 45 Rue des Saints Pères, F-75006 Paris, France 3 CMLA, ENS Cachan, PRES UniverSud Paris, 61 Avenue du Président Wilson, F-94235 Cachan Cedex, France 4 Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1 - 27100 Pavia

Received  July 2012 Revised  October 2012 Published  December 2012

In this work, we consider the non-reactive fully elastic Boltzmann equations for mixtures in the diffusive scaling. We mainly use a Hilbert expansion of the distribution functions. After briefly recalling the H-theorem, the lower-order non trivial equality obtained from the Boltzmann equations leads to a linear functional equation in the velocity variable. This equation is solved thanks to the Fredholm alternative. Since we consider multicomponent mixtures, the classical techniques introduced by Grad cannot be applied, and we propose a new method to treat the terms involving particles with different masses.
Citation: Laurent Boudin, Bérénice Grec, Milana Pavić, Francesco Salvarani. Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinetic and Related Models, 2013, 6 (1) : 137-157. doi: 10.3934/krm.2013.6.137
##### References:
 [1] C. Bardos, F. Golse and C. D. Levermore, Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 727-732. [2] C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), 323-344. [3] C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753. [4] S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations, J. Statist. Phys., 101 (2000), 1087-1136. [5] M. Bennoune, M. Lemou and L. Mieussens, An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit, Contin. Mech. Thermodyn., 21 (2009), 401-421. [6] M. Bisi and L. Desvillettes, Incompressible Navier-Stokes equations from kinetic models for a mixture of rarefied gases, Work in progress. [7] L. Boudin, B. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, Submitted. [8] J.-F. Bourgat, L. Desvillettes, P. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, European J. Mech. B Fluids, 13 (1994), 237-254. [9] C. Cercignani, "The Boltzmann Equation and Its Applications," volume 67 of Applied Mathematical Sciences. Springer-Verlag, New York, 1988. [10] S. Chapman and T. G. Cowling, "The mathematical Theory of Nonuniform Gases," Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 1990. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, In co-operation with D. Burnett, With a foreword by Carlo Cercignani. [11] P. Degond, Macroscopic limits of the Boltzmann equation: A review, in "Modeling and Computational Methods for Kinetic Equations," Model. Simul. Sci. Eng. Technol., 3-57. Birkhäuser Boston, Boston, MA, (2004). [12] L. Desvillettes, R. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236. [13] C. Dogbe, Fluid dynamic limits for gas mixture. I. Formal derivations, Math. Models Methods Appl. Sci., 18 (2008), 1633-1672. [14] J. B. Duncan and H. L. Toor, An experimental study of three component gas diffusion, AIChE Journal, 8 (1962), 38-41. [15] F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161. [16] F. Golse and L. Saint-Raymond, The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl. (9), 91 (2009), 508-552. [17] H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181. [18] H. Grad, Asymptotic theory of the Boltzmann equation. II, in "Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962" I, 26-59. Academic Press, New York, (1963). [19] D. E. Greene, Mathematical aspects of kinetic model equations for binary gas mixtures, J. Mathematical Phys., 16 (1975), 776-782. [20] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479. [21] F. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728. [22] F. Huang, Y. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phys., 295 (2010), 293-326. [23] S. Jin and Q. Li, A BGK-penalization asymptotic-preserving scheme for the multispecies Boltzmann equation, Technical report, 2012. Submitted. [24] R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci., 52 (1997), 861-911. [25] P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II, Arch. Ration. Mech. Anal., 158 (2001), 173-193, 195-211. [26] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc., 157 (1866), 49-88. [27] T. F. Morse, Kinetic model equations for a gas mixture, Phys. Fluids, 7 (1964), 2012-2013. [28] J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28. [29] A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573. [30] L. Sirovich, Kinetic modeling of gas mixtures, Phys. Fluids, 5 (1962), 908-918. [31] J. Stefan, Ueber das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen, Akad. Wiss. Wien, 63 (1871), 63-124. [32] S.-H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equation, Comm. Pure Appl. Math., 58 (2005), 409-443.

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##### References:
 [1] C. Bardos, F. Golse and C. D. Levermore, Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 727-732. [2] C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys., 63 (1991), 323-344. [3] C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753. [4] S. Bastea, R. Esposito, J. L. Lebowitz and R. Marra, Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations, J. Statist. Phys., 101 (2000), 1087-1136. [5] M. Bennoune, M. Lemou and L. Mieussens, An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit, Contin. Mech. Thermodyn., 21 (2009), 401-421. [6] M. Bisi and L. Desvillettes, Incompressible Navier-Stokes equations from kinetic models for a mixture of rarefied gases, Work in progress. [7] L. Boudin, B. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, Submitted. [8] J.-F. Bourgat, L. Desvillettes, P. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, European J. Mech. B Fluids, 13 (1994), 237-254. [9] C. Cercignani, "The Boltzmann Equation and Its Applications," volume 67 of Applied Mathematical Sciences. Springer-Verlag, New York, 1988. [10] S. Chapman and T. G. Cowling, "The mathematical Theory of Nonuniform Gases," Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 1990. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, In co-operation with D. Burnett, With a foreword by Carlo Cercignani. [11] P. Degond, Macroscopic limits of the Boltzmann equation: A review, in "Modeling and Computational Methods for Kinetic Equations," Model. Simul. Sci. Eng. Technol., 3-57. Birkhäuser Boston, Boston, MA, (2004). [12] L. Desvillettes, R. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236. [13] C. Dogbe, Fluid dynamic limits for gas mixture. I. Formal derivations, Math. Models Methods Appl. Sci., 18 (2008), 1633-1672. [14] J. B. Duncan and H. L. Toor, An experimental study of three component gas diffusion, AIChE Journal, 8 (1962), 38-41. [15] F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161. [16] F. Golse and L. Saint-Raymond, The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials, J. Math. Pures Appl. (9), 91 (2009), 508-552. [17] H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids, 6 (1963), 147-181. [18] H. Grad, Asymptotic theory of the Boltzmann equation. II, in "Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962" I, 26-59. Academic Press, New York, (1963). [19] D. E. Greene, Mathematical aspects of kinetic model equations for binary gas mixtures, J. Mathematical Phys., 16 (1975), 776-782. [20] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479. [21] F. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728. [22] F. Huang, Y. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phys., 295 (2010), 293-326. [23] S. Jin and Q. Li, A BGK-penalization asymptotic-preserving scheme for the multispecies Boltzmann equation, Technical report, 2012. Submitted. [24] R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci., 52 (1997), 861-911. [25] P.-L. Lions and N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II, Arch. Ration. Mech. Anal., 158 (2001), 173-193, 195-211. [26] J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc., 157 (1866), 49-88. [27] T. F. Morse, Kinetic model equations for a gas mixture, Phys. Fluids, 7 (1964), 2012-2013. [28] J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28. [29] A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573. [30] L. Sirovich, Kinetic modeling of gas mixtures, Phys. Fluids, 5 (1962), 908-918. [31] J. Stefan, Ueber das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen, Akad. Wiss. Wien, 63 (1871), 63-124. [32] S.-H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equation, Comm. Pure Appl. Math., 58 (2005), 409-443.
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