C. Bardos, F. Golse, and C. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, Journal of Statistical Physics, vol.16, issue.1-2, p.323344, 1991.
DOI : 10.1007/BF01026608

C. Bardos, F. Golse, and C. Levermore, Fluid dynamic limits of kinetic equations II convergence proofs for the boltzmann equation, Communications on Pure and Applied Mathematics, vol.82, issue.162, p.667753, 1993.
DOI : 10.1002/cpa.3160460503

D. Bothe, On the Maxwell-Stefan approach to multicomponent diusion, Parabolic problems, p.8193, 2011.

L. Boudin, D. Götz, and B. Grec, Diusion models of multicomponent mixtures in the lung, CEMRACS 2009: Mathematical modelling in medicine of ESAIM Proc., pages 90103. EDP Sci., Les Ulis, 2010.

L. Boudin, B. Grec, M. Pavi¢, and F. Salvarani, Diusion asymptotics of a kinetic model for gaseous mixtures, Kinet. Relat. Models, vol.6, issue.1, p.137157, 2013.

L. Boudin, B. Grec, and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diusion equations, Discrete Contin. Dyn. Syst. Ser. B, vol.17, issue.5, p.14271440, 2012.

L. Boudin, B. Grec, and F. Salvarani, The Maxwell-Stefan diusion limit for a kinetic model of mixtures, Acta Appl. Math, vol.136, p.7990, 2015.

L. Boudin and F. Salvarani, Compactness of Linearized Kinetic Operators, 2015.
DOI : 10.1007/978-3-319-32144-8_4

URL : https://hal.archives-ouvertes.fr/hal-01242573

J. Bourgat, L. Desvillettes, P. L. Tallec, and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, European J. Mech. B Fluids, vol.13, issue.2, p.237254, 1994.

M. Briant, Stability of global equilibrium for the multi-species Boltzmann equation in L ? settings. ArXiv e-prints, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01492053

M. Briant and E. Daus, The Boltzmann equation for a multi-species mixture close to global equilibrium. ArXiv e-prints, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01492048

S. Brull, V. Pavan, and J. Schneider, Derivation of a BGK model for mixtures, European Journal of Mechanics - B/Fluids, vol.33, p.7486, 2012.
DOI : 10.1016/j.euromechflu.2011.12.003

URL : https://hal.archives-ouvertes.fr/hal-01459970

H. K. Chang, Multicomponent diusion in the lung, Fed. Proc, vol.39, issue.10, p.27592764, 1980.

L. Chen and A. , Analysis of a multidimensional parabolic population model with strong cross-diusion

X. Chen and A. , Analysis of an Incompressible Navier???Stokes???Maxwell???Stefan System, Communications in Mathematical Physics, vol.253, issue.1, p.471497, 2015.
DOI : 10.1007/s00220-015-2472-z

C. F. Curtiss, Symmetric gaseous diusion coecients, J. Chem. Phys, vol.49, issue.7, p.29172919, 1968.

E. S. Daus, A. Jüngel, C. Mouhot, and N. Zamponi, Hypocoercivity for a Linearized Multispecies Boltzmann System, SIAM Journal on Mathematical Analysis, vol.48, issue.1, p.538568, 2016.
DOI : 10.1137/15M1017934

URL : https://hal.archives-ouvertes.fr/hal-01220673

L. Desvillettes, T. Lepoutre, and A. Moussa, Entropy, duality, and cross diusion, SIAM J. Math. Anal, vol.46, issue.1, p.820853, 2014.

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, European Journal of Mechanics - B/Fluids, vol.24, issue.2, p.219236, 2005.
DOI : 10.1016/j.euromechflu.2004.07.004

J. B. Duncan and H. L. Toor, An experimental study of three component gas diffusion, AIChE Journal, vol.8, issue.1, pp.38-41, 1962.
DOI : 10.1002/aic.690080112

A. Fick, Ueber Diusion, Ann. der Physik, vol.170, pp.5986-1855

V. Garzó, A. Santos, and J. J. Brey, A kinetic model for a multicomponent gas, Physics of Fluids A: Fluid Dynamics, vol.1, issue.2, p.380383, 1989.
DOI : 10.1063/1.857458

V. Giovangigli, Convergent iterative methods for multicomponent diusion, Impact Comput. Sci. Engrg, vol.3, issue.3, p.244276, 1991.

V. Giovangigli, Multicomponent ow modeling. Modeling and Simulation in Science, Engineering and Technology, 1999.

F. Golse and L. Saint-raymond, The Navier???Stokes limit of the Boltzmann equation for bounded collision kernels, Inventiones mathematicae, vol.203, issue.1, p.81161, 2004.
DOI : 10.1007/s00222-003-0316-5

URL : https://hal.archives-ouvertes.fr/hal-00021089

T. Graham, Researches on the arseniates, phosphates, and modications of phosphoric acid, Phil. Trans. R. Soc. Lond, vol.123, p.253284, 1833.

H. Hutridurga and F. Salvarani, On the Maxwell-Stefan diusion limit for a mixture of monatomic gases. ArXiv e-prints, 2015.

A. Jüngel and I. V. Stelzer, Existence Analysis of Maxwell--Stefan Systems for Multicomponent Mixtures, SIAM Journal on Mathematical Analysis, vol.45, issue.4, p.24212440, 2013.
DOI : 10.1137/120898164

R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chemical Engineering Science, vol.52, issue.6, p.861911, 1997.
DOI : 10.1016/S0009-2509(96)00458-7

P. , L. Tallec, and J. Perlat, Numerical analysis of Levermore's moment system, 1997.

P. , L. Tallec, and J. P. Perlat, Boundary conditions and existence results for Levermore's moments system, Math. Models Methods Appl. Sci, vol.10, issue.1, p.127152, 2000.

T. Lepoutre, M. Pierre, and G. Rolland, Global well-posedness of a conservative relaxed cross diusion system

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, vol.23, issue.5-6, pp.5-610211065, 1996.
DOI : 10.1007/BF02179552

Y. Lou and S. Martínez, Evolution of cross-diusion and self-diusion, J. Biol. Dyn, vol.3, issue.4, p.410429, 2009.

Y. Lou and W. Ni, Diusion, self-diusion and cross-diusion, J. Dierential Equations, vol.131, issue.1, p.79131, 1996.

J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. R. Soc, vol.157, p.4988, 1866.

I. Müller and T. Ruggeri, Rational extended thermodynamics
DOI : 10.1007/978-1-4612-2210-1

. Springer-verlag, With supplementary chapters by H, 1998.

M. Pavi¢-ƒoli¢ and S. Simi¢, Moment closure hierarchies for rareed gases, Theor. Appl. Mech, vol.42, issue.4, p.261276, 2015.

J. Perrin, Mouvement brownien et réalité moléculaire, Ann. Chim. Phys, vol.123, p.5104, 1909.

J. Schneider, Entropic approximation in kinetic theory, ESAIM: Mathematical Modelling and Numerical Analysis, vol.38, issue.3, p.541561, 2004.
DOI : 10.1051/m2an:2004025

URL : https://hal.archives-ouvertes.fr/hal-01313788

L. Schwartz, Les tenseurs. Hermann, Paris, second edition Suivi de Torseurs sur un espace ane. [With Torsors on an ane space, 1981.

N. Shigesada, K. Kawasaki, and E. Teramoto, Spatial segregation of interacting species, Journal of Theoretical Biology, vol.79, issue.1, p.8399, 1979.
DOI : 10.1016/0022-5193(79)90258-3

J. Stefan, Ueber das Gleichgewicht und die Bewegung insbesondere die Diusion von Gasgemengen, Akad. Wiss. Wien, vol.63, p.63124, 1871.

F. Williams, Combustion Theory, 1985.
URL : https://hal.archives-ouvertes.fr/hal-00014918

L. B. Cnrs, I. , L. Jacques, and V. P. , Équipe-projet Reo, Paris, F-75005, France E-mail address: laurent, Louis Lions (LJLL CNRS UMR 7598)