Structure of entropies in dissipative multicomponent fluids

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June 2013, 6(2): 373-406. doi: 10.3934/krm.2013.6.373

Structure of entropies in dissipative multicomponent fluids

Vincent Giovangigli ^1, and Lionel Matuszewski ^2,

1.	CMAP, CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France
2.	ONERA, Centre de Palaiseau, 91198 Palaiseau cedex, France

Received June 2012 Revised December 2012 Published February 2013

Abstract
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We investigate the structure of mathematical entropies for dissipative multicomponent fluid models derived from the kinetic theory of gases. The corresponding governing equations notably involve nonideal thermochemistry as well as diffusion fluxes driven by chemical potential gradients and temperature gradients. We obtain the general form of mathematical entropies compatible with the hyperbolic structure of the system of partial differential equations assuming a natural nondegeneracy condition. We next establish that entropies compatible with the hyperbolic-parabolic structure are unique up to an affine transform when they are independent on mass and heat diffusion parameters.

Keywords: Entropy, dissipative, multicomponent., hyperbolic.

Mathematics Subject Classification: Primary: 35L65, 76T30, 35M30; Secondary: 80A3.

Citation: Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic and Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373

References:

[1]	R. Aris, Prolegomena to the rational analysis of systems of chemical reactions, Archiv. Rat. Mech. Anal., 19 (1965), 81-99.
[2]	R. J. Bearman and J. G. Kirkwood, The statistical mechanics of transport processes. XI. Equations of transport in multicomponent systems, J. Chem. Phys., 28 (1958), 136-145.
[3]	H. Van Beijeren and M. H. Ernst, The modified enskog equations, Phys. A, 68 (1973), 437-456.
[4]	H. Van Beijeren and M. H. Ernst, The modified enskog equations for mixtures, Phys. A, 70 (1973), 225-242.
[5]	R. Bendahklia, V. Giovangigli and D. Rosner, Soret effects in laminar counterflow spray diffusion flames, Comb. Theory Mod., 6 (2002), 1-17.
[6]	G. Billet, V. Giovangigli and G. de Gassowski, Impact of volume viscosity on a shock-hydrogen bubble interaction, Comb. Theory Mod., 12 (2008), 221-248. doi: 10.1080/13647830701545875.
[7]	D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pure Appl., 87 (2007), 57-90. doi: 10.1016/j.matpur.2006.11.001.
[8]	S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases," Third edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.
[9]	G. Q. Chen, C. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602.
[10]	J.-P. Croisille and P. Delorme, Kinetic symmetrizations and pressure laws for the Euler equations, Physica D, 57 (1992), 395-416. doi: 10.1016/0167-2789(92)90010-K.
[11]	C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.
[12]	P. Degond, S. Genieys and A. Jüengel, A system of parabolic equations in nonequilibrium thermodynamics including thermal and electric effects, J. Math. Pure Appl., 160 (1997), 991-1015.
[13]	S. R. de Groot and P. Mazur, "Non-Equilibrium Thermodynamics," Dover publications, Inc., New York, 1984.
[14]	A. Ern and V. Giovangigli, Thermal diffusion effects in hydrogen-air and methane-air flames, Comb. Theory Mod., 2-4 (1998), 349-372.
[15]	A. Ern and V. Giovangigli, "Multicomponent Transport Algorithms," Lecture Notes in Physics, New Series m: Monographs, 24, Springer-Verlag, Berlin, 1994.
[16]	A. Ern and V. Giovangigli, Thermal conduction and thermal diffusion in dilute polyatomic gas mixtures, Physica A, 214 (1995), 526-546.
[17]	A. Ern and V. Giovangigli, Structure of transport linear systems in dilute isotropic gas mixtures, Phys. Rev. E (3), 53 (1996), 485-492. doi: 10.1103/PhysRevE.53.485.
[18]	A. Ern and V. Giovangigli, Optimized transport algorithms for flame codes, Comb. Sci. Tech., 118 (1996), 387-395.
[19]	A. Ern and V. Giovangigli, Projected iterative algorithms with application to multicomponent transport, Linear Algebra Appl., 250 (1997), 289-315. doi: 10.1016/0024-3795(95)00502-1.
[20]	A. Ern and V. Giovangigli, The Kinetic equilibrium regime, Physica A, 260 (1998), 49-72.
[21]	L. C. Evans, A survey of entropy methods for partial differential equations, Bulletin of the AMS (N.S.), 41 (2004), 409-438. doi: 10.1090/S0273-0979-04-01032-8.
[22]	E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.
[23]	J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases," North-Holland Publishing Company, Amsterdam, 1972.
[24]	K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688.
[25]	W. H. Furry, On the elementary explanation of diffusion phenomena in gases, Am. J. Phys., 16 (1948), 63-78.
[26]	V. Giovangigli, Convergent iterative methods for multicomponent diffusion, Impact Comput. Sci. Eng., 3 (1991), 244-276. doi: 10.1016/0899-8248(91)90010-R.
[27]	V. Giovangigli, "Multicomponent Flow Modeling," Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1580-6.
[28]	V. Giovangigli, Persistence of Boltzmann entropy in fluid models, Disc. Cont. Dyn. Syst., 24 (2009), 95-114. doi: 10.3934/dcds.2009.24.95.
[29]	V. Giovangigli, Higher order entropies, Arch. Rat. Mech. Anal., 187 (2008), 221-285. doi: 10.1007/s00205-007-0065-5.
[30]	V. Giovangigli, Higher order entropies for compressible fluid models, Math. Mod. Meth. Appl. Sci., 19 (2009), 67-125. doi: 10.1142/S021820250900336X.
[31]	V. Giovangigli, Multicomponent transport algorithms for partially ionized mixtures, J. Comp. Phys., 229 (2010), 4117-4142. doi: 10.1016/j.jcp.2010.02.001.
[32]	V. Giovangigli and M. Massot, Asymptotic stability of equilibrium states for multicomponent reactive flows, Math. Mod. Meth. App. Sci., 8 (1998), 251-297. doi: 10.1142/S0218202598000123.
[33]	V. Giovangigli and M. Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry, Math. Meth. Appl. Sci., 27 (2004), 739-768. doi: 10.1002/mma.429.
[34]	V. Giovangigli and L. Matuszewski, Supercritical fluid thermodynamics from equations of state, Phys. D, 241 (2012), 649-670. doi: 10.1016/j.physd.2011.12.002.
[35]	V. Giovangigli and L. Matuszewski, Mathematical modeling of supercritical multicomponent reactive fluids, to appear, M3AS, (2012).
[36]	V. Giovangigli, L. Matuszewski and F. Dupoirieux, Detailed modeling of planar transcritical $H_2$-$O_2$-$N_2$ flames, Combustion Theory and Modelling, 15 (2011), 141-182.
[37]	V. Giovangigli and B. Tran, Mathematical analysis of a Saint-Venant model with variable temperature, Math. Mod. Meth. Appl. Sci., 20 (2010), 1251-1297. doi: 10.1142/S0218202510004593.
[38]	A. Glitzky, K. Gröger and R. Hünlich, Free energy and dissipation rate for reaction diffusion processes of electrically charged species, Appl. Anal., 60 (1996), 201-217. doi: 10.1080/00036819608840428.
[39]	S. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl., 2 (1961), 947-949.
[40]	E. A. Guggenheim, "Thermodynamics," North Holland, Amsterdam, 1962.
[41]	J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, "Molecular Theory of Gases and Liquids," Wiley, New York, 1954.
[42]	T. J. R. Hughes, L. P. Franca and M. Mallet, A new finite element formulation for computational fluid dynamics. I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comp. Meth. Appl. Mech. Eng., 54 (1986), 223-234. doi: 10.1016/0045-7825(86)90127-1.
[43]	J. H. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics, J. Chem. Phys., 18 (1950), 817-829.
[44]	A. Jüengel and I. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures, arXiv:1211.2394, (2012).
[45]	S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics," Doctoral Thesis, Kyoto University, 1984.
[46]	S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservations laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194. doi: 10.1017/S0308210500018308.
[47]	S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math. J. (2), 40 (1988), 449-464. doi: 10.2748/tmj/1178227986.
[48]	S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal., 174 (2004), 345-364. doi: 10.1007/s00205-004-0330-9.
[49]	J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes," Springer-Verlag, New York, 1987.
[50]	F. J. Krambeck, The mathematical structure of chemical kinetics in homogeneous single-phase systems, Arch. Rational Mech. Anal., 38 (1970), 317-347. doi: 10.1007/BF00251527.
[51]	V. I. Kurochkin, S. F. Makarenko and G. A. Tirskii, Transport coefficients and the Onsager relations in the kinetic theroy of dense gas mixtures, J. Appl. Mech. Tech. Phys., 25 (1984), 218-225.
[52]	P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys., 163 (1994), 415-431.
[53]	M. R. Marcelin, "Sur la Mécanique des Phénomènes Irréversibles," Comptes Rendus de l'Académie des Sciences de Paris, Séance du 5 décembre 1910, (1910), 1052-1055.
[54]	J. Meixner, Zur Thermodynamik der irreversiblen Prozesse in Gasen mit chemisch reagierenden, dissoziierenden und anregbaren Komponenten, Ann. der Phys., 43 (1943), 244-270.
[55]	H. Mori, Statistical-mechanical theory of transport in fluids, Phys. Rev., 112 (1958), 1829-1842.
[56]	I. Prigogine, "Etude Thermodynamique des Phénomènes Irréversibles," Dunod, Paris, 1947.
[57]	J. Pousin, "Modélisation et Analyse Numérique de Couches Limites Réactives d'Air," Doctorat es Sciences, Ecole Polytechnique Fédérale de Lausanne, 1112, 1993.
[58]	T. Ruggeri, Thermodynamics and Symmetric Hyperbolic Systems. Nonlinear hyperbolic equations in applied sciences,, Rend. Sem. Mat. Univ. Politec. Torino, 1988 (): 167.
[59]	N. Z. Shapiro and L. S. Shapley, Mass action law and the Gibbs free energy function, SIAM J. Appl. Math., 13 (1965), 353-375.
[60]	D. Serre, Systèmes de Lois de Conservation. I et II, Diderot Editeur, Paris, 1996.
[61]	D. Serre, The structure of dissipative viscous system of conservation laws, Physica D, 239 (2010), 1381-1386. doi: 10.1016/j.physd.2009.03.014.
[62]	Y. Shizuta and S. Kawashima, Systems of Equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.
[63]	R. Taylor and R. Krishna, "Multicomponent Mass Transfer," John Wiley, New York, 1993.
[64]	T. Umeda, S. Kawashima and Y. Shizuta, On the Decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.
[65]	J. Van de Ree, On the definition of the diffusion coefficients in reacting gases, Physica, 36 (1967), 118-126.
[66]	A. I. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. (N.S.), 87(129) (1972), 504-528.
[67]	L. G. Vulkov, On the conservation laws of the Compressible euler equations, Applicable Analysis, 64 (1997), 255-271. doi: 10.1080/00036819708840534.
[68]	L. Waldmann, Transporterscheinungen in Gasen von mittlerem Druck, in "1958 Handbuch der Physik," Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 295-514.
[69]	J. Wei, An axiomatic treatment of chemical reaction systems, J. Chem. Phys., 36 (1962), 1578-1584.
[70]	F. A. Williams, "Combustion Theory," Menlo Park, 1985.
[71]	W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Rat. Mech. Anal., 172 (2004), 247-266. doi: 10.1007/s00205-003-0304-3.

show all references

References:

[1]	R. Aris, Prolegomena to the rational analysis of systems of chemical reactions, Archiv. Rat. Mech. Anal., 19 (1965), 81-99.
[2]	R. J. Bearman and J. G. Kirkwood, The statistical mechanics of transport processes. XI. Equations of transport in multicomponent systems, J. Chem. Phys., 28 (1958), 136-145.
[3]	H. Van Beijeren and M. H. Ernst, The modified enskog equations, Phys. A, 68 (1973), 437-456.
[4]	H. Van Beijeren and M. H. Ernst, The modified enskog equations for mixtures, Phys. A, 70 (1973), 225-242.
[5]	R. Bendahklia, V. Giovangigli and D. Rosner, Soret effects in laminar counterflow spray diffusion flames, Comb. Theory Mod., 6 (2002), 1-17.
[6]	G. Billet, V. Giovangigli and G. de Gassowski, Impact of volume viscosity on a shock-hydrogen bubble interaction, Comb. Theory Mod., 12 (2008), 221-248. doi: 10.1080/13647830701545875.
[7]	D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pure Appl., 87 (2007), 57-90. doi: 10.1016/j.matpur.2006.11.001.
[8]	S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases," Third edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.
[9]	G. Q. Chen, C. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602.
[10]	J.-P. Croisille and P. Delorme, Kinetic symmetrizations and pressure laws for the Euler equations, Physica D, 57 (1992), 395-416. doi: 10.1016/0167-2789(92)90010-K.
[11]	C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.
[12]	P. Degond, S. Genieys and A. Jüengel, A system of parabolic equations in nonequilibrium thermodynamics including thermal and electric effects, J. Math. Pure Appl., 160 (1997), 991-1015.
[13]	S. R. de Groot and P. Mazur, "Non-Equilibrium Thermodynamics," Dover publications, Inc., New York, 1984.
[14]	A. Ern and V. Giovangigli, Thermal diffusion effects in hydrogen-air and methane-air flames, Comb. Theory Mod., 2-4 (1998), 349-372.
[15]	A. Ern and V. Giovangigli, "Multicomponent Transport Algorithms," Lecture Notes in Physics, New Series m: Monographs, 24, Springer-Verlag, Berlin, 1994.
[16]	A. Ern and V. Giovangigli, Thermal conduction and thermal diffusion in dilute polyatomic gas mixtures, Physica A, 214 (1995), 526-546.
[17]	A. Ern and V. Giovangigli, Structure of transport linear systems in dilute isotropic gas mixtures, Phys. Rev. E (3), 53 (1996), 485-492. doi: 10.1103/PhysRevE.53.485.
[18]	A. Ern and V. Giovangigli, Optimized transport algorithms for flame codes, Comb. Sci. Tech., 118 (1996), 387-395.
[19]	A. Ern and V. Giovangigli, Projected iterative algorithms with application to multicomponent transport, Linear Algebra Appl., 250 (1997), 289-315. doi: 10.1016/0024-3795(95)00502-1.
[20]	A. Ern and V. Giovangigli, The Kinetic equilibrium regime, Physica A, 260 (1998), 49-72.
[21]	L. C. Evans, A survey of entropy methods for partial differential equations, Bulletin of the AMS (N.S.), 41 (2004), 409-438. doi: 10.1090/S0273-0979-04-01032-8.
[22]	E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.
[23]	J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases," North-Holland Publishing Company, Amsterdam, 1972.
[24]	K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688.
[25]	W. H. Furry, On the elementary explanation of diffusion phenomena in gases, Am. J. Phys., 16 (1948), 63-78.
[26]	V. Giovangigli, Convergent iterative methods for multicomponent diffusion, Impact Comput. Sci. Eng., 3 (1991), 244-276. doi: 10.1016/0899-8248(91)90010-R.
[27]	V. Giovangigli, "Multicomponent Flow Modeling," Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1580-6.
[28]	V. Giovangigli, Persistence of Boltzmann entropy in fluid models, Disc. Cont. Dyn. Syst., 24 (2009), 95-114. doi: 10.3934/dcds.2009.24.95.
[29]	V. Giovangigli, Higher order entropies, Arch. Rat. Mech. Anal., 187 (2008), 221-285. doi: 10.1007/s00205-007-0065-5.
[30]	V. Giovangigli, Higher order entropies for compressible fluid models, Math. Mod. Meth. Appl. Sci., 19 (2009), 67-125. doi: 10.1142/S021820250900336X.
[31]	V. Giovangigli, Multicomponent transport algorithms for partially ionized mixtures, J. Comp. Phys., 229 (2010), 4117-4142. doi: 10.1016/j.jcp.2010.02.001.
[32]	V. Giovangigli and M. Massot, Asymptotic stability of equilibrium states for multicomponent reactive flows, Math. Mod. Meth. App. Sci., 8 (1998), 251-297. doi: 10.1142/S0218202598000123.
[33]	V. Giovangigli and M. Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry, Math. Meth. Appl. Sci., 27 (2004), 739-768. doi: 10.1002/mma.429.
[34]	V. Giovangigli and L. Matuszewski, Supercritical fluid thermodynamics from equations of state, Phys. D, 241 (2012), 649-670. doi: 10.1016/j.physd.2011.12.002.
[35]	V. Giovangigli and L. Matuszewski, Mathematical modeling of supercritical multicomponent reactive fluids, to appear, M3AS, (2012).
[36]	V. Giovangigli, L. Matuszewski and F. Dupoirieux, Detailed modeling of planar transcritical $H_2$-$O_2$-$N_2$ flames, Combustion Theory and Modelling, 15 (2011), 141-182.
[37]	V. Giovangigli and B. Tran, Mathematical analysis of a Saint-Venant model with variable temperature, Math. Mod. Meth. Appl. Sci., 20 (2010), 1251-1297. doi: 10.1142/S0218202510004593.
[38]	A. Glitzky, K. Gröger and R. Hünlich, Free energy and dissipation rate for reaction diffusion processes of electrically charged species, Appl. Anal., 60 (1996), 201-217. doi: 10.1080/00036819608840428.
[39]	S. Godunov, An interesting class of quasilinear systems, Sov. Math. Dokl., 2 (1961), 947-949.
[40]	E. A. Guggenheim, "Thermodynamics," North Holland, Amsterdam, 1962.
[41]	J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, "Molecular Theory of Gases and Liquids," Wiley, New York, 1954.
[42]	T. J. R. Hughes, L. P. Franca and M. Mallet, A new finite element formulation for computational fluid dynamics. I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comp. Meth. Appl. Mech. Eng., 54 (1986), 223-234. doi: 10.1016/0045-7825(86)90127-1.
[43]	J. H. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics, J. Chem. Phys., 18 (1950), 817-829.
[44]	A. Jüengel and I. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures, arXiv:1211.2394, (2012).
[45]	S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics," Doctoral Thesis, Kyoto University, 1984.
[46]	S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservations laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194. doi: 10.1017/S0308210500018308.
[47]	S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math. J. (2), 40 (1988), 449-464. doi: 10.2748/tmj/1178227986.
[48]	S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal., 174 (2004), 345-364. doi: 10.1007/s00205-004-0330-9.
[49]	J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes," Springer-Verlag, New York, 1987.
[50]	F. J. Krambeck, The mathematical structure of chemical kinetics in homogeneous single-phase systems, Arch. Rational Mech. Anal., 38 (1970), 317-347. doi: 10.1007/BF00251527.
[51]	V. I. Kurochkin, S. F. Makarenko and G. A. Tirskii, Transport coefficients and the Onsager relations in the kinetic theroy of dense gas mixtures, J. Appl. Mech. Tech. Phys., 25 (1984), 218-225.
[52]	P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys., 163 (1994), 415-431.
[53]	M. R. Marcelin, "Sur la Mécanique des Phénomènes Irréversibles," Comptes Rendus de l'Académie des Sciences de Paris, Séance du 5 décembre 1910, (1910), 1052-1055.
[54]	J. Meixner, Zur Thermodynamik der irreversiblen Prozesse in Gasen mit chemisch reagierenden, dissoziierenden und anregbaren Komponenten, Ann. der Phys., 43 (1943), 244-270.
[55]	H. Mori, Statistical-mechanical theory of transport in fluids, Phys. Rev., 112 (1958), 1829-1842.
[56]	I. Prigogine, "Etude Thermodynamique des Phénomènes Irréversibles," Dunod, Paris, 1947.
[57]	J. Pousin, "Modélisation et Analyse Numérique de Couches Limites Réactives d'Air," Doctorat es Sciences, Ecole Polytechnique Fédérale de Lausanne, 1112, 1993.
[58]	T. Ruggeri, Thermodynamics and Symmetric Hyperbolic Systems. Nonlinear hyperbolic equations in applied sciences,, Rend. Sem. Mat. Univ. Politec. Torino, 1988 (): 167.
[59]	N. Z. Shapiro and L. S. Shapley, Mass action law and the Gibbs free energy function, SIAM J. Appl. Math., 13 (1965), 353-375.
[60]	D. Serre, Systèmes de Lois de Conservation. I et II, Diderot Editeur, Paris, 1996.
[61]	D. Serre, The structure of dissipative viscous system of conservation laws, Physica D, 239 (2010), 1381-1386. doi: 10.1016/j.physd.2009.03.014.
[62]	Y. Shizuta and S. Kawashima, Systems of Equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.
[63]	R. Taylor and R. Krishna, "Multicomponent Mass Transfer," John Wiley, New York, 1993.
[64]	T. Umeda, S. Kawashima and Y. Shizuta, On the Decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.
[65]	J. Van de Ree, On the definition of the diffusion coefficients in reacting gases, Physica, 36 (1967), 118-126.
[66]	A. I. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. (N.S.), 87(129) (1972), 504-528.
[67]	L. G. Vulkov, On the conservation laws of the Compressible euler equations, Applicable Analysis, 64 (1997), 255-271. doi: 10.1080/00036819708840534.
[68]	L. Waldmann, Transporterscheinungen in Gasen von mittlerem Druck, in "1958 Handbuch der Physik," Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 295-514.
[69]	J. Wei, An axiomatic treatment of chemical reaction systems, J. Chem. Phys., 36 (1962), 1578-1584.
[70]	F. A. Williams, "Combustion Theory," Menlo Park, 1985.
[71]	W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Rat. Mech. Anal., 172 (2004), 247-266. doi: 10.1007/s00205-003-0304-3.

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