American Institute of Mathematical Sciences

Advanced
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

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 & 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.   Google Scholar

[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.   Google Scholar

[3]

H. Van Beijeren and M. H. Ernst, The modified enskog equations,, Phys. A, 68 (1973), 437.   Google Scholar

[4]

H. Van Beijeren and M. H. Ernst, The modified enskog equations for mixtures,, Phys. A, 70 (1973), 225.   Google Scholar

[5]

R. Bendahklia, V. Giovangigli and D. Rosner, Soret effects in laminar counterflow spray diffusion flames,, Comb. Theory Mod., 6 (2002), 1.   Google Scholar

[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.  doi: 10.1080/13647830701545875.  Google Scholar

[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.  doi: 10.1016/j.matpur.2006.11.001.  Google Scholar

[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, (1970).   Google Scholar

[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.  doi: 10.1002/cpa.3160470602.  Google Scholar

[10]

J.-P. Croisille and P. Delorme, Kinetic symmetrizations and pressure laws for the Euler equations,, Physica D, 57 (1992), 395.  doi: 10.1016/0167-2789(92)90010-K.  Google Scholar

[11]

C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325 (2000).  doi: 10.1007/3-540-29089-3_14.  Google Scholar

[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.   Google Scholar

[13]

S. R. de Groot and P. Mazur, "Non-Equilibrium Thermodynamics,", Dover publications, (1984).   Google Scholar

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A. Ern and V. Giovangigli, Thermal diffusion effects in hydrogen-air and methane-air flames,, Comb. Theory Mod., 2-4 (1998), 2.   Google Scholar

[15]

A. Ern and V. Giovangigli, "Multicomponent Transport Algorithms,", Lecture Notes in Physics, 24 (1994).   Google Scholar

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A. Ern and V. Giovangigli, Thermal conduction and thermal diffusion in dilute polyatomic gas mixtures,, Physica A, 214 (1995), 526.   Google Scholar

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A. Ern and V. Giovangigli, Structure of transport linear systems in dilute isotropic gas mixtures,, Phys. Rev. E (3), 53 (1996), 485.  doi: 10.1103/PhysRevE.53.485.  Google Scholar

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A. Ern and V. Giovangigli, Optimized transport algorithms for flame codes,, Comb. Sci. Tech., 118 (1996), 387.   Google Scholar

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A. Ern and V. Giovangigli, Projected iterative algorithms with application to multicomponent transport,, Linear Algebra Appl., 250 (1997), 289.  doi: 10.1016/0024-3795(95)00502-1.  Google Scholar

[20]

A. Ern and V. Giovangigli, The Kinetic equilibrium regime,, Physica A, 260 (1998), 49.   Google Scholar

[21]

L. C. Evans, A survey of entropy methods for partial differential equations,, Bulletin of the AMS (N.S.), 41 (2004), 409.  doi: 10.1090/S0273-0979-04-01032-8.  Google Scholar

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[24]

K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension,, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686.   Google Scholar

[25]

W. H. Furry, On the elementary explanation of diffusion phenomena in gases,, Am. J. Phys., 16 (1948), 63.   Google Scholar

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V. Giovangigli, Convergent iterative methods for multicomponent diffusion,, Impact Comput. Sci. Eng., 3 (1991), 244.  doi: 10.1016/0899-8248(91)90010-R.  Google Scholar

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V. Giovangigli, "Multicomponent Flow Modeling,", Modeling and Simulation in Science, (1999).  doi: 10.1007/978-1-4612-1580-6.  Google Scholar

[28]

V. Giovangigli, Persistence of Boltzmann entropy in fluid models,, Disc. Cont. Dyn. Syst., 24 (2009), 95.  doi: 10.3934/dcds.2009.24.95.  Google Scholar

[29]

V. Giovangigli, Higher order entropies,, Arch. Rat. Mech. Anal., 187 (2008), 221.  doi: 10.1007/s00205-007-0065-5.  Google Scholar

[30]

V. Giovangigli, Higher order entropies for compressible fluid models,, Math. Mod. Meth. Appl. Sci., 19 (2009), 67.  doi: 10.1142/S021820250900336X.  Google Scholar

[31]

V. Giovangigli, Multicomponent transport algorithms for partially ionized mixtures,, J. Comp. Phys., 229 (2010), 4117.  doi: 10.1016/j.jcp.2010.02.001.  Google Scholar

[32]

V. Giovangigli and M. Massot, Asymptotic stability of equilibrium states for multicomponent reactive flows,, Math. Mod. Meth. App. Sci., 8 (1998), 251.  doi: 10.1142/S0218202598000123.  Google Scholar

[33]

V. Giovangigli and M. Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry,, Math. Meth. Appl. Sci., 27 (2004), 739.  doi: 10.1002/mma.429.  Google Scholar

[34]

V. Giovangigli and L. Matuszewski, Supercritical fluid thermodynamics from equations of state,, Phys. D, 241 (2012), 649.  doi: 10.1016/j.physd.2011.12.002.  Google Scholar

[35]

V. Giovangigli and L. Matuszewski, Mathematical modeling of supercritical multicomponent reactive fluids,, to appear, (2012).   Google Scholar

[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.   Google Scholar

[37]

V. Giovangigli and B. Tran, Mathematical analysis of a Saint-Venant model with variable temperature,, Math. Mod. Meth. Appl. Sci., 20 (2010), 1251.  doi: 10.1142/S0218202510004593.  Google Scholar

[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.  doi: 10.1080/00036819608840428.  Google Scholar

[39]

S. Godunov, An interesting class of quasilinear systems,, Sov. Math. Dokl., 2 (1961), 947.   Google Scholar

[40]

E. A. Guggenheim, "Thermodynamics,", North Holland, (1962).   Google Scholar

[41]

J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, "Molecular Theory of Gases and Liquids,", Wiley, (1954).   Google Scholar

[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.  doi: 10.1016/0045-7825(86)90127-1.  Google Scholar

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[44]

A. Jüengel and I. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures,, , (2012).   Google Scholar

[45]

S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics,", Doctoral Thesis, (1984).   Google Scholar

[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.  doi: 10.1017/S0308210500018308.  Google Scholar

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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.  doi: 10.2748/tmj/1178227986.  Google Scholar

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S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of conservation laws,, Arch. Rat. Mech. Anal., 174 (2004), 345.  doi: 10.1007/s00205-004-0330-9.  Google Scholar

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J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes,", Springer-Verlag, (1987).   Google Scholar

[50]

F. J. Krambeck, The mathematical structure of chemical kinetics in homogeneous single-phase systems,, Arch. Rational Mech. Anal., 38 (1970), 317.  doi: 10.1007/BF00251527.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[53]

M. R. Marcelin, "Sur la Mécanique des Phénomènes Irréversibles,", Comptes Rendus de l'Académie des Sciences de Paris, (1910), 1052.   Google Scholar

[54]

J. Meixner, Zur Thermodynamik der irreversiblen Prozesse in Gasen mit chemisch reagierenden, dissoziierenden und anregbaren Komponenten,, Ann. der Phys., 43 (1943), 244.   Google Scholar

[55]

H. Mori, Statistical-mechanical theory of transport in fluids,, Phys. Rev., 112 (1958), 1829.   Google Scholar

[56]

I. Prigogine, "Etude Thermodynamique des Phénomènes Irréversibles,", Dunod, (1947).   Google Scholar

[57]

J. Pousin, "Modélisation et Analyse Numérique de Couches Limites Réactives d'Air,", Doctorat es Sciences, 1112 (1993).   Google Scholar

[58]

T. Ruggeri, Thermodynamics and Symmetric Hyperbolic Systems. Nonlinear hyperbolic equations in applied sciences,, Rend. Sem. Mat. Univ. Politec. Torino, 1988 (): 167.   Google Scholar

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N. Z. Shapiro and L. S. Shapley, Mass action law and the Gibbs free energy function,, SIAM J. Appl. Math., 13 (1965), 353.   Google Scholar

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D. Serre, Systèmes de Lois de Conservation. I et II,, Diderot Editeur, (1996).   Google Scholar

[61]

D. Serre, The structure of dissipative viscous system of conservation laws,, Physica D, 239 (2010), 1381.  doi: 10.1016/j.physd.2009.03.014.  Google Scholar

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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.   Google Scholar

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R. Taylor and R. Krishna, "Multicomponent Mass Transfer,", John Wiley, (1993).   Google Scholar

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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.  doi: 10.1007/BF03167068.  Google Scholar

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J. Van de Ree, On the definition of the diffusion coefficients in reacting gases,, Physica, 36 (1967), 118.   Google Scholar

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show all references

References:
[1]

R. Aris, Prolegomena to the rational analysis of systems of chemical reactions,, Archiv. Rat. Mech. Anal., 19 (1965), 81.   Google Scholar

[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.   Google Scholar

[3]

H. Van Beijeren and M. H. Ernst, The modified enskog equations,, Phys. A, 68 (1973), 437.   Google Scholar

[4]

H. Van Beijeren and M. H. Ernst, The modified enskog equations for mixtures,, Phys. A, 70 (1973), 225.   Google Scholar

[5]

R. Bendahklia, V. Giovangigli and D. Rosner, Soret effects in laminar counterflow spray diffusion flames,, Comb. Theory Mod., 6 (2002), 1.   Google Scholar

[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.  doi: 10.1080/13647830701545875.  Google Scholar

[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.  doi: 10.1016/j.matpur.2006.11.001.  Google Scholar

[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, (1970).   Google Scholar

[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.  doi: 10.1002/cpa.3160470602.  Google Scholar

[10]

J.-P. Croisille and P. Delorme, Kinetic symmetrizations and pressure laws for the Euler equations,, Physica D, 57 (1992), 395.  doi: 10.1016/0167-2789(92)90010-K.  Google Scholar

[11]

C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325 (2000).  doi: 10.1007/3-540-29089-3_14.  Google Scholar

[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.   Google Scholar

[13]

S. R. de Groot and P. Mazur, "Non-Equilibrium Thermodynamics,", Dover publications, (1984).   Google Scholar

[14]

A. Ern and V. Giovangigli, Thermal diffusion effects in hydrogen-air and methane-air flames,, Comb. Theory Mod., 2-4 (1998), 2.   Google Scholar

[15]

A. Ern and V. Giovangigli, "Multicomponent Transport Algorithms,", Lecture Notes in Physics, 24 (1994).   Google Scholar

[16]

A. Ern and V. Giovangigli, Thermal conduction and thermal diffusion in dilute polyatomic gas mixtures,, Physica A, 214 (1995), 526.   Google Scholar

[17]

A. Ern and V. Giovangigli, Structure of transport linear systems in dilute isotropic gas mixtures,, Phys. Rev. E (3), 53 (1996), 485.  doi: 10.1103/PhysRevE.53.485.  Google Scholar

[18]

A. Ern and V. Giovangigli, Optimized transport algorithms for flame codes,, Comb. Sci. Tech., 118 (1996), 387.   Google Scholar

[19]

A. Ern and V. Giovangigli, Projected iterative algorithms with application to multicomponent transport,, Linear Algebra Appl., 250 (1997), 289.  doi: 10.1016/0024-3795(95)00502-1.  Google Scholar

[20]

A. Ern and V. Giovangigli, The Kinetic equilibrium regime,, Physica A, 260 (1998), 49.   Google Scholar

[21]

L. C. Evans, A survey of entropy methods for partial differential equations,, Bulletin of the AMS (N.S.), 41 (2004), 409.  doi: 10.1090/S0273-0979-04-01032-8.  Google Scholar

[22]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Lecture Series in Mathematics and its Applications, 26 (2004).   Google Scholar

[23]

J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases,", North-Holland Publishing Company, (1972).   Google Scholar

[24]

K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension,, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686.   Google Scholar

[25]

W. H. Furry, On the elementary explanation of diffusion phenomena in gases,, Am. J. Phys., 16 (1948), 63.   Google Scholar

[26]

V. Giovangigli, Convergent iterative methods for multicomponent diffusion,, Impact Comput. Sci. Eng., 3 (1991), 244.  doi: 10.1016/0899-8248(91)90010-R.  Google Scholar

[27]

V. Giovangigli, "Multicomponent Flow Modeling,", Modeling and Simulation in Science, (1999).  doi: 10.1007/978-1-4612-1580-6.  Google Scholar

[28]

V. Giovangigli, Persistence of Boltzmann entropy in fluid models,, Disc. Cont. Dyn. Syst., 24 (2009), 95.  doi: 10.3934/dcds.2009.24.95.  Google Scholar

[29]

V. Giovangigli, Higher order entropies,, Arch. Rat. Mech. Anal., 187 (2008), 221.  doi: 10.1007/s00205-007-0065-5.  Google Scholar

[30]

V. Giovangigli, Higher order entropies for compressible fluid models,, Math. Mod. Meth. Appl. Sci., 19 (2009), 67.  doi: 10.1142/S021820250900336X.  Google Scholar

[31]

V. Giovangigli, Multicomponent transport algorithms for partially ionized mixtures,, J. Comp. Phys., 229 (2010), 4117.  doi: 10.1016/j.jcp.2010.02.001.  Google Scholar

[32]

V. Giovangigli and M. Massot, Asymptotic stability of equilibrium states for multicomponent reactive flows,, Math. Mod. Meth. App. Sci., 8 (1998), 251.  doi: 10.1142/S0218202598000123.  Google Scholar

[33]

V. Giovangigli and M. Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry,, Math. Meth. Appl. Sci., 27 (2004), 739.  doi: 10.1002/mma.429.  Google Scholar

[34]

V. Giovangigli and L. Matuszewski, Supercritical fluid thermodynamics from equations of state,, Phys. D, 241 (2012), 649.  doi: 10.1016/j.physd.2011.12.002.  Google Scholar

[35]

V. Giovangigli and L. Matuszewski, Mathematical modeling of supercritical multicomponent reactive fluids,, to appear, (2012).   Google Scholar

[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.   Google Scholar

[37]

V. Giovangigli and B. Tran, Mathematical analysis of a Saint-Venant model with variable temperature,, Math. Mod. Meth. Appl. Sci., 20 (2010), 1251.  doi: 10.1142/S0218202510004593.  Google Scholar

[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.  doi: 10.1080/00036819608840428.  Google Scholar

[39]

S. Godunov, An interesting class of quasilinear systems,, Sov. Math. Dokl., 2 (1961), 947.   Google Scholar

[40]

E. A. Guggenheim, "Thermodynamics,", North Holland, (1962).   Google Scholar

[41]

J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, "Molecular Theory of Gases and Liquids,", Wiley, (1954).   Google Scholar

[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.  doi: 10.1016/0045-7825(86)90127-1.  Google Scholar

[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.   Google Scholar

[44]

A. Jüengel and I. Stelzer, Existence analysis of Maxwell-Stefan systems for multicomponent mixtures,, , (2012).   Google Scholar

[45]

S. Kawashima, "Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics,", Doctoral Thesis, (1984).   Google Scholar

[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.  doi: 10.1017/S0308210500018308.  Google Scholar

[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.  doi: 10.2748/tmj/1178227986.  Google Scholar

[48]

S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of conservation laws,, Arch. Rat. Mech. Anal., 174 (2004), 345.  doi: 10.1007/s00205-004-0330-9.  Google Scholar

[49]

J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes,", Springer-Verlag, (1987).   Google Scholar

[50]

F. J. Krambeck, The mathematical structure of chemical kinetics in homogeneous single-phase systems,, Arch. Rational Mech. Anal., 38 (1970), 317.  doi: 10.1007/BF00251527.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[53]

M. R. Marcelin, "Sur la Mécanique des Phénomènes Irréversibles,", Comptes Rendus de l'Académie des Sciences de Paris, (1910), 1052.   Google Scholar

[54]

J. Meixner, Zur Thermodynamik der irreversiblen Prozesse in Gasen mit chemisch reagierenden, dissoziierenden und anregbaren Komponenten,, Ann. der Phys., 43 (1943), 244.   Google Scholar

[55]

H. Mori, Statistical-mechanical theory of transport in fluids,, Phys. Rev., 112 (1958), 1829.   Google Scholar

[56]

I. Prigogine, "Etude Thermodynamique des Phénomènes Irréversibles,", Dunod, (1947).   Google Scholar

[57]

J. Pousin, "Modélisation et Analyse Numérique de Couches Limites Réactives d'Air,", Doctorat es Sciences, 1112 (1993).   Google Scholar

[58]

T. Ruggeri, Thermodynamics and Symmetric Hyperbolic Systems. Nonlinear hyperbolic equations in applied sciences,, Rend. Sem. Mat. Univ. Politec. Torino, 1988 (): 167.   Google Scholar

[59]

N. Z. Shapiro and L. S. Shapley, Mass action law and the Gibbs free energy function,, SIAM J. Appl. Math., 13 (1965), 353.   Google Scholar

[60]

D. Serre, Systèmes de Lois de Conservation. I et II,, Diderot Editeur, (1996).   Google Scholar

[61]

D. Serre, The structure of dissipative viscous system of conservation laws,, Physica D, 239 (2010), 1381.  doi: 10.1016/j.physd.2009.03.014.  Google Scholar

[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.   Google Scholar

[63]

R. Taylor and R. Krishna, "Multicomponent Mass Transfer,", John Wiley, (1993).   Google Scholar

[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.  doi: 10.1007/BF03167068.  Google Scholar

[65]

J. Van de Ree, On the definition of the diffusion coefficients in reacting gases,, Physica, 36 (1967), 118.   Google Scholar

[66]

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