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May  2020, 19(5): 2549-2573. doi: 10.3934/cpaa.2020112

Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions

1. 

Université de Paris, Université Paris Descartes, Laboratoire MAP5, CNRS UMR 8145, F-75006 Paris, FRANCE

2. 

Sorbonne Université, Université Paris–Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, Équipe Reo, F-75005 Paris, FRANCE

*Corresponding author

Received  November 2018 Revised  August 2019 Published  March 2020

Fund Project: This work was partially funded by the French ANR-13-BS01-0004 project Kibord headed by L. Desvillettes. The first and fourth authors have been partially funded by Université Sorbonne Paris Cité, in the framework of the "Investissements d'Avenir", convention ANR-11-IDEX-0005

We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hard potential, or hard-sphere collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species operator $ \mathbf{L} $ is known to possess an explicit spectral gap $ \lambda_{ \mathbf{L}} $, in the global equilibrium weighted $ L^2 $ space. We study a new operator $ \mathbf{ L^{\varepsilon}} $ obtained by linearizing the Boltzmann operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order $ \varepsilon $, $ \varepsilon >0 $. This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of $ \mathbf{ L^{\varepsilon}} $ in the global equilibrium weighted $ L^2 $ space. More precisely, we consider the explicit upper bound that has been proved for the entropy production functional associated to $ \mathbf{L} $ and we show that the same estimate holds for the entropy production functional associated to $ \mathbf{ L^{\varepsilon}} $, up to a correction of order $ \varepsilon $.

Citation: Andrea Bondesan, Laurent Boudin, Marc Briant, Bérénice Grec. Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2549-2573. doi: 10.3934/cpaa.2020112
References:
[1]

P. AndriesK. Aoki and B. Perthame, A consistent BGK-type model for gas mixtures, J. Statist. Phys., 106 (2002), 993-1018.  doi: 10.1023/A:1014033703134.

[2]

C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials, Rev. Mat. Iberoam., 21 (2005), 819-841.  doi: 10.4171/RMI/436.

[3]

A. V. Bobylev, The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044. 

[4]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in Soviet Scientific Reviews C, Mathematical Physics Reviews, Vol. 7, Harwood Academic Publ., Chur, (1988), 111–233.

[5]

L. BoudinB. Grec and V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Anal., 159 (2017), 40-61.  doi: 10.1016/j.na.2017.01.010.

[6]

L. BoudinB. GrecM. Pavić and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinet. Relat. Models, 6 (2013), 137-157.  doi: 10.3934/krm.2013.6.137.

[7]

L. BoudinB. Grec and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1427-1440.  doi: 10.3934/dcdsb.2012.17.1427.

[8]

L. BoudinB. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, Acta Appl. Math., 136 (2015), 79-90.  doi: 10.1007/s10440-014-9886-z.

[9]

J. F. BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Eur. J. Mech. B Fluids, 13 (1994), 237-254. 

[10]

M. Briant, Stability of global equilibrium for the multi-species {B}oltzmann equation in {$L^\infty$} settings, Discrete Contin. Dyn. Syst., 36 (2016), 6669-6688.  doi: 10.3934/dcds.2016090.

[11]

M. Briant and E. S. Daus, The Boltzmann equation for a multi-species mixture close to global equilibrium, Arch. Ration. Mech. Anal., 222 (2016), 1367-1443.  doi: 10.1007/s00205-016-1023-x.

[12]

S. BrullV. Pavan and J. Schneider, Derivation of a BGK model for mixtures, Eur. J. Mech. B Fluids, 33 (2012), 74-86.  doi: 10.1016/j.euromechflu.2011.12.003.

[13]

T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler., Vol. 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957.

[14] C. Cercignani, Mathematical Methods in Kinetic Theory, 2, edition, Plenum Press, New York, 1990.  doi: 10.1007/978-1-4899-7291-0.
[15]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, Vol. 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[16]

E. S. DausA. JüngelC. Mouhot and N. Zamponi, Hypocoercivity for a linearized multispecies Boltzmann system, SIAM J. Math. Anal., 48 (2016), 538-568.  doi: 10.1137/15M1017934.

[17]

L. DesvillettesR. 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.  doi: 10.1016/j.euromechflu.2004.07.004.

[18]

V. GarzóA. Santos and J. J. Brey, A kinetic model for a multicomponent gas, Phys. Fluids A, 1 (1989), 380-383.  doi: 10.1063/1.857458.

[19]

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.

[20]

H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205–294.

[21]

H. Grad, Asymptotic theory of the Boltzmann equation. II, in Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), I, Academic Press, New York, (1963), 26–59.

[22]

D. E. Greene, Mathematical aspects of kinetic model equations for binary gas mixtures, J. Math. Phys., 16 (1975), 776-782.  doi: 10.1063/1.522631.

[23]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.

[24]

D. Hilbert, Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen, Chelsea Publishing Company, New York, N.Y., 1953.

[25]

T. F. Morse, Kinetic model equations for a gas mixture, Phys. Fluids, 7 (1964), 2012-2013.  doi: 10.1063/1.1711112.

[26]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Commun. Partial Differ. Equ., 31 (2006), 1321-1348.  doi: 10.1080/03605300600635004.

[27]

C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Commun. Math. Phys., 261 (2006), 629-672.  doi: 10.1007/s00220-005-1455-x.

[28]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.

[29]

C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math. Pures Appl., 87 (2007), 515-535.  doi: 10.1016/j.matpur.2007.03.003.

[30]

J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28. 

[31]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.

[32]

L. Sirovich, Kinetic modeling of gas mixtures, Phys. Fluids, 5 (1962), 908-918.  doi: 10.1063/1.1706706.

[33]

S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes Ser., Vol. 8, (eds. Liu Bie Ju), Center of Mathematical Sciences, City University of Hong Kong, Hong Kong.

[34]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0.

show all references

References:
[1]

P. AndriesK. Aoki and B. Perthame, A consistent BGK-type model for gas mixtures, J. Statist. Phys., 106 (2002), 993-1018.  doi: 10.1023/A:1014033703134.

[2]

C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials, Rev. Mat. Iberoam., 21 (2005), 819-841.  doi: 10.4171/RMI/436.

[3]

A. V. Bobylev, The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules, Dokl. Akad. Nauk SSSR, 225 (1975), 1041-1044. 

[4]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in Soviet Scientific Reviews C, Mathematical Physics Reviews, Vol. 7, Harwood Academic Publ., Chur, (1988), 111–233.

[5]

L. BoudinB. Grec and V. Pavan, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections, Nonlinear Anal., 159 (2017), 40-61.  doi: 10.1016/j.na.2017.01.010.

[6]

L. BoudinB. GrecM. Pavić and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinet. Relat. Models, 6 (2013), 137-157.  doi: 10.3934/krm.2013.6.137.

[7]

L. BoudinB. Grec and F. Salvarani, A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1427-1440.  doi: 10.3934/dcdsb.2012.17.1427.

[8]

L. BoudinB. Grec and F. Salvarani, The Maxwell-Stefan diffusion limit for a kinetic model of mixtures, Acta Appl. Math., 136 (2015), 79-90.  doi: 10.1007/s10440-014-9886-z.

[9]

J. F. BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Eur. J. Mech. B Fluids, 13 (1994), 237-254. 

[10]

M. Briant, Stability of global equilibrium for the multi-species {B}oltzmann equation in {$L^\infty$} settings, Discrete Contin. Dyn. Syst., 36 (2016), 6669-6688.  doi: 10.3934/dcds.2016090.

[11]

M. Briant and E. S. Daus, The Boltzmann equation for a multi-species mixture close to global equilibrium, Arch. Ration. Mech. Anal., 222 (2016), 1367-1443.  doi: 10.1007/s00205-016-1023-x.

[12]

S. BrullV. Pavan and J. Schneider, Derivation of a BGK model for mixtures, Eur. J. Mech. B Fluids, 33 (2012), 74-86.  doi: 10.1016/j.euromechflu.2011.12.003.

[13]

T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz, Publ. Sci. Inst. Mittag-Leffler., Vol. 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957.

[14] C. Cercignani, Mathematical Methods in Kinetic Theory, 2, edition, Plenum Press, New York, 1990.  doi: 10.1007/978-1-4899-7291-0.
[15]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, Vol. 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[16]

E. S. DausA. JüngelC. Mouhot and N. Zamponi, Hypocoercivity for a linearized multispecies Boltzmann system, SIAM J. Math. Anal., 48 (2016), 538-568.  doi: 10.1137/15M1017934.

[17]

L. DesvillettesR. 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.  doi: 10.1016/j.euromechflu.2004.07.004.

[18]

V. GarzóA. Santos and J. J. Brey, A kinetic model for a multicomponent gas, Phys. Fluids A, 1 (1989), 380-383.  doi: 10.1063/1.857458.

[19]

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.

[20]

H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205–294.

[21]

H. Grad, Asymptotic theory of the Boltzmann equation. II, in Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), I, Academic Press, New York, (1963), 26–59.

[22]

D. E. Greene, Mathematical aspects of kinetic model equations for binary gas mixtures, J. Math. Phys., 16 (1975), 776-782.  doi: 10.1063/1.522631.

[23]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.

[24]

D. Hilbert, Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen, Chelsea Publishing Company, New York, N.Y., 1953.

[25]

T. F. Morse, Kinetic model equations for a gas mixture, Phys. Fluids, 7 (1964), 2012-2013.  doi: 10.1063/1.1711112.

[26]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Commun. Partial Differ. Equ., 31 (2006), 1321-1348.  doi: 10.1080/03605300600635004.

[27]

C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Commun. Math. Phys., 261 (2006), 629-672.  doi: 10.1007/s00220-005-1455-x.

[28]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.

[29]

C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math. Pures Appl., 87 (2007), 515-535.  doi: 10.1016/j.matpur.2007.03.003.

[30]

J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28. 

[31]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.

[32]

L. Sirovich, Kinetic modeling of gas mixtures, Phys. Fluids, 5 (1962), 908-918.  doi: 10.1063/1.1706706.

[33]

S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes Ser., Vol. 8, (eds. Liu Bie Ju), Center of Mathematical Sciences, City University of Hong Kong, Hong Kong.

[34]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0.

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