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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 & 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[13]

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[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. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[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. Google Scholar

[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.  Google Scholar

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