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December  2016, 36(12): 6669-6688. doi: 10.3934/dcds.2016090

Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings

Marc Briant 1,

1. 

Sorbonne Universités, UPMC Univ. Paris 06/ CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  March 2016 Revised  July 2016 Published  October 2016

We prove the stability of global equilibrium in a multi-species mixture, where the different species can have different masses, on the $3$-dimensional torus. We establish stability estimates in $L^\infty_{x,v}(w)$ where $w=w(v)$ is either polynomial or exponential, with explicit threshold. Along the way we extend recent estimates and stability results for the mono-species Boltzmann operator not only to the multi-species case but also to more general hard potential and Maxwellian kernels.
Keywords: Boltzmann equation, perturbative theory, Multi-species mixture, exponential trend to equilibrium., stability in $L^\infty$.
Mathematics Subject Classification: Primary: 35B40, 35Q20; Secondary: 76P05, 82C4.
Citation: Marc Briant. Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6669-6688. doi: 10.3934/dcds.2016090
References:
[1]

C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials, Rev. Mat. Iberoamericana, 21 (2005), 819-841, URL http://projecteuclid.org/getRecord?id=euclid.rmi/1136999132. doi: 10.4171/RMI/436.  Google Scholar

[2]

A. V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres, Eur. J. Mech. B Fluids, 18 (1999), 869-887. doi: 10.1016/S0997-7546(99)00121-1.  Google Scholar

[3]

A. Bobylev, I. Gamba and V. Panferov, Moment inequalities and high-energy tails for boltzmann equations with inelastic interactions, Journal of Statistical Physics, 116 (2004), 1651-1682. doi: 10.1023/B:JOSS.0000041751.11664.ea.  Google Scholar

[4]

L. Boudin, B. Grec, M. Pavić and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinetic and Related Models, 6 (2013), 137-157, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=8004. doi: 10.3934/krm.2013.6.137.  Google Scholar

[5]

L. Boudin, B. Grec and F. Salvarani, The maxwell-stefan diffusion limit for a kinetic model of mixtures, Acta Applicandae Mathematicae, 136 (2015), 79-90. doi: 10.1007/s10440-014-9886-z.  Google Scholar

[6]

M. Briant, Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions, Preprint 2015. Google Scholar

[7]

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

[8]

M. Briant and Y. Guo, Asymptotic stability of the boltzmann equation with maxwell boundary conditions, Preprint 2016. Google Scholar

[9]

C. Cercignani, The Boltzmann Equation and Its Applications, vol. 67 of Applied Mathematical Sciences, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[10]

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

[11]

E. S. Daus, A. Jüngel, C. 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

[12]

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, Eur. J. Mech. B Fluids, 24 (2005), 219-236. doi: 10.1016/j.euromechflu.2004.07.004.  Google Scholar

[13]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.  Google Scholar

[14]

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, 1958, 205-294.  Google Scholar

[15]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, arXiv:1006.5523, Preprint 2013. Google Scholar

[16]

O. E. Lanford III, Time evolution of large classical systems, in Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), Springer, Berlin, Lecture Notes in Phys., 38 (1975), 1-111.  Google Scholar

[17]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348. doi: 10.1080/03605300600635004.  Google Scholar

[18]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64pp. doi: 10.1142/S0129055X14500019.  Google Scholar

[19]

S. Ukai and T. Yang, Mathematical Theory of the Boltzmann Equation, 2006,, Lecture Notes Series, ().   Google Scholar

[20]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics, North-Holland, Amsterdam, 1 (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

show all references

References:
[1]

C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials, Rev. Mat. Iberoamericana, 21 (2005), 819-841, URL http://projecteuclid.org/getRecord?id=euclid.rmi/1136999132. doi: 10.4171/RMI/436.  Google Scholar

[2]

A. V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres, Eur. J. Mech. B Fluids, 18 (1999), 869-887. doi: 10.1016/S0997-7546(99)00121-1.  Google Scholar

[3]

A. Bobylev, I. Gamba and V. Panferov, Moment inequalities and high-energy tails for boltzmann equations with inelastic interactions, Journal of Statistical Physics, 116 (2004), 1651-1682. doi: 10.1023/B:JOSS.0000041751.11664.ea.  Google Scholar

[4]

L. Boudin, B. Grec, M. Pavić and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinetic and Related Models, 6 (2013), 137-157, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=8004. doi: 10.3934/krm.2013.6.137.  Google Scholar

[5]

L. Boudin, B. Grec and F. Salvarani, The maxwell-stefan diffusion limit for a kinetic model of mixtures, Acta Applicandae Mathematicae, 136 (2015), 79-90. doi: 10.1007/s10440-014-9886-z.  Google Scholar

[6]

M. Briant, Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions, Preprint 2015. Google Scholar

[7]

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

[8]

M. Briant and Y. Guo, Asymptotic stability of the boltzmann equation with maxwell boundary conditions, Preprint 2016. Google Scholar

[9]

C. Cercignani, The Boltzmann Equation and Its Applications, vol. 67 of Applied Mathematical Sciences, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[10]

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

[11]

E. S. Daus, A. Jüngel, C. 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

[12]

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, Eur. J. Mech. B Fluids, 24 (2005), 219-236. doi: 10.1016/j.euromechflu.2004.07.004.  Google Scholar

[13]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.  Google Scholar

[14]

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, 1958, 205-294.  Google Scholar

[15]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, arXiv:1006.5523, Preprint 2013. Google Scholar

[16]

O. E. Lanford III, Time evolution of large classical systems, in Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), Springer, Berlin, Lecture Notes in Phys., 38 (1975), 1-111.  Google Scholar

[17]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348. doi: 10.1080/03605300600635004.  Google Scholar

[18]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64pp. doi: 10.1142/S0129055X14500019.  Google Scholar

[19]

S. Ukai and T. Yang, Mathematical Theory of the Boltzmann Equation, 2006,, Lecture Notes Series, ().   Google Scholar

[20]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics, North-Holland, Amsterdam, 1 (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

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