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

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.
Citation: Marc Briant. Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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. 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., (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. 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., (2016). Google Scholar

[9]

C. Cercignani, The Boltzmann Equation and Its Applications, vol. 67 of Applied Mathematical Sciences,, Springer-Verlag, (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, (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. 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. 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, (2013). Google Scholar

[14]

H. Grad, Principles of the kinetic theory of gases,, in Handbuch der Physik (herausgegeben von S. Flügge), (1958), 205. Google Scholar

[15]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem,, , (2013). Google Scholar

[16]

O. E. Lanford III, Time evolution of large classical systems,, in Dynamical systems, 38 (1975), 1. Google Scholar

[17]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators,, Comm. Partial Differential Equations, 31 (2006), 1321. 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). 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, 1 (2002), 71. 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. 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. 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. 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. 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. 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., (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. 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., (2016). Google Scholar

[9]

C. Cercignani, The Boltzmann Equation and Its Applications, vol. 67 of Applied Mathematical Sciences,, Springer-Verlag, (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, (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. 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. 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, (2013). Google Scholar

[14]

H. Grad, Principles of the kinetic theory of gases,, in Handbuch der Physik (herausgegeben von S. Flügge), (1958), 205. Google Scholar

[15]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem,, , (2013). Google Scholar

[16]

O. E. Lanford III, Time evolution of large classical systems,, in Dynamical systems, 38 (1975), 1. Google Scholar

[17]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators,, Comm. Partial Differential Equations, 31 (2006), 1321. 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). 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, 1 (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

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