August  2021, 14(4): 639-679. doi: 10.3934/krm.2021018

Inelastic Boltzmann equation driven by a particle thermal bath

Pontificia Universidade Católica do Rio de Janeiro1, Gávea, Rio de Janeiro, 22451-900, Brazil

* Corresponding author: Rafael Sanabria

1Research conducted while affiliated as a PhD. student at PUC-Rio

Received  September 2020 Revised  April 2021 Published  August 2021 Early access  June 2021

We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient $ \alpha\in(0,1) $, under the thermalization induced by a host medium with fixed $ e\in(0,1] $ and a fixed Maxwellian distribution. When the restitution coefficient $ \alpha $ is close to 1 we prove existence and uniqueness of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.

Citation: Rafael Sanabria. Inelastic Boltzmann equation driven by a particle thermal bath. Kinetic & Related Models, 2021, 14 (4) : 639-679. doi: 10.3934/krm.2021018
References:
[1]

R. J. AlonsoV. Bagland and B. Lods, Convergence to self-similarity for ballistic annihilation dynamics, J. Math. Pures Appl., 138 (2020), 88-163.  doi: 10.1016/j.matpur.2019.09.008.  Google Scholar

[2]

R. J. Alonso, I. M. Gamba and M. Tasković, Exponentially-tailed regularity and decay rate to equilibrium for the boltzmann equation, arXiv preprint, arXiv: 1711.06596. Google Scholar

[3]

R. J. Alonso and B. Lods, Uniqueness and regularity of steady states of the boltzmann equation for viscoelastic hard-spheres driven by a thermal bath, Commun. Math. Sci., 11 (2013), 851-906.  doi: 10.4310/CMS.2013.v11.n4.a1.  Google Scholar

[4]

L. Arlotti and B. Lods, Integral representation of the linear Boltzmann operator for granular gas dynamics with applications, J. Stat. Phys., 129 (2007), 517-536.  doi: 10.1007/s10955-007-9402-1.  Google Scholar

[5]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[6]

M. BisiJ. A. Cañizo and B. Lods, Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath, SIAM J. Math. Anal., 43 (2011), 2640-2674.  doi: 10.1137/110837437.  Google Scholar

[7]

M. BisiJ. A. Cañizo and B. Lods, Entropy dissipation estimates for the linear Boltzmann operator, J. Funct. Anal., 269 (2015), 1028-1069.  doi: 10.1016/j.jfa.2015.05.002.  Google Scholar

[8]

M. BisiJ. A. Carrillo and B. Lods, Equilibrium solution to the inelastic Boltzmann equation driven by a particle bath, J. Stat. Phys., 133 (2008), 841-870.  doi: 10.1007/s10955-008-9636-6.  Google Scholar

[9] N. V. Brilliantov and T. Pöschel, Kinetic Theory of Granular Gases, Oxford University Press, 2004.  doi: 10.1093/acprof:oso/9780198530381.001.0001.  Google Scholar
[10]

J. A. Cañizo and B. Lods, Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath, Nonlinearity, 29 (2016), 1687-1715.  doi: 10.1088/0951-7715/29/5/1687.  Google Scholar

[11]

R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[12]

I. M. GambaV. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282.  doi: 10.1007/s00205-009-0250-9.  Google Scholar

[13]

I. M. GambaV. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys., 246 (2004), 503-541.  doi: 10.1007/s00220-004-1051-5.  Google Scholar

[14]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of Non-Symmetric Operators and Exponential $H$-Theorem, Mém. Soc. Math. Fr. (N.S.), 2017. doi: 10.24033/msmf.461.  Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators, vol. 132, Springer Science & Business Media, 2013. Google Scholar

[16]

B. LodsC. Mouhot and G. Toscani, Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models, Kinet. Relat. Models, 1 (2008), 223-248.  doi: 10.3934/krm.2008.1.223.  Google Scholar

[17]

S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres, part Ⅱ: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.  doi: 10.1007/s10955-006-9097-8.  Google Scholar

[18]

S. Mischler and C. Mouhot, Stability, convergence to self-similarity and elastic limit for the boltzmann equation for inelastic hard spheres, Comm. Math. Phys., 288 (2009), 431-502.  doi: 10.1007/s00220-009-0773-9.  Google Scholar

[19]

S. Mischler and C. Mouhot, Stability, convergence to the steady state and elastic limit for the boltzmann equation for diffusively excited granular media, Discrete Contin. Dyn. Syst., 24 (2009), 159-185.  doi: 10.3934/dcds.2009.24.159.  Google Scholar

[20]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.  Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

I. Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Funct. Anal., 270 (2016), 1922-1970.  doi: 10.1016/j.jfa.2015.09.025.  Google Scholar

[23]

C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-74.  doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

show all references

References:
[1]

R. J. AlonsoV. Bagland and B. Lods, Convergence to self-similarity for ballistic annihilation dynamics, J. Math. Pures Appl., 138 (2020), 88-163.  doi: 10.1016/j.matpur.2019.09.008.  Google Scholar

[2]

R. J. Alonso, I. M. Gamba and M. Tasković, Exponentially-tailed regularity and decay rate to equilibrium for the boltzmann equation, arXiv preprint, arXiv: 1711.06596. Google Scholar

[3]

R. J. Alonso and B. Lods, Uniqueness and regularity of steady states of the boltzmann equation for viscoelastic hard-spheres driven by a thermal bath, Commun. Math. Sci., 11 (2013), 851-906.  doi: 10.4310/CMS.2013.v11.n4.a1.  Google Scholar

[4]

L. Arlotti and B. Lods, Integral representation of the linear Boltzmann operator for granular gas dynamics with applications, J. Stat. Phys., 129 (2007), 517-536.  doi: 10.1007/s10955-007-9402-1.  Google Scholar

[5]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[6]

M. BisiJ. A. Cañizo and B. Lods, Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath, SIAM J. Math. Anal., 43 (2011), 2640-2674.  doi: 10.1137/110837437.  Google Scholar

[7]

M. BisiJ. A. Cañizo and B. Lods, Entropy dissipation estimates for the linear Boltzmann operator, J. Funct. Anal., 269 (2015), 1028-1069.  doi: 10.1016/j.jfa.2015.05.002.  Google Scholar

[8]

M. BisiJ. A. Carrillo and B. Lods, Equilibrium solution to the inelastic Boltzmann equation driven by a particle bath, J. Stat. Phys., 133 (2008), 841-870.  doi: 10.1007/s10955-008-9636-6.  Google Scholar

[9] N. V. Brilliantov and T. Pöschel, Kinetic Theory of Granular Gases, Oxford University Press, 2004.  doi: 10.1093/acprof:oso/9780198530381.001.0001.  Google Scholar
[10]

J. A. Cañizo and B. Lods, Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath, Nonlinearity, 29 (2016), 1687-1715.  doi: 10.1088/0951-7715/29/5/1687.  Google Scholar

[11]

R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[12]

I. M. GambaV. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282.  doi: 10.1007/s00205-009-0250-9.  Google Scholar

[13]

I. M. GambaV. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys., 246 (2004), 503-541.  doi: 10.1007/s00220-004-1051-5.  Google Scholar

[14]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of Non-Symmetric Operators and Exponential $H$-Theorem, Mém. Soc. Math. Fr. (N.S.), 2017. doi: 10.24033/msmf.461.  Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators, vol. 132, Springer Science & Business Media, 2013. Google Scholar

[16]

B. LodsC. Mouhot and G. Toscani, Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models, Kinet. Relat. Models, 1 (2008), 223-248.  doi: 10.3934/krm.2008.1.223.  Google Scholar

[17]

S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres, part Ⅱ: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.  doi: 10.1007/s10955-006-9097-8.  Google Scholar

[18]

S. Mischler and C. Mouhot, Stability, convergence to self-similarity and elastic limit for the boltzmann equation for inelastic hard spheres, Comm. Math. Phys., 288 (2009), 431-502.  doi: 10.1007/s00220-009-0773-9.  Google Scholar

[19]

S. Mischler and C. Mouhot, Stability, convergence to the steady state and elastic limit for the boltzmann equation for diffusively excited granular media, Discrete Contin. Dyn. Syst., 24 (2009), 159-185.  doi: 10.3934/dcds.2009.24.159.  Google Scholar

[20]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.  Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

I. Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, J. Funct. Anal., 270 (2016), 1922-1970.  doi: 10.1016/j.jfa.2015.09.025.  Google Scholar

[23]

C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-74.  doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

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