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February  2020, 13(1): 97-128. doi: 10.3934/krm.2020004

Hypocoercivity of linear kinetic equations via Harris's Theorem

1. 

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

2. 

CEREMADE, Université Paris Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris, France

3. 

BCAM Basque Center for Applied Mathematics, Alameda Mazarredo, 14, 48009 Bilbao, Spain

*Corresponding author: Chuqi Cao

The authors would like to thank S. Mischler, C. Mouhot, J. Féjoz and R. Ortega for some useful discussion and ideas for some parts in the paper

Received  February 2019 Revised  August 2019 Published  December 2019

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $ (x,v) \in \mathbb{T}^d \times \mathbb{R}^d $ or on the whole space $ (x,v) \in \mathbb{R}^d \times \mathbb{R}^d $ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $ L^1 $ or weighted $ L^1 $ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem.

Citation: José A. Cañizo, Chuqi Cao, Josephine Evans, Havva Yoldaş. Hypocoercivity of linear kinetic equations via Harris's Theorem. Kinetic & Related Models, 2020, 13 (1) : 97-128. doi: 10.3934/krm.2020004
References:
[1]

D. BakryP. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759.  doi: 10.1016/j.jfa.2007.11.002.  Google Scholar

[2]

V. BansayeB. Cloez and P. Gabriel, Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions, Acta Applicandae Mathematicae, (2017), 1-44.  doi: 10.1007/s10440-019-00253-5.  Google Scholar

[3]

A. BensousanJ. L. Lions and G. C. Papanicolau, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.  doi: 10.2977/prims/1195188427.  Google Scholar

[4]

P. G. Bergman and J. L. Lebowitz, New approach to nonequilibrium processes, Phys. Rev. (2), 99 (1955), 578-587.  doi: 10.1103/PhysRev.99.578.  Google Scholar

[5]

E. Bernard and F. Salvarani, On the exponential decay to equilibrium of the degenerate linear Boltzmann equation, J. Funct. Anal., 265 (2013), 1934-1954.  doi: 10.1016/j.jfa.2013.06.012.  Google Scholar

[6]

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

[7]

M. Briant, Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, Kinet. Relat. Models, 8 (2015), 281-308.  doi: 10.3934/krm.2015.8.281.  Google Scholar

[8]

M. Briant, Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains, Arch. Ration. Mech. Anal., 218 (2015), 985-1041.  doi: 10.1007/s00205-015-0874-x.  Google Scholar

[9]

J. A. CañizoJ. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.  Google Scholar

[10]

J. A. CañizoA. Einav and B. Lods, On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials, J. Math. Anal. Appl., 462 (2018), 801-839.  doi: 10.1016/j.jmaa.2017.12.052.  Google Scholar

[11]

M. J. CáceresJ. A. Carrillo and T. Goudon, Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles, Comm. Partial Differential Equations, 28 (2003), 969-989.  doi: 10.1081/PDE-120021182.  Google Scholar

[12]

J. A. Cañizo and H. Yoldaş, Asymptotic behaviour of neuron population models structured by elapsed-time, Nonlinearity, 32 (2019), 464-495.  doi: 10.1088/1361-6544/aaea9c.  Google Scholar

[13]

C. Cao, The kinetic Fokker-Planck equation with weak confinement force, preprint, arXiv: 1801.10354. Google Scholar

[14]

E. A. CarlenR. EspositoJ. L. LebowitzR. Marra and C. Mouhot, Approach to the steady state in kinetic models with thermal reservoirs at different temperatures, J. Stat. Phys., 172 (2018), 522-543.  doi: 10.1007/s10955-018-2074-1.  Google Scholar

[15]

P. Cattiaux and A. Guillin, Functional inequalities via Lyapunov conditions, in Optimal Transportation (London Math. Soc. Lecture Note Ser.), Cambridge Univ. Press, Cambridge, 413 (2014), 274–287.  Google Scholar

[16]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[17]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[18]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[19]

R. DoucG. Fort and A. Guillin, Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Process. Appl., 119 (2009), 897-923.  doi: 10.1016/j.spa.2008.03.007.  Google Scholar

[20]

G. Dumont and P. Gabriel, The mean-field equation of a leaky integrate- and-fire neural network: Measure solutions and steady states, arXiv: 1710.05596. Google Scholar

[21]

W. E and D. Li, The Andersen thermostat in molecular dynamics, Comm. Pure Appl. Math., 61 (2008), 96-136.  doi: 10.1002/cpa.20198.  Google Scholar

[22]

J. Evans, Hypocoercivity in Phi-entropy for the linear relaxation Boltzmann equation on the torus, arXiv: 1702.04168. Google Scholar

[23]

N. Fournier and S. Méléard, A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules, J. Statist. Phys., 104 (2001), 359-385.  doi: 10.1023/A:1010322130480.  Google Scholar

[24]

P. Gabriel, Measure solutions to the conservative renewal equation, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 62 (2018), 68–78. doi: 10.1051/proc/201862186206.  Google Scholar

[25]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp.  Google Scholar

[26]

M. Hairer, Lecture notes: Convergence of Markov processes, 2016. Google Scholar

[27]

M. Hairer and J C. Mattingly, Yet another look at Harrisergodic theorem for Markov chains, in Progr. Probab., (eds. Seminar on Stochastic Analysis, Random Fields and Applications VI), Birkhäuser/Springer, Basel, 63 (2011), 109–117. doi: 10.1007/978-3-0348-0021-1_7.  Google Scholar

[28]

T. E. Harris, The existence of stationary measures for certain Markov processes, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, (1956), 113–124.  Google Scholar

[29]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.  doi: 10.1007/s00205-003-0276-3.  Google Scholar

[30]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.   Google Scholar

[31]

J. L. Lebowitz and P. G. Bergmann, Irreversible Gibbsian ensembles, Ann. Physics, 1 (1957), 1-23.  doi: 10.1016/0003-4916(57)90002-7.  Google Scholar

[32]

B. Lods and M. Mokhtar-Kharroubi, Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in $L^1$-spaces, Math. Methods Appl. Sci., 40 (2017), 6527-6555.  doi: 10.1002/mma.4473.  Google Scholar

[33]

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

[34]

J. C. MattinglyA. M. Stuart and D. J. Higham, Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl., 101 (2002), 185-232.  doi: 10.1016/S0304-4149(02)00150-3.  Google Scholar

[35] S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511626630.  Google Scholar
[36]

M. Mokhtar-Kharroubi, On L1 exponential trend to equilibrium for conservative linear kinetic equations on the torus, J. Funct. Anal., 226 (2014), 6418-6455.  doi: 10.1016/j.jfa.2014.03.019.  Google Scholar

[37]

C. Mouhot, Quantitative lower bounds for the full Boltzmann equation. Ⅰ. Periodic boundary conditions, Comm. Partial Differential Equations, 30 (2005), 881-917.  doi: 10.1081/PDE-200059299.  Google Scholar

[38]

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

[39]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

show all references

References:
[1]

D. BakryP. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759.  doi: 10.1016/j.jfa.2007.11.002.  Google Scholar

[2]

V. BansayeB. Cloez and P. Gabriel, Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions, Acta Applicandae Mathematicae, (2017), 1-44.  doi: 10.1007/s10440-019-00253-5.  Google Scholar

[3]

A. BensousanJ. L. Lions and G. C. Papanicolau, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.  doi: 10.2977/prims/1195188427.  Google Scholar

[4]

P. G. Bergman and J. L. Lebowitz, New approach to nonequilibrium processes, Phys. Rev. (2), 99 (1955), 578-587.  doi: 10.1103/PhysRev.99.578.  Google Scholar

[5]

E. Bernard and F. Salvarani, On the exponential decay to equilibrium of the degenerate linear Boltzmann equation, J. Funct. Anal., 265 (2013), 1934-1954.  doi: 10.1016/j.jfa.2013.06.012.  Google Scholar

[6]

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

[7]

M. Briant, Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions, Kinet. Relat. Models, 8 (2015), 281-308.  doi: 10.3934/krm.2015.8.281.  Google Scholar

[8]

M. Briant, Instantaneous filling of the vacuum for the full Boltzmann equation in convex domains, Arch. Ration. Mech. Anal., 218 (2015), 985-1041.  doi: 10.1007/s00205-015-0874-x.  Google Scholar

[9]

J. A. CañizoJ. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, 123 (2013), 141-156.  doi: 10.1007/s10440-012-9758-3.  Google Scholar

[10]

J. A. CañizoA. Einav and B. Lods, On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials, J. Math. Anal. Appl., 462 (2018), 801-839.  doi: 10.1016/j.jmaa.2017.12.052.  Google Scholar

[11]

M. J. CáceresJ. A. Carrillo and T. Goudon, Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles, Comm. Partial Differential Equations, 28 (2003), 969-989.  doi: 10.1081/PDE-120021182.  Google Scholar

[12]

J. A. Cañizo and H. Yoldaş, Asymptotic behaviour of neuron population models structured by elapsed-time, Nonlinearity, 32 (2019), 464-495.  doi: 10.1088/1361-6544/aaea9c.  Google Scholar

[13]

C. Cao, The kinetic Fokker-Planck equation with weak confinement force, preprint, arXiv: 1801.10354. Google Scholar

[14]

E. A. CarlenR. EspositoJ. L. LebowitzR. Marra and C. Mouhot, Approach to the steady state in kinetic models with thermal reservoirs at different temperatures, J. Stat. Phys., 172 (2018), 522-543.  doi: 10.1007/s10955-018-2074-1.  Google Scholar

[15]

P. Cattiaux and A. Guillin, Functional inequalities via Lyapunov conditions, in Optimal Transportation (London Math. Soc. Lecture Note Ser.), Cambridge Univ. Press, Cambridge, 413 (2014), 274–287.  Google Scholar

[16]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[17]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[18]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[19]

R. DoucG. Fort and A. Guillin, Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Process. Appl., 119 (2009), 897-923.  doi: 10.1016/j.spa.2008.03.007.  Google Scholar

[20]

G. Dumont and P. Gabriel, The mean-field equation of a leaky integrate- and-fire neural network: Measure solutions and steady states, arXiv: 1710.05596. Google Scholar

[21]

W. E and D. Li, The Andersen thermostat in molecular dynamics, Comm. Pure Appl. Math., 61 (2008), 96-136.  doi: 10.1002/cpa.20198.  Google Scholar

[22]

J. Evans, Hypocoercivity in Phi-entropy for the linear relaxation Boltzmann equation on the torus, arXiv: 1702.04168. Google Scholar

[23]

N. Fournier and S. Méléard, A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules, J. Statist. Phys., 104 (2001), 359-385.  doi: 10.1023/A:1010322130480.  Google Scholar

[24]

P. Gabriel, Measure solutions to the conservative renewal equation, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 62 (2018), 68–78. doi: 10.1051/proc/201862186206.  Google Scholar

[25]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp.  Google Scholar

[26]

M. Hairer, Lecture notes: Convergence of Markov processes, 2016. Google Scholar

[27]

M. Hairer and J C. Mattingly, Yet another look at Harrisergodic theorem for Markov chains, in Progr. Probab., (eds. Seminar on Stochastic Analysis, Random Fields and Applications VI), Birkhäuser/Springer, Basel, 63 (2011), 109–117. doi: 10.1007/978-3-0348-0021-1_7.  Google Scholar

[28]

T. E. Harris, The existence of stationary measures for certain Markov processes, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, (1956), 113–124.  Google Scholar

[29]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.  doi: 10.1007/s00205-003-0276-3.  Google Scholar

[30]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.   Google Scholar

[31]

J. L. Lebowitz and P. G. Bergmann, Irreversible Gibbsian ensembles, Ann. Physics, 1 (1957), 1-23.  doi: 10.1016/0003-4916(57)90002-7.  Google Scholar

[32]

B. Lods and M. Mokhtar-Kharroubi, Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in $L^1$-spaces, Math. Methods Appl. Sci., 40 (2017), 6527-6555.  doi: 10.1002/mma.4473.  Google Scholar

[33]

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

[34]

J. C. MattinglyA. M. Stuart and D. J. Higham, Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl., 101 (2002), 185-232.  doi: 10.1016/S0304-4149(02)00150-3.  Google Scholar

[35] S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511626630.  Google Scholar
[36]

M. Mokhtar-Kharroubi, On L1 exponential trend to equilibrium for conservative linear kinetic equations on the torus, J. Funct. Anal., 226 (2014), 6418-6455.  doi: 10.1016/j.jfa.2014.03.019.  Google Scholar

[37]

C. Mouhot, Quantitative lower bounds for the full Boltzmann equation. Ⅰ. Periodic boundary conditions, Comm. Partial Differential Equations, 30 (2005), 881-917.  doi: 10.1081/PDE-200059299.  Google Scholar

[38]

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

[39]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

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