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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

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

[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.

[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.

[39]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

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

[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.

[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.

[39]

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

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