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March  2016, 9(1): 1-49. doi: 10.3934/krm.2016.9.1

Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules

Kleber Carrapatoso 1,

1. 

CEREMADE, Université Paris-Dauphine, UMR CNRS 7534, F-75775 Paris, France

Received  April 2014 Revised  September 2015 Published  October 2015

We prove a quantitative propagation of chaos and entropic chaos, uniformly in time, for the spatially homogeneous Landau equation in the case of Maxwellian molecules. We improve the results of Fontbona, Guérin and Méléard [9] and Fournier [10] where the propagation of chaos is proved for finite time. Moreover, we prove a quantitative estimate on the rate of convergence to equilibrium uniformly in the number of particles.
Keywords: Landau equation, trend to equilibrium., entropic chaos, Maxwellian molecules, grazing collisions, Chaos, propagation of chaos.
Mathematics Subject Classification: Primary: 82C40, 76P05, 60K35, 54C7.
Citation: Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic & Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1
References:
[1]

A. A. Arsen'ev and O. E. Buryak, On the connection betwenn a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik, 69 (1991), 465-478.  Google Scholar

[2]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in Mathematical physics reviews, Vol. 7, Soviet Sci. Rev. Sect. C Math. Phys. Rev., Harwood Academic Publ., Chur, 7 (1988), 111-233.  Google Scholar

[3]

A. V. Boblylev, M. Pulvirenti and C. Saffirio, From particle systems to the Landau equation: A consistency result, Comm. Math. Phys., 319 (2013), 683-702. doi: 10.1007/s00220-012-1633-6.  Google Scholar

[4]

E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122. doi: 10.3934/krm.2010.3.85.  Google Scholar

[5]

K. Carrapatoso, Quantitative and qualitative Kac's chaos on the Boltzmann's sphere, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 993-1039. doi: 10.1214/14-AIHP612.  Google Scholar

[6]

P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Mod. Meth. in Appl. Sci., 2 (1992), 167-182. doi: 10.1142/S0218202592000119.  Google Scholar

[7]

L. Desvillettes, On the asymptotics of the Boltzmann equation when the collisions become grazing, Transp. Theory and Stat. Phys., 21 (1992), 259-276. doi: 10.1080/00411459208203923.  Google Scholar

[8]

A. Einav, A counter-example to Cercignani's conjecture for the d-dimensional Kac model, J. Statist. Phys., 148 (2012), 1076-1103. doi: 10.1007/s10955-012-0565-z.  Google Scholar

[9]

J. Fontbona, H. Guérin and S. Méléard, Measurability of optimal transportation and convergence rate for Landau type interacting particle systems, Probab. Theory Related Fields, 143 (2009), 329-351. doi: 10.1007/s00440-007-0128-4.  Google Scholar

[10]

N. Fournier, Particle approximation of some Landau equations, Kinet. Relat. Models, 2 (2009), 451-464. doi: 10.3934/krm.2009.2.451.  Google Scholar

[11]

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

[12]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.  Google Scholar

[13]

H. Guérin, Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach, Stochastic Process. Appl., 101 (2002), 303-325. doi: 10.1016/S0304-4149(02)00107-2.  Google Scholar

[14]

H. Guérin, Solving Landau equation for some soft potentials through a probabilistic approach, Ann. Appl. Probab., 13 (2003), 515-539. doi: 10.1214/aoap/1050689592.  Google Scholar

[15]

M. Hauray and S. Mischler, On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157. doi: 10.1016/j.jfa.2014.02.030.  Google Scholar

[16]

R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum, Comm. Math. Phys., 105 (1986), 189-203. doi: 10.1007/BF01211098.  Google Scholar

[17]

M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III, University of California Press, Berkeley and Los Angeles, 1956, 171-197.  Google Scholar

[18]

M. Kiessling and C. Lancellotti, On the master equation approach to kinetic theory: Linear and nonlinear Fokker-Planck equations, Transport Theory and Statistical Physics, 33 (2004), 379-401. doi: 10.1081/TT-200053929.  Google Scholar

[19]

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

[20]

H. P. McKean Jr., An exponential formula for solving Boltmann's equation for a Maxwellian gas, J. Combinatorial Theory, 2 (1967), 358-382. doi: 10.1016/S0021-9800(67)80035-8.  Google Scholar

[21]

E. Miot, M. Pulvirenti and C. Saffirio, On the Kac model for the Landau equation, Kinet. Relat. Models, 4 (2011), 333-344. doi: 10.3934/krm.2011.4.333.  Google Scholar

[22]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Inv. Math., 193 (2013), 1-147. doi: 10.1007/s00222-012-0422-3.  Google Scholar

[23]

S. Mischler, C. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59. doi: 10.1007/s00440-013-0542-8.  Google Scholar

[24]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, vol. 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1979.  Google Scholar

[25]

A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lecture Notes in Math., Springer, Berlin, 1991, 165-251. doi: 10.1007/BFb0085169.  Google Scholar

[26]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Z. Wahrsch. Verw. Gebiete, 46 (): 67.  doi: 10.1007/BF00535689.  Google Scholar

[27]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756.  Google Scholar

[28]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.  Google Scholar

[29]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., 8 (1998), 957-983. doi: 10.1142/S0218202598000433.  Google Scholar

[30]

C. Villani, Decrease of the Fisher information for solutions of the spatially homogeneous Landau equation with Maxwellian molecules, Math. Mod. Meth. Appl. Sci., 10 (2000), 153-161. doi: 10.1142/S0218202500000100.  Google Scholar

[31]

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

[32]

C. Villani, Optimal Transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, Old and new. doi: 10.1007/978-3-540-71050-9.  Google Scholar

show all references

References:
[1]

A. A. Arsen'ev and O. E. Buryak, On the connection betwenn a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation, Math. USSR Sbornik, 69 (1991), 465-478.  Google Scholar

[2]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in Mathematical physics reviews, Vol. 7, Soviet Sci. Rev. Sect. C Math. Phys. Rev., Harwood Academic Publ., Chur, 7 (1988), 111-233.  Google Scholar

[3]

A. V. Boblylev, M. Pulvirenti and C. Saffirio, From particle systems to the Landau equation: A consistency result, Comm. Math. Phys., 319 (2013), 683-702. doi: 10.1007/s00220-012-1633-6.  Google Scholar

[4]

E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122. doi: 10.3934/krm.2010.3.85.  Google Scholar

[5]

K. Carrapatoso, Quantitative and qualitative Kac's chaos on the Boltzmann's sphere, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 993-1039. doi: 10.1214/14-AIHP612.  Google Scholar

[6]

P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Mod. Meth. in Appl. Sci., 2 (1992), 167-182. doi: 10.1142/S0218202592000119.  Google Scholar

[7]

L. Desvillettes, On the asymptotics of the Boltzmann equation when the collisions become grazing, Transp. Theory and Stat. Phys., 21 (1992), 259-276. doi: 10.1080/00411459208203923.  Google Scholar

[8]

A. Einav, A counter-example to Cercignani's conjecture for the d-dimensional Kac model, J. Statist. Phys., 148 (2012), 1076-1103. doi: 10.1007/s10955-012-0565-z.  Google Scholar

[9]

J. Fontbona, H. Guérin and S. Méléard, Measurability of optimal transportation and convergence rate for Landau type interacting particle systems, Probab. Theory Related Fields, 143 (2009), 329-351. doi: 10.1007/s00440-007-0128-4.  Google Scholar

[10]

N. Fournier, Particle approximation of some Landau equations, Kinet. Relat. Models, 2 (2009), 451-464. doi: 10.3934/krm.2009.2.451.  Google Scholar

[11]

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

[12]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.  Google Scholar

[13]

H. Guérin, Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach, Stochastic Process. Appl., 101 (2002), 303-325. doi: 10.1016/S0304-4149(02)00107-2.  Google Scholar

[14]

H. Guérin, Solving Landau equation for some soft potentials through a probabilistic approach, Ann. Appl. Probab., 13 (2003), 515-539. doi: 10.1214/aoap/1050689592.  Google Scholar

[15]

M. Hauray and S. Mischler, On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157. doi: 10.1016/j.jfa.2014.02.030.  Google Scholar

[16]

R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum, Comm. Math. Phys., 105 (1986), 189-203. doi: 10.1007/BF01211098.  Google Scholar

[17]

M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III, University of California Press, Berkeley and Los Angeles, 1956, 171-197.  Google Scholar

[18]

M. Kiessling and C. Lancellotti, On the master equation approach to kinetic theory: Linear and nonlinear Fokker-Planck equations, Transport Theory and Statistical Physics, 33 (2004), 379-401. doi: 10.1081/TT-200053929.  Google Scholar

[19]

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

[20]

H. P. McKean Jr., An exponential formula for solving Boltmann's equation for a Maxwellian gas, J. Combinatorial Theory, 2 (1967), 358-382. doi: 10.1016/S0021-9800(67)80035-8.  Google Scholar

[21]

E. Miot, M. Pulvirenti and C. Saffirio, On the Kac model for the Landau equation, Kinet. Relat. Models, 4 (2011), 333-344. doi: 10.3934/krm.2011.4.333.  Google Scholar

[22]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Inv. Math., 193 (2013), 1-147. doi: 10.1007/s00222-012-0422-3.  Google Scholar

[23]

S. Mischler, C. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59. doi: 10.1007/s00440-013-0542-8.  Google Scholar

[24]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, vol. 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1979.  Google Scholar

[25]

A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lecture Notes in Math., Springer, Berlin, 1991, 165-251. doi: 10.1007/BFb0085169.  Google Scholar

[26]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Z. Wahrsch. Verw. Gebiete, 46 (): 67.  doi: 10.1007/BF00535689.  Google Scholar

[27]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756.  Google Scholar

[28]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106.  Google Scholar

[29]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., 8 (1998), 957-983. doi: 10.1142/S0218202598000433.  Google Scholar

[30]

C. Villani, Decrease of the Fisher information for solutions of the spatially homogeneous Landau equation with Maxwellian molecules, Math. Mod. Meth. Appl. Sci., 10 (2000), 153-161. doi: 10.1142/S0218202500000100.  Google Scholar

[31]

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

[32]

C. Villani, Optimal Transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, Old and new. doi: 10.1007/978-3-540-71050-9.  Google Scholar

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