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Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules
1. | CEREMADE, Université Paris-Dauphine, UMR CNRS 7534, F-75775 Paris, France |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
N. Fournier, Particle approximation of some Landau equations, Kinet. Relat. Models, 2 (2009), 451-464.
doi: 10.3934/krm.2009.2.451. |
[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. |
[12] |
H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Z. Wahrsch. Verw. Gebiete, 46 (): 67.
doi: 10.1007/BF00535689. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
N. Fournier, Particle approximation of some Landau equations, Kinet. Relat. Models, 2 (2009), 451-464.
doi: 10.3934/krm.2009.2.451. |
[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. |
[12] |
H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Z. Wahrsch. Verw. Gebiete, 46 (): 67.
doi: 10.1007/BF00535689. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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