December  2013, 6(4): 919-943. doi: 10.3934/krm.2013.6.919

Empirical measures and Vlasov hierarchies

1. 

Ecole Polytechnique, Centre de Mathématiques Laurent Schwartz, 91128 Palaiseau Cedex, France

2. 

University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce road, Cambridge CB30WA, United Kingdom

3. 

Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy

Received  August 2013 Revised  September 2013 Published  November 2013

The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the $N$-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent $1$ on the infinite mean field hierarchy. This last result amplifies Spohn's uniqueness theorem [H. Spohn, H. Neunzert, Math. Meth. Appl. Sci. 3 (1981), 445--455].
Citation: François Golse, Clément Mouhot, Valeria Ricci. Empirical measures and Vlasov hierarchies. Kinetic and Related Models, 2013, 6 (4) : 919-943. doi: 10.3934/krm.2013.6.919
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, $2^{nd}$ edition, Birkhäuser Verlag AG, Basel, Boston, Berlin, 2008.

[2]

C. Bardos, F. Golse, A. Gottlieb and N. Mauser, On the derivation of nonlinear Schrödinger and Vlasov equations, in Dispersive Transport Equations and Multiscale Models, (Minneapolis, MN, 2000), IMA Vol. Math. Appl., Springer, New York, 136 (2004), 1-23. doi: 10.1007/978-1-4419-8935-2_1.

[3]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $1/N$ limit of interacting classical particles, Comm. in Math. Phys., 56 (1977), 101-113. doi: 10.1007/BF01611497.

[4]

R. L. Dobrushin, Vlasov equations, Func. Anal. Appl., 13 (1979), 115-123.

[5]

F. A. Grünbaum, Propagation of chaos for the Boltzmann equation, Arch. Rational Mech. Anal., 42 (1971), 323-345.

[6]

G. Hardy, J. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, 1952.

[7]

M. Hauray and P.-E. Jabin, $N$-particle approximation of the Vlasov equation with singular potential, Arch. Rational Mech. Anal., 183 (2007), 489-524. doi: 10.1007/s00205-006-0021-9.

[8]

M. Hauray and P.-E. Jabin, Particle Approximation of Vlasov Equations with Singular Forces: Propagation of Chaos, preprint, arXiv:1107.3821.

[9]

E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), 470-501. doi: 10.1090/S0002-9947-1955-0076206-8.

[10]

J. Horowitz and R. Karandikar, Mean rates of convergence of empirical measures in the Wasserstein metric, J. Comput. Appl. Math., 55 (1994), 261-273. doi: 10.1016/0377-0427(94)90033-7.

[11]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, New York,1973.

[12]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, 96. Springer Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[13]

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

[14]

S. Mischler, C. Mouhot and B. Wennberg, A New Approach to Quantitative Propagation of Chaos for Drift, Diffusion and Jump Processes, preprint, arXiv:1101.4727.

[15]

H. Narnhofer and G. Sewell, Vlasov hydrodynamics of a quantum mechanical model, Comm. Math. Phys., 79 (1981), 9-24. doi: 10.1007/BF01208282.

[16]

H. Neunzert and J. Wick, Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen, (German), in Numerische Behandlungen nichtlinearer Integrodifferential- und Differentialgleichungen, (eds. R. Ansorge and W. Tornig), Lecture Notes in Math., Springer, Berlin, 395 (1974), 275-290. doi: 10.1007/BFb0060678.

[17]

L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9), Supplemento 2 (Convegno Internazionale di Meccanica Statistica), 6 (1949), 279-287. doi: 10.1007/BF02780991.

[18]

H. Spohn, On the Vlasov hierarchy, Math. Meth. in the Appl. Sci., 3 (1981), 445-455. doi: 10.1002/mma.1670030131.

[19]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027.

[20]

S. Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem, Recent topics in mathematics moving toward science and engineering, Japan J. Indust. Appl. Math., 18 (2001), 383-392. doi: 10.1007/BF03168581.

[21]

S. Ukai and T. Okabe, On classical solutions in the large in time of two-dimensional Vlasov's equation, Osaka J. Math., 15 (1978), 245-261.

[22]

C. Villani, Topics on Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, $2^{nd}$ edition, Birkhäuser Verlag AG, Basel, Boston, Berlin, 2008.

[2]

C. Bardos, F. Golse, A. Gottlieb and N. Mauser, On the derivation of nonlinear Schrödinger and Vlasov equations, in Dispersive Transport Equations and Multiscale Models, (Minneapolis, MN, 2000), IMA Vol. Math. Appl., Springer, New York, 136 (2004), 1-23. doi: 10.1007/978-1-4419-8935-2_1.

[3]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $1/N$ limit of interacting classical particles, Comm. in Math. Phys., 56 (1977), 101-113. doi: 10.1007/BF01611497.

[4]

R. L. Dobrushin, Vlasov equations, Func. Anal. Appl., 13 (1979), 115-123.

[5]

F. A. Grünbaum, Propagation of chaos for the Boltzmann equation, Arch. Rational Mech. Anal., 42 (1971), 323-345.

[6]

G. Hardy, J. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, 1952.

[7]

M. Hauray and P.-E. Jabin, $N$-particle approximation of the Vlasov equation with singular potential, Arch. Rational Mech. Anal., 183 (2007), 489-524. doi: 10.1007/s00205-006-0021-9.

[8]

M. Hauray and P.-E. Jabin, Particle Approximation of Vlasov Equations with Singular Forces: Propagation of Chaos, preprint, arXiv:1107.3821.

[9]

E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), 470-501. doi: 10.1090/S0002-9947-1955-0076206-8.

[10]

J. Horowitz and R. Karandikar, Mean rates of convergence of empirical measures in the Wasserstein metric, J. Comput. Appl. Math., 55 (1994), 261-273. doi: 10.1016/0377-0427(94)90033-7.

[11]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, New York,1973.

[12]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, 96. Springer Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0.

[13]

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

[14]

S. Mischler, C. Mouhot and B. Wennberg, A New Approach to Quantitative Propagation of Chaos for Drift, Diffusion and Jump Processes, preprint, arXiv:1101.4727.

[15]

H. Narnhofer and G. Sewell, Vlasov hydrodynamics of a quantum mechanical model, Comm. Math. Phys., 79 (1981), 9-24. doi: 10.1007/BF01208282.

[16]

H. Neunzert and J. Wick, Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen, (German), in Numerische Behandlungen nichtlinearer Integrodifferential- und Differentialgleichungen, (eds. R. Ansorge and W. Tornig), Lecture Notes in Math., Springer, Berlin, 395 (1974), 275-290. doi: 10.1007/BFb0060678.

[17]

L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9), Supplemento 2 (Convegno Internazionale di Meccanica Statistica), 6 (1949), 279-287. doi: 10.1007/BF02780991.

[18]

H. Spohn, On the Vlasov hierarchy, Math. Meth. in the Appl. Sci., 3 (1981), 445-455. doi: 10.1002/mma.1670030131.

[19]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027.

[20]

S. Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem, Recent topics in mathematics moving toward science and engineering, Japan J. Indust. Appl. Math., 18 (2001), 383-392. doi: 10.1007/BF03168581.

[21]

S. Ukai and T. Okabe, On classical solutions in the large in time of two-dimensional Vlasov's equation, Osaka J. Math., 15 (1978), 245-261.

[22]

C. Villani, Topics on Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.

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