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 & 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, (2008).   Google Scholar

[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, 136 (2004), 1.  doi: 10.1007/978-1-4419-8935-2_1.  Google Scholar

[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.  doi: 10.1007/BF01611497.  Google Scholar

[4]

R. L. Dobrushin, Vlasov equations,, Func. Anal. Appl., 13 (1979), 115.   Google Scholar

[5]

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

[6]

G. Hardy, J. Littlewood and G. Polya, Inequalities,, Cambridge Univ. Press, (1952).   Google Scholar

[7]

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

[8]

M. Hauray and P.-E. Jabin, Particle Approximation of Vlasov Equations with Singular Forces: Propagation of Chaos, preprint,, , ().   Google Scholar

[9]

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

[10]

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

[11]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics,, McGraw-Hill, ().   Google Scholar

[12]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids,, Applied Mathematical Sciences, (1994).  doi: 10.1007/978-1-4612-4284-0.  Google Scholar

[13]

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

[14]

S. Mischler, C. Mouhot and B. Wennberg, A New Approach to Quantitative Propagation of Chaos for Drift, Diffusion and Jump Processes, preprint,, , ().   Google Scholar

[15]

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

[16]

H. Neunzert and J. Wick, Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen, (German),, in Numerische Behandlungen nichtlinearer Integrodifferential- und Differentialgleichungen, 395 (1974), 275.  doi: 10.1007/BFb0060678.  Google Scholar

[17]

L. Onsager, Statistical hydrodynamics,, Nuovo Cimento (9), 6 (1949), 279.  doi: 10.1007/BF02780991.  Google Scholar

[18]

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

[19]

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

[20]

S. Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem,, Recent topics in mathematics moving toward science and engineering, 18 (2001), 383.  doi: 10.1007/BF03168581.  Google Scholar

[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.   Google Scholar

[22]

C. Villani, Topics on Optimal Transportation,, Graduate Studies in Mathematics, (2003).   Google Scholar

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, (2008).   Google Scholar

[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, 136 (2004), 1.  doi: 10.1007/978-1-4419-8935-2_1.  Google Scholar

[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.  doi: 10.1007/BF01611497.  Google Scholar

[4]

R. L. Dobrushin, Vlasov equations,, Func. Anal. Appl., 13 (1979), 115.   Google Scholar

[5]

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

[6]

G. Hardy, J. Littlewood and G. Polya, Inequalities,, Cambridge Univ. Press, (1952).   Google Scholar

[7]

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

[8]

M. Hauray and P.-E. Jabin, Particle Approximation of Vlasov Equations with Singular Forces: Propagation of Chaos, preprint,, , ().   Google Scholar

[9]

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

[10]

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

[11]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics,, McGraw-Hill, ().   Google Scholar

[12]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids,, Applied Mathematical Sciences, (1994).  doi: 10.1007/978-1-4612-4284-0.  Google Scholar

[13]

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

[14]

S. Mischler, C. Mouhot and B. Wennberg, A New Approach to Quantitative Propagation of Chaos for Drift, Diffusion and Jump Processes, preprint,, , ().   Google Scholar

[15]

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

[16]

H. Neunzert and J. Wick, Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen, (German),, in Numerische Behandlungen nichtlinearer Integrodifferential- und Differentialgleichungen, 395 (1974), 275.  doi: 10.1007/BFb0060678.  Google Scholar

[17]

L. Onsager, Statistical hydrodynamics,, Nuovo Cimento (9), 6 (1949), 279.  doi: 10.1007/BF02780991.  Google Scholar

[18]

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

[19]

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

[20]

S. Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem,, Recent topics in mathematics moving toward science and engineering, 18 (2001), 383.  doi: 10.1007/BF03168581.  Google Scholar

[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.   Google Scholar

[22]

C. Villani, Topics on Optimal Transportation,, Graduate Studies in Mathematics, (2003).   Google Scholar

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