December  2014, 7(4): 661-711. doi: 10.3934/krm.2014.7.661

A review of the mean field limits for Vlasov equations

1. 

CSCAMM and Dpt of Mathematics, University of Maryland, College Park, MD 20742, United States

Received  August 2014 Revised  September 2014 Published  November 2014

We review some classical and more recent results on the mean field limit and propagation of chaos for systems of many particles, leading to Vlasov or macroscopic equations.
Citation: Pierre-Emmanuel Jabin. A review of the mean field limits for Vlasov equations. Kinetic and Related Models, 2014, 7 (4) : 661-711. doi: 10.3934/krm.2014.7.661
References:
[1]

S. J. Aarseth, Gravitational N-Body Simulations, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511535246.

[2]

L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2.

[3]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, In Calculus of variations and nonlinear partial differential equations, volume 1927 of {Lecture Notes in Math., pages 1-41. Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_1.

[4]

H. Andréasson, M. Kunze and G. Rein, Global existence for the spherically symmetric Einstein-Vlasov system with outgoing matter, Comm. Partial Differential Equations, 33 (2008), 656-668. doi: 10.1080/03605300701454883.

[5]

A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations, Ž. Vyčisl. Mat. i Mat. Fiz., 15 (1975), 136-147.

[6]

R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975.

[7]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.

[8]

C. Bardos, L. Erdös, F. Golse, N. Mauser and H.-T. Yau, Derivation of the Schrödinger-Poisson equation from the quantum n-body problem, C. R. Math. Acad. Sci. Paris, 334 (2002), 515-520. doi: 10.1016/S1631-073X(02)02253-7.

[9]

C. Bardos, F. Golse, A. D. Gottlieb and N. J. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl. (9), 82 (2003), 665-683. doi: 10.1016/S0021-7824(03)00023-0.

[10]

J. Barré, M. Hauray and P. E. Jabin, Stability of trajectories for N-particle dynamics with a singular potential, Journal of Statistical Mechanics: Theory and Experiment, 7, July 2010.

[11]

J. Barré and P. E. Jabin, Free transport limit for $N$-particles dynamics with singular and short range potential, J. Stat. Phys., 131 (2008), 1085-1101. doi: 10.1007/s10955-008-9526-y.

[12]

J. Batt, $N$-particle approximation to the nonlinear Vlasov-Poisson system, In Proceedings of the Third World Congress of Nonlinear Analysts, Part 3 (Catania, 2000), 47 (2001), 1445-1456. doi: 10.1016/S0362-546X(01)00280-2.

[13]

J. Batt and G. Rein, Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 411-416.

[14]

J. Bedrossian and N. Masmoudi, Asymptotic stability for the Couette flow in the 2D Euler equations, Appl. Math. Res. Express. AMRX, (2014), 157-175.

[15]

L. Berlyand, P. Jabin and M. Potomkin, Complexity reduction in many particle systems with random initial data, Submitted to J. Uncertainty Quantification.

[16]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009.

[17]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83. doi: 10.1002/cpa.20334.

[18]

C. Birdsall and A. Langdon, Plasma Physics Via Computer Simulation, Series in plasma physics. Adam Hilger, 1991. doi: 10.1887/0750301171.

[19]

N. N. Bogoliubov, Kinetic equations, Journal of Experimental and Theoretical Physics (in Russian), 16 (1946), 691-702.

[20]

N. N. Bogoliubov, Kinetic equations, Journal of Physics USSR, 10 (1946), 265-274.

[21]

E. Boissard, Problèmes D'interaction Discret-continu et Distances de Wasserstein, PhD thesis, Université de Toulouse III, 2011.

[22]

E. Boissard, Simple bounds for convergence of empirical and occupation measures in 1-Wasserstein distance, Electron. J. Probab., 16 (2011), 2296-2333. doi: 10.1214/EJP.v16-958.

[23]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.

[24]

F. Bolley, A. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Related Fields, 137 (2007), 541-593. doi: 10.1007/s00440-006-0004-7.

[25]

M. Born and H. S. Green, A general kinetic theory of liquids i. the molecular distribution functions, Proc. Roy. Soc. A, 188 (1946), 10-18. doi: 10.1098/rspa.1946.0093.

[26]

F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal., 157 (2001), 75-90. doi: 10.1007/PL00004237.

[27]

F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15. doi: 10.1007/s00205-003-0265-6.

[28]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, L. Desvillettes and B. Perthame eds, Gauthier-Villars, Paris, 2000.

[29]

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

[30]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262.

[31]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. II, Comm. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602.

[32]

E. Caglioti and F. Rousset, Long time behavior of particle systems in the mean field limit, Commun. Math. Sci., 1 (2007), 11-19. doi: 10.4310/CMS.2007.v5.n5.a3.

[33]

E. Caglioti and F. Rousset, Quasi-stationary states for particle systems in the mean-field limit, J. Stat. Phys., 129 (2007), 241-263. doi: 10.1007/s10955-007-9390-1.

[34]

E. Caglioti and F. Rousset, Long time estimates in the mean field limit, Arch. Ration. Mech. Anal., 190 (2008), 517-547. doi: 10.1007/s00205-008-0157-x.

[35]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbb R^2$, Commun. Math. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8.

[36]

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.

[37]

J. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and wasserstein distances, In Collective Dynamics from Bacteria to Crowds, volume 553 of CISM International Centre for Mechanical Sciences, pages 1-46. Springer Vienna, 2014. doi: 10.1007/978-3-7091-1785-9_1.

[38]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211.

[39]

J. A. Carrillo, S. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials, Nonlinear Anal., 100 (2014), 122-147. doi: 10.1016/j.na.2014.01.010.

[40]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[41]

T. Champion, L. De Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal., 40 (2008), 1-20. doi: 10.1137/07069938X.

[42]

P.-H. Chavanis, Hamiltonian and Brownian systems with long-range interactions. V. Stochastic kinetic equations and theory of fluctuations, Phys. A., 387 (2008), 5716-5740. doi: 10.1016/j.physa.2008.06.016.

[43]

J.-Y. Chemin, Perfect Incompressible Fluids, volume 14 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie.

[44]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.

[45]

G.-H. Cottet, J. Goodman and T. Y. Hou, Convergence of the grid-free point vortex method for the three-dimensional Euler equations, SIAM J. Numer. Anal., 28 (1991), 291-307. doi: 10.1137/0728016.

[46]

G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46. doi: 10.1515/CRELLE.2008.016.

[47]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[48]

F. Cucker and S. Smale, On the mathematics of emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.

[49]

M. Cullen, W. Gangbo and G. Pisante, The semigeostrophic equations discretized in reference and dual variables, Arch. Ration. Mech. Anal., 185 (2007), 341-363. doi: 10.1007/s00205-006-0040-6.

[50]

C. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations, In Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., pages 277-382. Elsevier/North-Holland, Amsterdam, 2007.

[51]

P. Degond, A. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456. doi: 10.1007/s00332-012-9157-y.

[52]

P. Degond and F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Statist. Comput., 11 (1990), 293-310. doi: 10.1137/0911018.

[53]

W. Dehnen, A very fast and momentum-conserving tree code, The Astrophysical Journal, 536 (2000), L39-L42.

[54]

J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 4 (1991), 553-586. doi: 10.1090/S0894-0347-1991-1102579-6.

[55]

L. Desvillettes, F. Golse and V. Ricci, The mean-field limit for solid particles in a Navier-Stokes flow, J. Stat. Phys., 131 (2008), 941-967. doi: 10.1007/s10955-008-9521-3.

[56]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, J. Differential Equations, 250 (2011), 1334-1362. doi: 10.1016/j.jde.2010.10.015.

[57]

R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.

[58]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, Invent. Math, 98 (1989), 511-547. doi: 10.1007/BF01393835.

[59]

V. Dobrić and J. E. Yukich, Asymptotics for transportation cost in high dimensions, J. Theoret. Probab., 8 (1995), 97-118. doi: 10.1007/BF02213456.

[60]

R. L. Dobrušin, Vlasov equations, Funktsional. Anal. i Prilozhen., 13 (1979), 48-58, 96.

[61]

L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-614. doi: 10.1007/s00222-006-0022-1.

[62]

L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math. (2), 172 (2010), 291-370. doi: 10.4007/annals.2010.172.291.

[63]

N. Fournier and A. Guillin, On the rate of convergence in wasserstein distance of the empirical measure, arXiv:1312.2128, 2014.

[64]

N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2d viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1425-1466. doi: 10.4171/JEMS/465.

[65]

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.

[66]

K. Ganguly, J. T. Lee and H. D. Victory, Jr., On simulation methods for Vlasov-Poisson systems with particles initially asymptotically distributed, SIAM J. Numer. Anal., 28 (1991), 1574-1609. doi: 10.1137/0728080.

[67]

K. Ganguly and H. D. Victory, Jr., On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal., 26 (1989), 249-288. doi: 10.1137/0726015.

[68]

F. Gao, Moderate deviations and large deviations for kernel density estimators, J. Theoret. Probab., 16 (2003), 401-418. doi: 10.1023/A:1023574711733.

[69]

I. Gasser, P.-E. Jabin and B. Perthame, Regularity and propagation of moments in some nonlinear Vlasov systems, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1259-1273. doi: 10.1017/S0308210500000676.

[70]

J. W. Gibbs, On the Fundamental Formulae of Dynamics, Amer. J. Math., 2 (1879), 49-64. doi: 10.2307/2369196.

[71]

J. W. Gibbs, Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundation of Thermodynamics, Dover publications, Inc., New York, 1960.

[72]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[73]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. I, II, Arch. Rational Mech. Anal., 141 (1998), 331-354, 355-374. doi: 10.1007/s002050050079.

[74]

F. Golse, On the dynamics of large particle systems in the mean field limit, arXiv:1301.5494, 2013.

[75]

F. Golse, C. Mouhot and V. Ricci, Empirical measures and Vlasov hierarchies, Kinet. Relat. Models, 6 (2013), 919-943. doi: 10.3934/krm.2013.6.919.

[76]

J. Goodman and T. Y. Hou, New stability estimates for the $2$-D vortex method, Comm. Pure Appl. Math., 44 (1991), 1015-1031. doi: 10.1002/cpa.3160440813.

[77]

J. Goodman, T. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the $2$-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 415-430. doi: 10.1002/cpa.3160430305.

[78]

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

[79]

L. Greengard and V. Rokhlin, A fast algorithm for particle simulation, Journal of Computational Physics, 73 (1987), 325-348. doi: 10.1016/0021-9991(87)90140-9.

[80]

L. Greengard and V. Rokhlin, Rapid evaluation of potential fields in three dimensions, Lecture Notes in Mathematics, 1360 (1988), 121-141. doi: 10.1007/BFb0089775.

[81]

Y. N. Grigoryev, V. A. Vshivkov and M. P. Fedoruk, Numerical "Particle-in-Cell" Methods: Theory and Applications, De Gruyter, 2002. doi: 10.1515/9783110916706.

[82]

O. Guéant, J.-M. Lasry, and P.-L. Lions, Mean field games and applications, In Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., (2011), 205-266. doi: 10.1007/978-3-642-14660-2_3.

[83]

V. Gyrya, L. Berlyand, I. Aranson and D. A. Karpeev, A model of hydrodynamics interaction between swimming bacteria, Bulletin of Mathematical Biology, 72 (2010), 148-183. doi: 10.1007/s11538-009-9442-6.

[84]

M. Hauray, On Liouville transport equation with force field in $BV_{loc}$, Comm. Partial Differential Equations, 29 (2004), 207-217. doi: 10.1081/PDE-120028850.

[85]

M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., 19 (2009), 1357-1384. doi: 10.1142/S0218202509003814.

[86]

M. Hauray, Mean field limit for the one dimensional vlasov-poisson equation, Séminaire Laurent Schwartz, École Polytechnique, 2013. arXiv:1309.2531.

[87]

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

[88]

M. Hauray and P.-E. Jabin, Particles Approximations of Vlasov Equations with Singular Forces: Propagation of Chaos, To appearAnn. Sci. Ec. Norm. Super., 2014.

[89]

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.

[90]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation, Journal of Artifical Societies and Social Simulation (JASSS), 5, no. 3, 2002.

[91]

M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194. doi: 10.1007/s002850050049.

[92]

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

[93]

A. Honig, B. Niethammer and F. Otto, On first-order corrections to the lsw theory i: Infinite systems, Journal of Statistical Physics, 119 (2005), 61-122. doi: 10.1007/s10955-004-2057-2.

[94]

A. Honig, B. Niethammer and F. Otto, On first-order corrections to the lsw theory ii: Finite systems, Journal of Statistical Physics, 119 (2005), 123-164. doi: 10.1007/s10955-004-2058-1.

[95]

E. Horst, Global strong solutions of Vlasov's equation-necessary and sufficient conditions for their existence, In Partial differential equations (Warsaw, 1984), volume 19 of Banach Center Publ., pages 143-153. PWN, Warsaw, 1987.

[96]

E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci., 16 (1993), 75-86. doi: 10.1002/mma.1670160202.

[97]

T. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the $3$-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 965-981. doi: 10.1002/cpa.3160430803.

[98]

T. Y. Hou, J. Lowengrub and M. J. Shelley, The convergence of an exact desingularization for vortex methods, SIAM J. Sci. Comput., 14 (1993), 1-18. doi: 10.1137/0914001.

[99]

R. Illner and M. Pulvirenti, Global validity of the boltzmann equation for two- and three-dimensional rare gas in vacuum, Comm. Math. Phys., 121 (1989), 143-146.

[100]

P. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004), 415-432. doi: 10.1007/s00220-004-1126-3.

[101]

P. Jabin and B. Perthame, Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid, In Modelling in applied sciences, a kinetic theory approach, Model. Simul. Sci. Eng. Technol., pages 111-147. Birkhauser Boston, 2000.

[102]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.2307/2153966.

[103]

J. H. Jeans, On the theory of star-streaming and the structure of the universe, Monthly Notices of the Royal Astronomical Society, 76 (1915), 70-84.

[104]

V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066.

[105]

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

[106]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.

[107]

M. K.-H. Kiessling, On the equilibrium statistical mechanics of isothermal classical self-gravitating matter, J. Statist. Phys., 55 (1989), 203-257. doi: 10.1007/BF01042598.

[108]

M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103.

[109]

M. K.-H. Kiessling and H. Spohn, A note on the eigenvalue density of random matrices, Comm. Math. Phys., 199 (1999), 683-695. doi: 10.1007/s002200050516.

[110]

J. G. Kirkwood, The statistical mechanical theory of transport processes i. general theory, The Journal of Chemical Physics, 14 (1946), p180.

[111]

J. G. Kirkwood, The statistical mechanical theory of transport processes i. transport in gases, The Journal of Chemical Physics, 15 (1947), p72.

[112]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, Communications in difference equations, pages 227-236, 2000.

[113]

C. Lancellotti, On the fluctuations about the Vlasov limit for $N$-particle systems with mean-field interactions, J. Stat. Phys., 136 (2009), 643-665. doi: 10.1007/s10955-009-9800-7.

[114]

L. Landau, On the vibrations of the electronic plasma, Akad. Nauk SSSR. Zhurnal Eksper. Teoret. Fiz., 16 (1946), 574-586.

[115]

O. E. Lanford, III, Time evolution of large classical systems, In Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), pages 1-111. Lecture Notes in Phys., Vol. 38. Springer, Berlin, 1975.

[116]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[117]

M. Lemou, F. Méhats and P. Raphaël, Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov-Poisson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 825-833. doi: 10.1016/j.anihpc.2006.07.003.

[118]

A. Lenard, On Bogoliubov's kinetic equation for a spatially homogeneous plasma, Ann. Physics, 10 (1960), 390-400.

[119]

P.-L. Lions and S. Mas-Gallic, Une méthode particulaire déterministe pour des équations diffusives non linéaires, C. R. Math. Acad. Sci. Paris, 332 (2001), 369-376. doi: 10.1016/S0764-4442(00)01795-X.

[120]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.

[121]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. (9), 86 (2006), 68-79. doi: 10.1016/j.matpur.2006.01.005.

[122]

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

[123]

R. J. McCann, Stable rotating binary stars and fluid in a tube, Houston J. Math., 32 (2006), 603-631.

[124]

H. P. McKean, Jr., Propagation of chaos for a class of non-linear parabolic equations, In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pages 41-57. Air Force Office Sci. Res., Arlington, Va., 1967.

[125]

J. Messer and H. Spohn, Statistical mechanics of the isothermal Lane-Emden equation, J. Statist. Phys., 29 (1982), 561-578. doi: 10.1007/BF01342187.

[126]

S. Mischler, Sur le programme de Kac concernant les limites de champ moyen, In Seminaire: Equations aux Dérivées Partielles. 2009-2010, Sémin. Équ. Dériv. Partielles, pages Exp. No. XXXIII, 19. École Polytech., Palaiseau, 2012.

[127]

S. Mischler and C. Mouhot, Kac's Program in Kinetic Theory, Invent. Math., 193 (2013), 1-147. doi: 10.1007/s00222-012-0422-3.

[128]

S. Mischler, C. Mouhot and B. Wennberg, A New Approach to Quantitative Chaos Propagation for Drift, Diffusion and Jump Process, Arxiv, 2013.

[129]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.

[130]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9.

[131]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.

[132]

H. Neunzert and J. Wick, The convergence of simulation methods in plasma physics, In Mathematical methods of plasmaphysics (Oberwolfach, 1979), volume 20 of Methoden Verfahren Math. Phys., pages 271-286. Lang, Frankfurt, 1980.

[133]

H. Osada, Propagation of chaos for the two-dimensional Navier-Stokes equation, In Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), pages 303-334. Academic Press, Boston, MA, 1987.

[134]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.

[135]

C. Pallard, Moment propagation for weak solutions to the Vlasov-Poisson system, Comm. Partial Differential Equations, 37 (2012), 1273-1285. doi: 10.1080/03605302.2011.606863.

[136]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.

[137]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.

[138]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.

[139]

F. Planchon, An extension of the Beale-Kato-Majda criterion for the Euler equations, Comm. Math. Phys., 232 (2003), 319-326. doi: 10.1007/s00220-002-0744-x.

[140]

M. Rascle and C. Ziti, Finite time blow-up in some models of chemotaxis, J. Math. Biol., 33 (1995), 388-414. doi: 10.1007/BF00176379.

[141]

N. Rougerie and S. Serfaty, Higher dimensional coulomb gases and renormalized energy functionals, arXiv:1307.2805, 2013.

[142]

D. G. Saari, Improbability of collisions in Newtonian gravitational systems. II, Trans. Amer. Math. Soc., 181 (1973), 351-368. doi: 10.1090/S0002-9947-1973-0321386-0.

[143]

D. G. Saari, A global existence theorem for the four-body problem of Newtonian mechanics, J. Differential Equations, 26 (1977), 80-111. doi: 10.1016/0022-0396(77)90100-0.

[144]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335. doi: 10.1080/03605309108820801.

[145]

S. Schochet, The weak vorticity formulation of the $2$-D Euler equations and concentration-cancellation, Comm. Partial Differential Equations, 20 (1995), 1077-1104. doi: 10.1080/03605309508821124.

[146]

S. Schochet, The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm. Pure Appl. Math., 49 (1996), 911-965. doi: 10.1002/(SICI)1097-0312(199609)49:9<911::AID-CPA2>3.0.CO;2-A.

[147]

Y. Sone, Molecular Gas Dynamics. Theory, Techniques, and Applications, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4573-1.

[148]

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

[149]

H. Spohn, Large Scale Dynamics of Interacting Particles, Springer Verlag, New York, 1991. doi: 10.1007/978-3-642-84371-6.

[150]

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

[151]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.

[152]

M. Trocheris, On the derivation of the one-dimensional Vlasov equation, Transport Theory Statist. Phys., 15 (1986), 597-628. doi: 10.1080/00411458608212706.

[153]

V. S. Varadarajan, On the convergence of sample probability distributions, Sankhyā, 19 (1958), 23-26.

[154]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical Review Letters, 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

[155]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[156]

A. A. Vlasov, On vibration properties of electron gas, Soviet Physics Uspekhi, 10 (1968), p291. doi: 10.1070/PU1968v010n06ABEH003709.

[157]

A. A. Vlasov, The vibrational properties of an electron gas, Sov. Phys. Usp., 10 (1968), 721-733. doi: 10.1070/PU1968v010n06ABEH003709.

[158]

S. Wollman, On the approximation of the Vlasov-Poisson system by particle methods, SIAM J. Numer. Anal., 37 (2000), 1369-1398 (electronic). doi: 10.1137/S0036142999298528.

[159]

H. Xia, H. Wang and Z. Xuan, Opinion dynamics: A multidisciplinary review and perspective on future research, International Journal of Knowledge and Systems Science (IJKSS), 2 (2011), 72-91. doi: 10.4018/jkss.2011100106.

[160]

Z. Xia, The existence of noncollision singularities in Newtonian systems, Ann. of Math. (2), 135 (1992), 411-468. doi: 10.2307/2946572.

[161]

V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., 2 (1995), 27-38. doi: 10.4310/MRL.1995.v2.n1.a4.

[162]

J. Yvon, La théorie statistique des fluides et l'équation d'état (in french), Actual. Sci. Indust., 203, 1935.

show all references

References:
[1]

S. J. Aarseth, Gravitational N-Body Simulations, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511535246.

[2]

L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2.

[3]

L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, In Calculus of variations and nonlinear partial differential equations, volume 1927 of {Lecture Notes in Math., pages 1-41. Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_1.

[4]

H. Andréasson, M. Kunze and G. Rein, Global existence for the spherically symmetric Einstein-Vlasov system with outgoing matter, Comm. Partial Differential Equations, 33 (2008), 656-668. doi: 10.1080/03605300701454883.

[5]

A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations, Ž. Vyčisl. Mat. i Mat. Fiz., 15 (1975), 136-147.

[6]

R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975.

[7]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.

[8]

C. Bardos, L. Erdös, F. Golse, N. Mauser and H.-T. Yau, Derivation of the Schrödinger-Poisson equation from the quantum n-body problem, C. R. Math. Acad. Sci. Paris, 334 (2002), 515-520. doi: 10.1016/S1631-073X(02)02253-7.

[9]

C. Bardos, F. Golse, A. D. Gottlieb and N. J. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl. (9), 82 (2003), 665-683. doi: 10.1016/S0021-7824(03)00023-0.

[10]

J. Barré, M. Hauray and P. E. Jabin, Stability of trajectories for N-particle dynamics with a singular potential, Journal of Statistical Mechanics: Theory and Experiment, 7, July 2010.

[11]

J. Barré and P. E. Jabin, Free transport limit for $N$-particles dynamics with singular and short range potential, J. Stat. Phys., 131 (2008), 1085-1101. doi: 10.1007/s10955-008-9526-y.

[12]

J. Batt, $N$-particle approximation to the nonlinear Vlasov-Poisson system, In Proceedings of the Third World Congress of Nonlinear Analysts, Part 3 (Catania, 2000), 47 (2001), 1445-1456. doi: 10.1016/S0362-546X(01)00280-2.

[13]

J. Batt and G. Rein, Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 411-416.

[14]

J. Bedrossian and N. Masmoudi, Asymptotic stability for the Couette flow in the 2D Euler equations, Appl. Math. Res. Express. AMRX, (2014), 157-175.

[15]

L. Berlyand, P. Jabin and M. Potomkin, Complexity reduction in many particle systems with random initial data, Submitted to J. Uncertainty Quantification.

[16]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009.

[17]

A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83. doi: 10.1002/cpa.20334.

[18]

C. Birdsall and A. Langdon, Plasma Physics Via Computer Simulation, Series in plasma physics. Adam Hilger, 1991. doi: 10.1887/0750301171.

[19]

N. N. Bogoliubov, Kinetic equations, Journal of Experimental and Theoretical Physics (in Russian), 16 (1946), 691-702.

[20]

N. N. Bogoliubov, Kinetic equations, Journal of Physics USSR, 10 (1946), 265-274.

[21]

E. Boissard, Problèmes D'interaction Discret-continu et Distances de Wasserstein, PhD thesis, Université de Toulouse III, 2011.

[22]

E. Boissard, Simple bounds for convergence of empirical and occupation measures in 1-Wasserstein distance, Electron. J. Probab., 16 (2011), 2296-2333. doi: 10.1214/EJP.v16-958.

[23]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.

[24]

F. Bolley, A. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Related Fields, 137 (2007), 541-593. doi: 10.1007/s00440-006-0004-7.

[25]

M. Born and H. S. Green, A general kinetic theory of liquids i. the molecular distribution functions, Proc. Roy. Soc. A, 188 (1946), 10-18. doi: 10.1098/rspa.1946.0093.

[26]

F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal., 157 (2001), 75-90. doi: 10.1007/PL00004237.

[27]

F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15. doi: 10.1007/s00205-003-0265-6.

[28]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, L. Desvillettes and B. Perthame eds, Gauthier-Villars, Paris, 2000.

[29]

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

[30]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262.

[31]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. II, Comm. Math. Phys., 174 (1995), 229-260. doi: 10.1007/BF02099602.

[32]

E. Caglioti and F. Rousset, Long time behavior of particle systems in the mean field limit, Commun. Math. Sci., 1 (2007), 11-19. doi: 10.4310/CMS.2007.v5.n5.a3.

[33]

E. Caglioti and F. Rousset, Quasi-stationary states for particle systems in the mean-field limit, J. Stat. Phys., 129 (2007), 241-263. doi: 10.1007/s10955-007-9390-1.

[34]

E. Caglioti and F. Rousset, Long time estimates in the mean field limit, Arch. Ration. Mech. Anal., 190 (2008), 517-547. doi: 10.1007/s00205-008-0157-x.

[35]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbb R^2$, Commun. Math. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8.

[36]

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.

[37]

J. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and wasserstein distances, In Collective Dynamics from Bacteria to Crowds, volume 553 of CISM International Centre for Mechanical Sciences, pages 1-46. Springer Vienna, 2014. doi: 10.1007/978-3-7091-1785-9_1.

[38]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211.

[39]

J. A. Carrillo, S. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials, Nonlinear Anal., 100 (2014), 122-147. doi: 10.1016/j.na.2014.01.010.

[40]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[41]

T. Champion, L. De Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal., 40 (2008), 1-20. doi: 10.1137/07069938X.

[42]

P.-H. Chavanis, Hamiltonian and Brownian systems with long-range interactions. V. Stochastic kinetic equations and theory of fluctuations, Phys. A., 387 (2008), 5716-5740. doi: 10.1016/j.physa.2008.06.016.

[43]

J.-Y. Chemin, Perfect Incompressible Fluids, volume 14 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie.

[44]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.

[45]

G.-H. Cottet, J. Goodman and T. Y. Hou, Convergence of the grid-free point vortex method for the three-dimensional Euler equations, SIAM J. Numer. Anal., 28 (1991), 291-307. doi: 10.1137/0728016.

[46]

G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46. doi: 10.1515/CRELLE.2008.016.

[47]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[48]

F. Cucker and S. Smale, On the mathematics of emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.

[49]

M. Cullen, W. Gangbo and G. Pisante, The semigeostrophic equations discretized in reference and dual variables, Arch. Ration. Mech. Anal., 185 (2007), 341-363. doi: 10.1007/s00205-006-0040-6.

[50]

C. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations, In Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., pages 277-382. Elsevier/North-Holland, Amsterdam, 2007.

[51]

P. Degond, A. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456. doi: 10.1007/s00332-012-9157-y.

[52]

P. Degond and F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Statist. Comput., 11 (1990), 293-310. doi: 10.1137/0911018.

[53]

W. Dehnen, A very fast and momentum-conserving tree code, The Astrophysical Journal, 536 (2000), L39-L42.

[54]

J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 4 (1991), 553-586. doi: 10.1090/S0894-0347-1991-1102579-6.

[55]

L. Desvillettes, F. Golse and V. Ricci, The mean-field limit for solid particles in a Navier-Stokes flow, J. Stat. Phys., 131 (2008), 941-967. doi: 10.1007/s10955-008-9521-3.

[56]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, J. Differential Equations, 250 (2011), 1334-1362. doi: 10.1016/j.jde.2010.10.015.

[57]

R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.

[58]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, Invent. Math, 98 (1989), 511-547. doi: 10.1007/BF01393835.

[59]

V. Dobrić and J. E. Yukich, Asymptotics for transportation cost in high dimensions, J. Theoret. Probab., 8 (1995), 97-118. doi: 10.1007/BF02213456.

[60]

R. L. Dobrušin, Vlasov equations, Funktsional. Anal. i Prilozhen., 13 (1979), 48-58, 96.

[61]

L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-614. doi: 10.1007/s00222-006-0022-1.

[62]

L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math. (2), 172 (2010), 291-370. doi: 10.4007/annals.2010.172.291.

[63]

N. Fournier and A. Guillin, On the rate of convergence in wasserstein distance of the empirical measure, arXiv:1312.2128, 2014.

[64]

N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2d viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1425-1466. doi: 10.4171/JEMS/465.

[65]

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.

[66]

K. Ganguly, J. T. Lee and H. D. Victory, Jr., On simulation methods for Vlasov-Poisson systems with particles initially asymptotically distributed, SIAM J. Numer. Anal., 28 (1991), 1574-1609. doi: 10.1137/0728080.

[67]

K. Ganguly and H. D. Victory, Jr., On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal., 26 (1989), 249-288. doi: 10.1137/0726015.

[68]

F. Gao, Moderate deviations and large deviations for kernel density estimators, J. Theoret. Probab., 16 (2003), 401-418. doi: 10.1023/A:1023574711733.

[69]

I. Gasser, P.-E. Jabin and B. Perthame, Regularity and propagation of moments in some nonlinear Vlasov systems, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1259-1273. doi: 10.1017/S0308210500000676.

[70]

J. W. Gibbs, On the Fundamental Formulae of Dynamics, Amer. J. Math., 2 (1879), 49-64. doi: 10.2307/2369196.

[71]

J. W. Gibbs, Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundation of Thermodynamics, Dover publications, Inc., New York, 1960.

[72]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[73]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. I, II, Arch. Rational Mech. Anal., 141 (1998), 331-354, 355-374. doi: 10.1007/s002050050079.

[74]

F. Golse, On the dynamics of large particle systems in the mean field limit, arXiv:1301.5494, 2013.

[75]

F. Golse, C. Mouhot and V. Ricci, Empirical measures and Vlasov hierarchies, Kinet. Relat. Models, 6 (2013), 919-943. doi: 10.3934/krm.2013.6.919.

[76]

J. Goodman and T. Y. Hou, New stability estimates for the $2$-D vortex method, Comm. Pure Appl. Math., 44 (1991), 1015-1031. doi: 10.1002/cpa.3160440813.

[77]

J. Goodman, T. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the $2$-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 415-430. doi: 10.1002/cpa.3160430305.

[78]

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

[79]

L. Greengard and V. Rokhlin, A fast algorithm for particle simulation, Journal of Computational Physics, 73 (1987), 325-348. doi: 10.1016/0021-9991(87)90140-9.

[80]

L. Greengard and V. Rokhlin, Rapid evaluation of potential fields in three dimensions, Lecture Notes in Mathematics, 1360 (1988), 121-141. doi: 10.1007/BFb0089775.

[81]

Y. N. Grigoryev, V. A. Vshivkov and M. P. Fedoruk, Numerical "Particle-in-Cell" Methods: Theory and Applications, De Gruyter, 2002. doi: 10.1515/9783110916706.

[82]

O. Guéant, J.-M. Lasry, and P.-L. Lions, Mean field games and applications, In Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., (2011), 205-266. doi: 10.1007/978-3-642-14660-2_3.

[83]

V. Gyrya, L. Berlyand, I. Aranson and D. A. Karpeev, A model of hydrodynamics interaction between swimming bacteria, Bulletin of Mathematical Biology, 72 (2010), 148-183. doi: 10.1007/s11538-009-9442-6.

[84]

M. Hauray, On Liouville transport equation with force field in $BV_{loc}$, Comm. Partial Differential Equations, 29 (2004), 207-217. doi: 10.1081/PDE-120028850.

[85]

M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., 19 (2009), 1357-1384. doi: 10.1142/S0218202509003814.

[86]

M. Hauray, Mean field limit for the one dimensional vlasov-poisson equation, Séminaire Laurent Schwartz, École Polytechnique, 2013. arXiv:1309.2531.

[87]

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

[88]

M. Hauray and P.-E. Jabin, Particles Approximations of Vlasov Equations with Singular Forces: Propagation of Chaos, To appearAnn. Sci. Ec. Norm. Super., 2014.

[89]

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.

[90]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation, Journal of Artifical Societies and Social Simulation (JASSS), 5, no. 3, 2002.

[91]

M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194. doi: 10.1007/s002850050049.

[92]

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

[93]

A. Honig, B. Niethammer and F. Otto, On first-order corrections to the lsw theory i: Infinite systems, Journal of Statistical Physics, 119 (2005), 61-122. doi: 10.1007/s10955-004-2057-2.

[94]

A. Honig, B. Niethammer and F. Otto, On first-order corrections to the lsw theory ii: Finite systems, Journal of Statistical Physics, 119 (2005), 123-164. doi: 10.1007/s10955-004-2058-1.

[95]

E. Horst, Global strong solutions of Vlasov's equation-necessary and sufficient conditions for their existence, In Partial differential equations (Warsaw, 1984), volume 19 of Banach Center Publ., pages 143-153. PWN, Warsaw, 1987.

[96]

E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci., 16 (1993), 75-86. doi: 10.1002/mma.1670160202.

[97]

T. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the $3$-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 965-981. doi: 10.1002/cpa.3160430803.

[98]

T. Y. Hou, J. Lowengrub and M. J. Shelley, The convergence of an exact desingularization for vortex methods, SIAM J. Sci. Comput., 14 (1993), 1-18. doi: 10.1137/0914001.

[99]

R. Illner and M. Pulvirenti, Global validity of the boltzmann equation for two- and three-dimensional rare gas in vacuum, Comm. Math. Phys., 121 (1989), 143-146.

[100]

P. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004), 415-432. doi: 10.1007/s00220-004-1126-3.

[101]

P. Jabin and B. Perthame, Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid, In Modelling in applied sciences, a kinetic theory approach, Model. Simul. Sci. Eng. Technol., pages 111-147. Birkhauser Boston, 2000.

[102]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.2307/2153966.

[103]

J. H. Jeans, On the theory of star-streaming and the structure of the universe, Monthly Notices of the Royal Astronomical Society, 76 (1915), 70-84.

[104]

V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 1032-1066.

[105]

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

[106]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.

[107]

M. K.-H. Kiessling, On the equilibrium statistical mechanics of isothermal classical self-gravitating matter, J. Statist. Phys., 55 (1989), 203-257. doi: 10.1007/BF01042598.

[108]

M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103.

[109]

M. K.-H. Kiessling and H. Spohn, A note on the eigenvalue density of random matrices, Comm. Math. Phys., 199 (1999), 683-695. doi: 10.1007/s002200050516.

[110]

J. G. Kirkwood, The statistical mechanical theory of transport processes i. general theory, The Journal of Chemical Physics, 14 (1946), p180.

[111]

J. G. Kirkwood, The statistical mechanical theory of transport processes i. transport in gases, The Journal of Chemical Physics, 15 (1947), p72.

[112]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, Communications in difference equations, pages 227-236, 2000.

[113]

C. Lancellotti, On the fluctuations about the Vlasov limit for $N$-particle systems with mean-field interactions, J. Stat. Phys., 136 (2009), 643-665. doi: 10.1007/s10955-009-9800-7.

[114]

L. Landau, On the vibrations of the electronic plasma, Akad. Nauk SSSR. Zhurnal Eksper. Teoret. Fiz., 16 (1946), 574-586.

[115]

O. E. Lanford, III, Time evolution of large classical systems, In Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), pages 1-111. Lecture Notes in Phys., Vol. 38. Springer, Berlin, 1975.

[116]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[117]

M. Lemou, F. Méhats and P. Raphaël, Uniqueness of the critical mass blow up solution for the four dimensional gravitational Vlasov-Poisson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 825-833. doi: 10.1016/j.anihpc.2006.07.003.

[118]

A. Lenard, On Bogoliubov's kinetic equation for a spatially homogeneous plasma, Ann. Physics, 10 (1960), 390-400.

[119]

P.-L. Lions and S. Mas-Gallic, Une méthode particulaire déterministe pour des équations diffusives non linéaires, C. R. Math. Acad. Sci. Paris, 332 (2001), 369-376. doi: 10.1016/S0764-4442(00)01795-X.

[120]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.

[121]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. (9), 86 (2006), 68-79. doi: 10.1016/j.matpur.2006.01.005.

[122]

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

[123]

R. J. McCann, Stable rotating binary stars and fluid in a tube, Houston J. Math., 32 (2006), 603-631.

[124]

H. P. McKean, Jr., Propagation of chaos for a class of non-linear parabolic equations, In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pages 41-57. Air Force Office Sci. Res., Arlington, Va., 1967.

[125]

J. Messer and H. Spohn, Statistical mechanics of the isothermal Lane-Emden equation, J. Statist. Phys., 29 (1982), 561-578. doi: 10.1007/BF01342187.

[126]

S. Mischler, Sur le programme de Kac concernant les limites de champ moyen, In Seminaire: Equations aux Dérivées Partielles. 2009-2010, Sémin. Équ. Dériv. Partielles, pages Exp. No. XXXIII, 19. École Polytech., Palaiseau, 2012.

[127]

S. Mischler and C. Mouhot, Kac's Program in Kinetic Theory, Invent. Math., 193 (2013), 1-147. doi: 10.1007/s00222-012-0422-3.

[128]

S. Mischler, C. Mouhot and B. Wennberg, A New Approach to Quantitative Chaos Propagation for Drift, Diffusion and Jump Process, Arxiv, 2013.

[129]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.

[130]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9.

[131]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.

[132]

H. Neunzert and J. Wick, The convergence of simulation methods in plasma physics, In Mathematical methods of plasmaphysics (Oberwolfach, 1979), volume 20 of Methoden Verfahren Math. Phys., pages 271-286. Lang, Frankfurt, 1980.

[133]

H. Osada, Propagation of chaos for the two-dimensional Navier-Stokes equation, In Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), pages 303-334. Academic Press, Boston, MA, 1987.

[134]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.

[135]

C. Pallard, Moment propagation for weak solutions to the Vlasov-Poisson system, Comm. Partial Differential Equations, 37 (2012), 1273-1285. doi: 10.1080/03605302.2011.606863.

[136]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.

[137]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.

[138]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.

[139]

F. Planchon, An extension of the Beale-Kato-Majda criterion for the Euler equations, Comm. Math. Phys., 232 (2003), 319-326. doi: 10.1007/s00220-002-0744-x.

[140]

M. Rascle and C. Ziti, Finite time blow-up in some models of chemotaxis, J. Math. Biol., 33 (1995), 388-414. doi: 10.1007/BF00176379.

[141]

N. Rougerie and S. Serfaty, Higher dimensional coulomb gases and renormalized energy functionals, arXiv:1307.2805, 2013.

[142]

D. G. Saari, Improbability of collisions in Newtonian gravitational systems. II, Trans. Amer. Math. Soc., 181 (1973), 351-368. doi: 10.1090/S0002-9947-1973-0321386-0.

[143]

D. G. Saari, A global existence theorem for the four-body problem of Newtonian mechanics, J. Differential Equations, 26 (1977), 80-111. doi: 10.1016/0022-0396(77)90100-0.

[144]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335. doi: 10.1080/03605309108820801.

[145]

S. Schochet, The weak vorticity formulation of the $2$-D Euler equations and concentration-cancellation, Comm. Partial Differential Equations, 20 (1995), 1077-1104. doi: 10.1080/03605309508821124.

[146]

S. Schochet, The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm. Pure Appl. Math., 49 (1996), 911-965. doi: 10.1002/(SICI)1097-0312(199609)49:9<911::AID-CPA2>3.0.CO;2-A.

[147]

Y. Sone, Molecular Gas Dynamics. Theory, Techniques, and Applications, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4573-1.

[148]

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

[149]

H. Spohn, Large Scale Dynamics of Interacting Particles, Springer Verlag, New York, 1991. doi: 10.1007/978-3-642-84371-6.

[150]

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

[151]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.

[152]

M. Trocheris, On the derivation of the one-dimensional Vlasov equation, Transport Theory Statist. Phys., 15 (1986), 597-628. doi: 10.1080/00411458608212706.

[153]

V. S. Varadarajan, On the convergence of sample probability distributions, Sankhyā, 19 (1958), 23-26.

[154]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical Review Letters, 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

[155]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[156]

A. A. Vlasov, On vibration properties of electron gas, Soviet Physics Uspekhi, 10 (1968), p291. doi: 10.1070/PU1968v010n06ABEH003709.

[157]

A. A. Vlasov, The vibrational properties of an electron gas, Sov. Phys. Usp., 10 (1968), 721-733. doi: 10.1070/PU1968v010n06ABEH003709.

[158]

S. Wollman, On the approximation of the Vlasov-Poisson system by particle methods, SIAM J. Numer. Anal., 37 (2000), 1369-1398 (electronic). doi: 10.1137/S0036142999298528.

[159]

H. Xia, H. Wang and Z. Xuan, Opinion dynamics: A multidisciplinary review and perspective on future research, International Journal of Knowledge and Systems Science (IJKSS), 2 (2011), 72-91. doi: 10.4018/jkss.2011100106.

[160]

Z. Xia, The existence of noncollision singularities in Newtonian systems, Ann. of Math. (2), 135 (1992), 411-468. doi: 10.2307/2946572.

[161]

V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., 2 (1995), 27-38. doi: 10.4310/MRL.1995.v2.n1.a4.

[162]

J. Yvon, La théorie statistique des fluides et l'équation d'état (in french), Actual. Sci. Indust., 203, 1935.

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