Propagation of chaos for a subcritical Keller–Segel model
Godinho, David ; Quiñinao, Cristobal
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 965-992 / Harvested from Numdam
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Cet article traite de l’équation de Keller–Segel dans un cadre sous-critique. À l’aide du système de particules en lien avec cette équation, nous montrons des résultats d’existence et d’unicité, puis la propagation du chaos pour ce dernier. Plus précisément, nous montrons que la mesure empirique du système tend vers l’unique solution de l’équation limite lorsque le nombre de particules tend vers l’infini.

This paper deals with a subcritical Keller–Segel equation. Starting from the stochastic particle system associated with it, we show well-posedness results and the propagation of chaos property. More precisely, we show that the empirical measure of the system tends towards the unique solution of the limit equation as the number of particles goes to infinity.

Publié le : 2015-01-01
DOI : https://doi.org/10.1214/14-AIHP606 
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     author = {Godinho, David and Qui\~ninao, Cristobal},
     title = {Propagation of chaos for a subcritical Keller--Segel model},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {51},
     year = {2015},
     pages = {965-992},
     doi = {10.1214/14-AIHP606},
     mrnumber = {3365970},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2015__51_3_965_0}
}

Godinho, David; Quiñinao, Cristobal. Propagation of chaos for a subcritical Keller–Segel model. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) pp. 965-992. doi : 10.1214/14-AIHP606. http://gdmltest.u-ga.fr/item/AIHPB_2015__51_3_965_0/

[1] A. Bhatt and R. Karandikar. Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21 (4) (1993) 2246–2268. | MR 1245309 | Zbl 0790.60062

[2] H. Brezis. Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, 1983. | MR 697382 | Zbl 0511.46001

[3] H. Brezis. Convergence in D ' and in L 1 under strict convexity. In Boundary Value Problems for Partial Differential Equations and Applications. RMA Res. Notes Appl. Math. 29 43–52. Masson, Paris, 1993. | MR 1260437 | Zbl 0813.49016

[4] A. Blanchet, J. Dolbeault and B. Perthame. Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions. Electron. J. Differential Equations 44 (2006) 1–32. | MR 2226917 | Zbl 1112.35023

[5] V. Calvez and L. Corrias. Blow-up dynamics of self-attracting diffusive particles driven by competing convexities. Discrete Contin. Dyn. Syst. Ser. B 18 (8) (2013) 2029–2050. | MR 3082309 | Zbl 1288.35110

[6] E. Carlen, M. Carvalho, J. Le Roux, M. Loss and C. Villani. Entropy and chaos in the Kac model. Kinet. Relat. Models 3 (1) (2010) 85–122. | MR 2580955 | Zbl 1186.76675

[7] R. Diperna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (3) (1989) 511–547. | MR 1022305 | Zbl 0696.34049

[8] N. Fournier, M. Hauray and S. Mischler. Propagation of chaos for the 2D viscous vortex model. Available at arXiv:1212.1437. | MR 3254330 | Zbl 1299.76040

[9] J. Haskovec and C. Schmeiser. Stochastic particle approximation to the global measure valued solutions of the Keller–Segel model in 2D. J. Stat. Phys. 135 (2009) 133–151. | MR 2505729 | Zbl 1173.82021

[10] J. Haskovec and C. Schmeiser. Convergence analysis of a stochastic particle approximation for measure valued solutions of the 2D Keller–Segel system. Comm. Partial Differential Equations 36 (2011) 940–960. | MR 2765424 | Zbl 1229.35111

[11] M. Hauray and S. Mischler. On Kac’s chaos and related problems. Available at arXiv:1205.4518. | MR 3188710 | Zbl 06326909

[12] D. Horstmann. From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105 (3) (2003) 103–165. | MR 2013508 | Zbl 1071.35001

[13] D. Horstmann. From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein. 106 (2) (2004) 51–69. | MR 2073515 | Zbl 1072.35007

[14] M. Kac. Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 1954–1955 III 171–197. Univ. California Press, Berkeley and Los Angeles, 1956. | MR 84985 | Zbl 0072.42802

[15] E. F. Keller and L. A. Segel. Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26 (1970) 399–415. | Zbl 1170.92306

[16] E. F. Keller and L. A. Segel. A model for chemotaxis. J. Theoret. Biol. 30 (1971) 225–234. | Zbl 1170.92307

[17] A. Stevens. A stochastic cellular automaton, modeling gliding and aggregation of myxobacteria. SIAM J. Appl. Math. 61 (2000) 172–182. | MR 1776392 | Zbl 0992.92005

[18] A. Stevens. The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61 (2000) 183–212. | MR 1776393 | Zbl 0963.60093

[19] A. S. Sznitman. Topics in propagation of chaos. In École d’Été de Probabilités de Saint-Flour XIX – 1989. Lecture Notes in Math. 1464. 165–251. Springer, Berlin, 1991. | MR 1108185 | Zbl 0732.60114

[20] S. Takanobu. On the existence and uniqueness of SDE describing an n-particle system interacting via a singular potential. Proc. Japan Acad. Ser. A Math. Sci. 61 (9) (1985) 287–290. | MR 831903 | Zbl 0579.60049

[21] C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI, 2003. | MR 1964483 | Zbl 1106.90001