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.
@article{AIHPB_2015__51_3_965_0, author = {Godinho, David and Qui\~ninao, Cristobal}, title = {Propagation of chaos for a subcritical {Keller{\textendash}Segel} model}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {965--992}, publisher = {Gauthier-Villars}, volume = {51}, number = {3}, year = {2015}, doi = {10.1214/14-AIHP606}, mrnumber = {3365970}, zbl = {1342.65234}, language = {en}, url = {http://www.numdam.org/articles/10.1214/14-AIHP606/} }
TY - JOUR AU - Godinho, David AU - Quiñinao, Cristobal TI - Propagation of chaos for a subcritical Keller–Segel model JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 965 EP - 992 VL - 51 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/14-AIHP606/ DO - 10.1214/14-AIHP606 LA - en ID - AIHPB_2015__51_3_965_0 ER -
%0 Journal Article %A Godinho, David %A Quiñinao, Cristobal %T Propagation of chaos for a subcritical Keller–Segel model %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 965-992 %V 51 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/14-AIHP606/ %R 10.1214/14-AIHP606 %G en %F 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) no. 3, pp. 965-992. doi : 10.1214/14-AIHP606. http://www.numdam.org/articles/10.1214/14-AIHP606/
[1] Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21 (4) (1993) 2246–2268. | MR | Zbl
and .[2] Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, 1983. | MR | Zbl
.[3] Convergence in and in 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 | Zbl
.[4] Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions. Electron. J. Differential Equations 44 (2006) 1–32. | MR | Zbl
, and .[5] Blow-up dynamics of self-attracting diffusive particles driven by competing convexities. Discrete Contin. Dyn. Syst. Ser. B 18 (8) (2013) 2029–2050. | MR | Zbl
and .[6] Entropy and chaos in the Kac model. Kinet. Relat. Models 3 (1) (2010) 85–122. | MR | Zbl
, , , and .[7] Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (3) (1989) 511–547. | MR | Zbl
and .[8] Propagation of chaos for the 2D viscous vortex model. Available at arXiv:1212.1437. | DOI | MR | Zbl
, and .[9] Stochastic particle approximation to the global measure valued solutions of the Keller–Segel model in 2D. J. Stat. Phys. 135 (2009) 133–151. | DOI | MR | Zbl
and .[10] 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. | DOI | MR | Zbl
and .[11] On Kac’s chaos and related problems. Available at arXiv:1205.4518. | DOI | MR | Zbl
and .[12] From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105 (3) (2003) 103–165. | MR | Zbl
.[13] From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein. 106 (2) (2004) 51–69. | MR | Zbl
.[14] 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 | Zbl
.[15] Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26 (1970) 399–415. | DOI | MR | Zbl
and .[16] A model for chemotaxis. J. Theoret. Biol. 30 (1971) 225–234. | DOI | Zbl
and .[17] A stochastic cellular automaton, modeling gliding and aggregation of myxobacteria. SIAM J. Appl. Math. 61 (2000) 172–182. | DOI | MR | Zbl
.[18] The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61 (2000) 183–212. | DOI | MR | Zbl
.[19] 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 | Zbl
.[20] On the existence and uniqueness of SDE describing an -particle system interacting via a singular potential. Proc. Japan Acad. Ser. A Math. Sci. 61 (9) (1985) 287–290. | MR | Zbl
.[21] Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI, 2003. | DOI | MR | Zbl
.Cité par Sources :