This note summarizes some results provided in the papers [14, 17], concerning the study of the fractional Keller-Segel model. This diffusion aggregation equation arises in the modeling of the chemotaxis motion of bacteria. The diffusion part consists in a fractional Laplacian, and the aggregation kernel is up to the Newtonian one. In the case where the aggregation and diffusion are well balanced, we present how this model can be obtained from an interacting particle system. Then we present some results about well-posedness of the model when the diffusion is not overtaken by the aggregation, and finite time blow-up in the opposite case.
@article{SLSEDP_2018-2019____A3_0,
author = {Salem, Samir and Lafleche, Laurent},
title = {Fractional {Keller-Segel} equations},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:3},
pages = {1--11},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2018-2019},
doi = {10.5802/slsedp.126},
language = {en},
url = {http://www.numdam.org/articles/10.5802/slsedp.126/}
}
TY - JOUR AU - Salem, Samir AU - Lafleche, Laurent TI - Fractional Keller-Segel equations JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:3 PY - 2018-2019 SP - 1 EP - 11 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/articles/10.5802/slsedp.126/ DO - 10.5802/slsedp.126 LA - en ID - SLSEDP_2018-2019____A3_0 ER -
%0 Journal Article %A Salem, Samir %A Lafleche, Laurent %T Fractional Keller-Segel equations %J Séminaire Laurent Schwartz — EDP et applications %Z talk:3 %D 2018-2019 %P 1-11 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/articles/10.5802/slsedp.126/ %R 10.5802/slsedp.126 %G en %F SLSEDP_2018-2019____A3_0
Salem, Samir; Lafleche, Laurent. Fractional Keller-Segel equations. Séminaire Laurent Schwartz — EDP et applications (2018-2019), Exposé no. 3, 11 p. doi : 10.5802/slsedp.126. http://www.numdam.org/articles/10.5802/slsedp.126/
[1] D. Applebaum. Lévy processes and stochastic calculus, Cambridge studies in advanced mathematics 93, (2004). | DOI
[2] P. Biler, T. Cieslak, G. Karch, J. Zienkiewicz. Local criteria for blowup in two-dimensional chemotaxis models, Discrete and Continuous Dynamical Systems - Series A, Vol.37, 1841-1856, (2017). | Zbl
[3] A. Blanchet, J. Dolbeaut, B. Perthame. Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electronic Journal of Differential Equations, Texas State University, Department of Mathematics, 44, 32 pp, (2006). | Zbl
[4] V. Calvez, N. Bournaveas. The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, Vol. 23, (2010). | DOI | MR | Zbl
[5] V. Calvez, J. Carrillo, F. Hoffmann. Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Analysis, (2016). | DOI | MR | Zbl
[6] E. Cinlar. Probability and stochastics, Graduate Texts in Mathematics 261, Springer (2011). | DOI | Zbl
[7] G. Egana, S. Mischler. Uniqueness and long time asymptotic for the Keller-Segel equation: the parabolic-elliptic case, Arch. Ration. Mech. Anal. 220 (2016). | DOI | MR | Zbl
[8] C. Escudero. The fractional Keller-Segel model, Nonlinearity, Vol. 19, (2006). | DOI | MR | Zbl
[9] N. Fournier, M. Hauray, S. Mischler. Propagation of chaos for the 2D viscous vortex model, J. Eur. Math. Soc., Vol. 16, No 7, 1423-1466, 2014. | DOI | MR | Zbl
[10] N. Fournier, B. Jourdain. Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes. Accepted at Ann. Appl. Probab. | DOI | MR | Zbl
[11] D. Godinho, C. Quininao. Propagation of chaos for a subcritical Keller-Segel Model, Annales de l’Institut Henri Poincaré (2013). | DOI | MR | Zbl
[12] H. Huang, J.-G. Liu. Well posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos, Kinetic and Related Models (2016). | DOI | MR | Zbl
[13] M. Kwasnicki. Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal. 20(1) (2017): 7–51. | DOI | MR | Zbl
[14] L. Lafleche, S. Salem. Fractional Keller-Segel equation : global well posedness and finite time blow up, preprint, . | arXiv | DOI | Zbl
[15] G. Loeper, Uniqueness of the solution to the Vlasov–Poisson system with bounded density, Journal de Mathématiques Pures et Appliquées Volume 86, Issue 1, July 2006, Pages 68-79. | DOI | MR | Zbl
[16] H. Osada. Propagation of chaos for the two-dimensional Navier-Stokes equation, Probabilistic methods in mathematical physics, 303-334, (1987). | DOI | MR
[17] S. Salem. Propagation of chaos for some two dimensionl fractional Keller-Segel equations in diffusion dominated and fair competition cases, preprint, . | arXiv | DOI | MR | Zbl
[18] A.-S. Sznitman, Topics in propagation of chaos, In École d’Été de Probabilités de Saint-Flour XIX–-1989, Lecture Notes in Math., volume 1464, pages 165–251. Springer, Berlin, 1991. | DOI
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