Propagation of chaos for some 2 dimensional fractional Keller Segel equation in dominated diffusion and fair competition cases

Date(s) : 03/07/2018   iCal
11 h 00 min - 12 h 00 min

In this work we deal with the local in time propagation of chaos without cut-off for some two dimensional fractional Keller Segel models. More precisely the diffusion considered here is given by the fractional Laplacian operator −(−∆) ^(a/2) with a ∈ (1, 2) and the singularity of the interaction is of order |x|^(1−α) with α ∈]1, a]. In the case α ∈ (1, a) we prove a complete propagation of chaos result, proving the Γ-l.s.c property of the fractional Fisher information, already known for the classical Fisher information, using a result of [4]. In the fair competition case ([1]) a = α, we only prove a convergence/consistency result in a sub-critical mass regime, similarly as the result obtained for the classical Keller-Segel equation in [2].

[1] J. Carrillo, V. Calvez, F. Hoffmann. Equilibria of homogeneous functionals in the fair-competition regime Nonlinear Analysis, (2016).
[2] N. Fournier, B. Jourdain. Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes. Accepted at Ann. Appl. Probab.
[3] 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.
[4] M. Hauray, S. Mischler. On Kac’s chaos and related problems, J. Funct. Anal., Volume 266, P. 60556157,(2014).

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