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

Abstract

In this work we deal with the local in time propagation of chaos without cut-off for some two dimensional fractional Keller Segel equations. More precisely the diffusion considered here is given by the fractional Laplacian operator $-(-\Delta {)}^{\frac{a}{2}}$ with $a\in (1,2)$ and the singularity of the interaction is of order $|x{|}^{1-\alpha }$ with $\alpha \in ]1,a]$. In the case $\alpha \in (1,a)$ we give a complete propagation of chaos result, proving the $\Gamma$-l.s.c property of the fractional Fisher information, already known for the classical Fisher information, using a result of Mischler and Hauray. In the fair competition case $a=\alpha$, we only prove a convergence/consistency result in a sub-critical mass regime, similarly as the result obtained for the classical Keller-Segel equation.

References in corpus (2)

• Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli
• Fractional Fokker-Planck Equation with General Confinement Force

Cited by in corpus (3)

• Propagation of chaos for the VPFP equation with a polynomial cut-off
• p-Laplacian Keller-Segel Equation: Fair Competition and Diffusion Dominated Cases
• Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up