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