  A simple proof of the existence of solutions for the two-dimensional
Keller-Segel model with measures with all the atoms less than $8\pi$ as the
initial data is given. This result has been obtained by Senba--Suzuki and
Bedrossian--Masmoudi using different arguments. Moreover, we show a uniform
bound for the existence time of solutions as well as an optimal
hypercontractivity estimate.

Analysis of PDEs