Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data

TL;DR

This paper provides a straightforward proof of the existence of solutions for the 2D Keller-Segel chemotaxis model with initial measures having atoms less than 8π, including bounds on existence time and hypercontractivity.

Contribution

It offers a new simple proof for the existence of solutions with measure initial data and establishes uniform bounds and optimal hypercontractivity estimates.

Findings

Solutions exist for initial measures with atoms less than 8π.

Uniform bounds on the existence time are established.

An optimal hypercontractivity estimate is derived.

Abstract

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.