# A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The two-dimensional case

## TL;DR

This paper introduces a stochastic McKean-Vlasov interpretation of the 2D Keller-Segel model, establishing conditions for well-posedness and global existence, and analyzing finite-time blow-up scenarios.

## Contribution

It extends previous stochastic interpretations to the two-dimensional case, providing new conditions for existence and blow-up analysis of the Keller-Segel system.

## Key findings

- Proved well-posedness of the McKean-Vlasov equation under certain constraints.
- Established global existence of solutions in the plane.
- Analyzed conditions leading to finite-time blow-up.

## Abstract

In Talay and Tomasevic [20] we proposed a new stochastic interpretation of the parabolic-parabolic Keller-Segel system without cutoff. It involved an original type of McKean-Vlasov interaction kernel which involved all the past time marginals of its probability distribution in a singular way. In the present paper, we study this McKean-Vlasov representation in the two-dimensional case. In this setting there exists a possibility of a blow-up in finite time for the Keller-Segel system if some parameters of the model are large. Indeed, we prove the well-posedness of the McKean-Vlasov equation under some constraints involving a parameter of the model and the initial datum. Under these constraints, we also prove the global existence for the Keller-Segel model in the plane. To obtain this result, we combine PDE analysis and stochastic analysis techniques.