# On Patlak-Keller-Segel system for several populations: a gradient flow approach

## TL;DR

This paper investigates the global existence of solutions for a multi-species Patlak-Keller-Segel system using a gradient flow framework in Wasserstein space, establishing conditions for solutions and their energy dissipation.

## Contribution

It introduces a gradient flow approach in Wasserstein space to prove global existence of solutions for multi-species Patlak-Keller-Segel systems under sub-critical initial mass conditions.

## Key findings

- Solutions exist globally under sub-critical mass conditions
- The system satisfies an energy dissipation inequality
- Gradient flow structure aids in analyzing solution existence

## Abstract

We study the global in time existence of solutions to the parabolic-elliptic Patlak-Keller-Segel system of multi-species populations. We prove that if the initial mass satisfies an appropriate notion of sub-criticality, then the system has a solution defined for all time. We explore the gradient flow structure in the Wasserstein space to study the question of existence. Moreover, we show that the obtained solution satisfies energy dissipation inequality.