# Ground States in the Diffusion-Dominated Regime

## TL;DR

This paper studies the behavior of particles under strong diffusion and attraction forces, proving the existence, symmetry, and regularity of stationary states in a Keller-Segel type model.

## Contribution

It establishes the existence and properties of global minimisers of the free energy in the diffusion-dominated regime, including symmetry, support, and regularity, and characterizes stationary states in one dimension.

## Key findings

- All stationary states are radially symmetric decreasing and compactly supported.
- Global minimisers exist, are bounded, smooth inside their support, and are radially symmetric.
- In one dimension, stationary states are uniquely characterized as optimisers of a specific functional inequality.

## Abstract

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric decreasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and $C^\infty$ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.