Phase transitions for nonlinear nonlocal aggregation-diffusion equations

TL;DR

This paper investigates phase transitions in nonlinear nonlocal aggregation-diffusion equations with porous medium diffusion, analyzing bifurcations, free energy minimizers, and the nature of phase transitions depending on parameters and potentials.

Contribution

It provides a comprehensive analysis of bifurcations, existence and uniqueness of minimizers, and classification of phase transitions in these equations, including the limit as the diffusion exponent grows large.

Findings

Existence of multiple bifurcations from homogeneous states.

Conditions for existence and uniqueness of free energy minimizers.

Classification of phase transitions as continuous or discontinuous.

Abstract

We are interested in studying the stationary solutions and phase transitions of aggregation equations with degenerate diffusion of porous medium-type, with exponent $1 < m < \infty$. We first prove the existence of possibly infinitely many bifurcations from the spatially homogeneous steady state. We then focus our attention on the associated free energy proving existence of minimisers and even uniqueness for sufficiently weak interactions. In the absence of uniqueness, we show that the system exhibits phase transitions: we classify values of $m$ and interaction potentials $W$ for which these phase transitions are continuous or discontinuous. Finally, we comment on the limit $m \to \infty$ and the influence that the presence of a phase transition has on this limit.