Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

José A Carrillo,Rishabh S Gvalani

doi:10.1007/s00220-021-03977-4

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 rightarrow infty and the influence that the presence of a phase transition has on this limit.

Highlights

We deal with the properties of the set of stationary states and long-time asymptotics for a general class of nonlinear aggregation-diffusion equations of the form
Our main goal is to develop a theory for the stationary solutions and phase transitions of (1.1) for general interactions W ∈ C2( ) and nonlinear diffusion in the periodic setting, something that has not been previously studied in the literature
It is similar in spirit to the proof in [KZ18] where regularity was proved for a degenerate diffusion equation posed on Rd with a potentially singular drift term

Summary

Introduction

We deal with the properties of the set of stationary states and long-time asymptotics for a general class of nonlinear aggregation-diffusion equations of the form. In the linear diffusion case m = 1, we refer to [CP10,CGPS20] where quite a complete picture of the appearance of bifurcations and of continuous and discontinuous phase transitions is present, under suitable assumptions on the interaction potential W. Our main goal is to develop a theory for the stationary solutions and phase transitions of (1.1) for general interactions W ∈ C2( ) and nonlinear diffusion in the periodic setting, something that has not been previously studied in the literature. In Lemmas 5.15 and 5.16 Proposition 5.18, to provide sufficient conditions for the existence of continuous or discontinuous phase transitions, where the proofs rely critically on the Hölder regularity obtained in Sect.

Preliminaries and Notation

Characterisation of Stationary Solutions and Bifurcations

Minimisers of the Free Energy and Phase Transitions

Numerical Experiments

Proof of Hölder Regularity