Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

Abstract

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as t → ∞.

Phase Transitions and Bump Solutions of the Keller--Segel Model with Volume Exclusion

“…In the whole space case, the existence of compactly supported radially decreasing global minimizers of the energy was proven in [17]. Taking advantage of this variational structure, the authors in [21] were able to show that all stationary states in the whole space must be radially decreasing and compactly supported about their center of mass. In short, in two dimensions for the classical Keller-Segel model, all stationary states with the right regularity in [21] are given by single bumps.…”

“…Taking advantage of this variational structure, the authors in [21] were able to show that all stationary states in the whole space must be radially decreasing and compactly supported about their center of mass. In short, in two dimensions for the classical Keller-Segel model, all stationary states with the right regularity in [21] are given by single bumps. This property generalizes to any case in which the uniqueness of radial stationary solutions is proven.…”

“…Moreover, it is easy to check that the distance between bumps in the multi bump solutions with more than one bump without boundary spikes diverges as L → ∞. The conclusion is that the only integrable steady states remaining in the limit are the single bump solutions as shown in [21]. In the bottom, we plot the energy and mass of each spike as L 0 varies.…”

Equilibria of an aggregation model with linear diffusion in domains with boundaries

“…The two modes of diffusion (linear vs. nonlinear) result in different features of equilibria/minimizers of the associated energy. In particular, nonlinear diffusion models admit compactly supported equilibria [11,10,15], in contrast with equilibria for linear diffusion which can only have full support within the domain (see Section 3 of the present paper).…”

“…We also mention that there has been extensive work on aggregation models with repulsive effects modelled by nonlinear diffusion (see for instance [4,15] and references therein). The two modes of diffusion (linear vs. nonlinear) result in different features of equilibria/minimizers of the associated energy.…”