Ground states in the diffusion-dominated regime

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 non-increasing 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 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.

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

“…Carrillo, Castorina, and Volzone extended the result to the Newtonian potential in two dimensions [50]. For kernels more singular than Newtonian, recent work by Carrillo, Hoffmann, Mainini, and Volzone showed existence of global minimizers for power law kernels W k with −d < k < 2 − d. Furthermore, the same work proved that, for all power law kernels −d < k ≤ 0, global minimizers are compactly supported [64]. If W is a integrable, bounded, and purely attractive kernel, Bedrossian showed that, for all m > 2 and any mass M, there exists a global minimizer of (5) [9].…”

“…In the particular case of the Newtonian interaction potential, uniqueness of steady states among radial functions was first shown by Lieb and Yau [115], using the specific structure of the Newtonian kernel to obtain an explicit ordinary differential equation for the mass distribution function of solutions. For general power-law kernels in the diffusion-dominated regime, uniqueness of steady states is known when d = 1 [64] but remains unclear for higher dimensions, aside from the Newtonian case. For integrable, purely attractive kernels, Kaib and Burger, Di Francesco, and Franek proved uniqueness of steady states at the critical power m = 2 when W dx < −2 using the Krein-Rutman theorem, under the additional assumptions that W is smooth and W (r) vanishes only at 0 [35,108].…”

Existence of ground states for aggregation-diffusion equations

“…Remark 5.2. The previous is result is a particular case of [41, Theorem II.1]; see also [27] for more comments. As a main difference, our proof exploits the extra rigidity stemming from the radially decreasing rearrangement, bypassing the need of exploiting the subadditivity of the energy of minimizers at different mass.…”

Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

“…Both the aggregation-diffusion equation and constrained aggregation equation have gradient flow structures with respect to the 2-Wasserstein metric. The aggregation-diffusion equation is formally the gradient flow of the sum of an interaction energy and Rényi entropy Over the past fifteen years, there has been significant work on aggregation-diffusion equations, analyzing dynamics of solutions, asymptotic behavior, and minimizers of the energy E m [8,9,12,17,19,22,26,28,29,31,34,37,49,70,71]. The vast majority of the literature has considered one of two choices of interaction potential: either purely attractive power-laws or repulsive-attractive power-laws, K(x) = |x| p /p or K(x) = |x| q /q − |x| p /p for 2 − d p < q 2, q > 0, (1.3) with the convention that |x| 0 /0 = log(|x|).…”

“…In the case of an attractive power-law interaction potential, K(x) = |x| p /p for −d < p < 0, hypothesis (LSC) is equivalent to the requirement that we are in the diffusion dominated regime, m 0 > 1−p/d (cf. [10,31,67]). More generally, for repulsive-attractive power-law interaction potentials K(x) = |x| q /q − |x| p /p with −d < p < q, hypothesis (LSC) merely requires that m 0 > 1.…”