[Paper Review] Equilibria of homogeneous functionals in the fair-competition regime

Vincent Calvez, José A. Carrillo+1 more|arXiv (Cornell University)|Oct 4, 2016

Mathematical Biology Tumor Growth6 references4 citations

TL;DR

This paper investigates equilibria of homogeneous functionals in the fair-competition regime, where repulsion via nonlinear power-law diffusion and attraction via a homogeneous kernel scale identically under dilations. It establishes that in the singular kernel case, global equilibria exist only at a critical parameter value and are optimizers of a modified HLS inequality, while in the smooth kernel case, no radially symmetric non-increasing stationary solutions exist, but positive self-similar solutions emerge when diffusion is not too strong.

ABSTRACT

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular/smooth kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse the regime in which both homogeneities scale the same with respect to dilations, that we coin as fair-competition. In the singular kernel case, we show that existence of global equilibria can only happen at a certain critical value and they are characterised as optimisers of a variant of HLS inequalities. We also study the existence of self-similar solutions for the sub-critical case, or equivalently of optimisers of rescaled free energies. These optimisers are shown to be compactly supported radially symmetric and non-increasing stationary solutions of the non-linear Keller-Segel equation. On the other hand, we show that no radially symmetric non-increasing stationary solutions exist in the smooth kernel case, implying that there is no criticality. However, we show the existence of positive self-similar solutions for all values of the parameter under the condition that diffusion is not too fast. We finally illustrate some of the open problems in the smooth kernel case by numerical experiments.

Motivation & Objective

• To analyze the existence and structure of equilibria for a class of non-local, non-convex free energy functionals modeling particle systems with competing diffusion and attraction.
• To identify the fair-competition regime where the homogeneities of diffusion and interaction scale identically under dilations.
• To determine whether minimizers of the free energy functional exist and characterize their properties, particularly radial symmetry and compact support.
• To investigate the existence of self-similar solutions and stationary states in both singular and smooth kernel cases.
• To clarify the role of criticality in the singular kernel case and the absence of radial symmetry in the smooth kernel case.

Proposed method

• Formalism of a free energy functional combining nonlinear diffusion and non-local interaction terms, parameterized by diffusion exponent $ m $ and interaction homogeneity $ k $.
• Analysis of the functional's critical points via the Euler-Lagrange equation derived from the first variation $ \mathcal{T}_{m,k}[ ho] $, leading to a stationary non-linear PDE.
• Use of variational methods and sharp inequalities (e.g., modified Hardy-Littlewood-Sobolev inequalities) to characterize minimizers in the singular kernel case.
• Asymptotic expansion techniques and hypergeometric function analysis to study the behavior of the kernel and its gradient near the origin in the singular regime.
• Application of gradient flow structure in the Wasserstein metric to connect the PDE dynamics to the free energy decay and equilibrium selection.
• Numerical experiments to illustrate open problems in the smooth kernel case, particularly regarding non-radial or non-compact solutions.

Experimental results

Research questions

• RQ1 Under what conditions does the free energy functional admit global minimizers in the fair-competition regime?
• RQ2 What characterizes the unique critical parameter value at which equilibria exist in the singular kernel case?
• RQ3 Why do no radially symmetric, non-increasing stationary solutions exist in the smooth kernel case despite the presence of self-similar dynamics?
• RQ4 Can positive self-similar solutions be constructed for all parameter values when diffusion is not too strong in the smooth kernel case?
• RQ5 What are the structural and qualitative differences between equilibria in the singular and smooth kernel regimes?

Key findings

• In the singular kernel case ($ -N < k < 2-N $), global equilibria exist only at a specific critical value of the interaction strength $ \chi $, and they are optimizers of a modified Hardy-Littlewood-Sobolev inequality.
• These minimizers are compactly supported, radially symmetric, and non-increasing, with a sharp threshold for existence at critical $ \chi $.
• In the sub-critical regime of the singular kernel, self-similar solutions exist and are characterized as optimizers of rescaled free energies.
• For the smooth kernel case ($ k > 0 $), no radially symmetric non-increasing stationary solutions exist, indicating absence of criticality.
• Positive self-similar solutions exist for all parameter values in the smooth kernel case as long as the diffusion exponent $ m $ is not too large (i.e., diffusion is not too fast).
• Numerical experiments suggest complex, non-radial behavior in the smooth kernel case, indicating open problems in the classification of stationary states.