[Paper Review] Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations

Matías G. Delgadino, Xukai Yan+1 more|arXiv (Cornell University)|Aug 26, 2019

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TL;DR

This paper establishes a sharp threshold for uniqueness of steady states in aggregation-diffusion equations with degenerate diffusion. For diffusion exponent $ m \geq 2 $, steady states are unique for any given mass; for $ 1 < m < 2 $, infinitely many radially decreasing steady states exist for certain smooth attractive potentials, resolving an open problem via a novel interpolation curve that proves convexity of interaction energy.

ABSTRACT

We consider a nonlocal aggregation equation with degenerate diffusion, which describes the mean-field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within the radial class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/non-uniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being $m = 2$. In the case $m \ge 2$, we show that for any attractive potential the steady state is unique for a fixed mass. In the case $1 &lt; m &lt; 2$, we construct examples of smooth attractive potentials, such that there are infinitely many radially decreasing steady states of the same mass. For the uniqueness proof, we develop a novel interpolation curve between two radially decreasing densities, and the key step is to show that the interaction energy is convex along this curve for any attractive interaction potential, which is of independent interest.

Motivation & Objective

• To resolve the open question of uniqueness of radially decreasing steady states for general attractive interaction potentials in aggregation-diffusion equations with degenerate diffusion.
• To determine the critical role of the diffusion exponent $ m $ in governing uniqueness or non-uniqueness of steady states for fixed mass.
• To develop a new interpolation curve between radially decreasing densities to prove convexity of the interaction energy, a result of independent interest.
• To construct explicit counterexamples showing infinitely many steady states when $ 1 < m < 2 $, even for smooth attractive potentials.

Proposed method

• Introduce a novel interpolation curve between two radially decreasing probability densities, parameterized by a scalar $ s \in [0,1] $, to analyze energy evolution.
• Prove that the interaction energy is convex along this curve for any attractive potential, using a new integral representation of the fractional Laplacian and regularity estimates.
• Apply the convexity of interaction energy to derive a necessary condition for minimizers, linking it to the diffusion exponent $ m $.
• Construct a sequence of smooth, radially symmetric, attractive potentials $ \widetilde{W} $ with $ W \sim |x|^k $ near origin, using a perturbation argument based on the fractional regularity lemma.
• Use the concentration-compactness principle and variational methods to establish existence of steady states, then apply the interpolation method to analyze their uniqueness.
• Leverage the fractional regularity lemma to control the behavior of $ (-\Delta)^s(W\phi) $ in $ L^1 $, ensuring the required regularity for the energy convexity argument.

Experimental results

Research questions

• RQ1 Is the steady state unique for a given mass when the diffusion exponent $ m \geq 2 $, for any smooth, attractive potential?
• RQ2 Can non-uniqueness of radially decreasing steady states occur when $ 1 < m < 2 $, even for smooth attractive potentials?
• RQ3 Is the interaction energy convex along a new interpolation curve connecting two radially decreasing densities, and does this convexity hold for all attractive potentials?
• RQ4 What is the critical value of $ m $ that separates uniqueness from non-uniqueness of steady states in the aggregation-diffusion equation?

Key findings

• For $ m \geq 2 $, the steady state is unique among radially decreasing densities for any fixed mass and any smooth, attractive potential.
• For $ 1 < m < 2 $, there exist smooth, radially symmetric, attractive potentials such that infinitely many radially decreasing steady states exist with the same mass.
• The interaction energy is convex along a novel interpolation curve between two radially decreasing densities, a result that holds for any attractive potential and is of independent mathematical interest.
• The critical exponent $ m = 2 $ separates the regimes of uniqueness ($ m \geq 2 $) and non-uniqueness ($ 1 < m < 2 $), resolving a long-standing open problem in the field.
• The construction of non-uniqueness relies on a perturbation of a potential $ W \sim |x|^k $ near the origin, with $ k \in (-n, 0) $, and the use of a fractional regularity lemma to control the behavior of the fractional Laplacian of $ W\phi $.
• The proof of convexity of interaction energy along the interpolation curve is based on a refined analysis of the fractional Laplacian and integral estimates involving $ H(t) $, leading to $ L^1 $ bounds on $ (-\Delta)^s f $ for $ s < \min\{\frac{n+k}{2}, 1\} $.