Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations

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 ≥ 2, we show that for any attractive potential the steady state is unique for a fixed mass. In the case 1 < m < 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.

Equilibration of aggregation-diffusion equations with weak interaction forces

“…where ρ ∞,1/2 denotes the unique steady state determined by the same W and m with total mass 1/2 (whose existence and uniqueness are guaranteed by the results in [3,13,19] with the previous assumptions).…”

“…Proposition 3.7 gives a lower bound on the energy dissipation rate for ρ which is not radially-decreasing, i.e., ρ(t, •) = ρ # (t, •), but it is useless for radially-decreasing distributions. To deal with this difficulty, we first give a quantitative version of Theorem 2.6 of [19] (in its 1D version): Proposition 3.8. If ρ(t0, x) is radially-decreasing at some t0 and suppρ(t0,…”

“…We first introduce the h(s)-representation of ρ(x), which can be viewed as a re-parametrization of the h-representation we introduced before. We start by defining the h(s) function for ρ(x): h[ρ](s) is defined by min{ρ(x), h(s)} dx = s, 0 ≤ s ≤ 1, (6.5) which coincide with the definition in [19] if ρ is radially decreasing. Then we define 7…”

“…• [19] shows the uniqueness of steady state for a general class of interaction potentials in the case m ≥ 2, and non-uniqueness in the case 1 < m < 2 for some potentials.…”

“…Based on the tightness, we also manage to obtain an explicit equilibration rate, by refining the energy dissipation rate estimates in [13] and [19], and connecting them via a perturbative argument. Although the current rate we obtain is algebraic, these estimates could give an exponential rate, provided a stronger result on tightness (for example, a uniform-in-time bound on the size of support of ρ).…”