The nonlocal attraction-repulsion transport equation with power kernels

Abstract

We study a nonlocal continuity equation on $\mathbb{R}^d$ in which a probability density is driven by the competition between attraction toward a prescribed background measure $ω$ and self-repulsion among particles, governed respectively by the power-law kernels $ψ_a(x) = |x|^{1+a}$ and $ψ_r(x) = |x|^{1+r}$ with exponents $a, r \in [0,1)$. We establish global Lagrangian well-posedness via a squared-radius regularization, obtaining uniform $L^\infty$ and moment bounds, $W^{n,\infty}$ regularity, and uniqueness in the Lagrangian class. When the initial data is compactly supported and attraction dominates ($a > r$, or $a = r$ with $ω(\mathbb{R}^d) > 1$), we prove that the support remains uniformly bounded at all time; a counterexample shows this fails for $a = r > 1$. For the attractive-dominant nonquadratic range $0 \leq r \leq a < 1$, we characterize zero-flux stationary states via a free-boundary problem involving a fractional Laplacian operator, reducing the stationarity condition to a fractional exterior Dirichlet problem. This characterization allow us to exhibit explicit examples of stationary measures in dimensions $d \in \{1,2,3\}$. Numerical particle simulations confirm agreement with the theoretical stationary profiles. Finally, we prove that every global solution with bounded energy and uniform moment bounds converges to a zero-flux stationary state.

Disclosure

“of the results in the MMD case a = r and ω(Rd ) = 1. For different and more specific results on the analysis of the MMD gradient flow for a = r, we refer to the work in progress by Rosenzweig, Slepčev, and Wang [26]. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process. During the preparation of this work the authors used ChaGPT in order to 1. search for references, 2. check for spelling and 3. generate the software to perform the numeric”

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2

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2026-08-24T00:34:05.733Z

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