The nonlocal attraction-repulsion transport equation with power kernels
Abstract
We study a nonlocal continuity equation on ℝd 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 ψa(x) = |x|1+a and ψr(x) = |x|1+r with exponents a, r ∈ [0,1). We establish global Lagrangian well-posedness via a squared-radius regularization, obtaining {finite-time L∞ and moment bounds}, Wn,∞ regularity, and uniqueness in the Lagrangian class. When the initial data is compactly supported and attraction dominates (a > r, or a = r with ω(ℝd) > 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 ≤ r ≤ 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 allows us to exhibit explicit examples of stationary measures in dimensions d ∈ {1,2,3}. Numerical particle simulations {are consistent} with the theoretical stationary profiles. Finally, for 0 ≤ r ≤ a < 1, we prove that every global solution whose energy is bounded from below and whose moments are bounded uniformly in time converges, along a sequence of times, to a zero-flux stationary state. Full narrow convergence follows whenever the omega-limit set contains a single stationary state.
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BibTeX
@misc{fornasier2026nonlocal,
title = {{The nonlocal attraction-repulsion transport equation with power kernels}},
author = {Massimo Fornasier and Hui Huang and Lukang Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.04424},
eprint = {2607.04424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04424},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The nonlocal attraction-repulsion transport equation with power kernels
AU - Massimo Fornasier
AU - Hui Huang
AU - Lukang Sun
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2607.04424
UR - https://arxiv.org/abs/2607.04424
AB - We study a nonlocal continuity equation on 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 ∈ [0,1). We establish global Lagrangian well-posedness via a squared-radius regularization, obtaining {finite-time L^∞ and moment bounds}, W^n,∞ regularity, and uniqueness in the Lagrangian class. When the initial data is compactly supported and attraction dominates (a > r, or a = r with ω(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 ≤ r ≤ 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 allows us to exhibit explicit examples of stationary measures in dimensions d ∈ {1,2,3}. Numerical particle simulations {are consistent} with the theoretical stationary profiles. Finally, for 0 ≤ r ≤ a < 1, we prove that every global solution whose energy is bounded from below and whose moments are bounded uniformly in time converges, along a sequence of times, to a zero-flux stationary state. Full narrow convergence follows whenever the omega-limit set contains a single stationary state.
ER -