Germany
author:
- Marco Di Francesco
- Massimo Fornasier
- 'Jan-Christian Hütter'
- Daniel Matthes
bibliography:
- 'BibKineticDitheringGF.bib'
title: Asymptotic Behavior of Gradient Flows Driven by Nonlocal Power Repulsion and Attraction Potentials in One Dimension
---

Introduction {#cha:introduction}
============

Setup and results
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We study existence, uniqueness and long-time behavior of non-negative weak solutions $\mu=\mu(t)$ of the PDE $$\label{eq:501}
  \partial_t\mu = \nabla \cdot \left[ \left(\nabla \psi_a \ast \omega - \nabla \psi_r \ast \mu \right)
    \mu\right], \quad \mu(0) = \mu_0,$$ in one space dimension. The functions $\psi_a,\psi_r : {\mathbb{R}}\rightarrow \mathbb{R}$, which are given by $$\begin{aligned}
  \label{eq:1}
  \psi_a(x) = |x|^{q_a}, \quad \psi_r(x)= |x|^{q_r},\quad \text{with parameters $q_a,\,q_r\in[1,2]$},\end{aligned}$$ represent the *attraction* and *repulsion kernels*, respectively, and $\omega\in L^\infty({\mathbb{R}})$ is a prescribed *reference profile* of compact support. Formally, equation is the gradient flow of the energy functional $$\label{contenergy}
  \mathcal{E}[\mu] = - \frac{1}{2} \int_{\mathbb R^d}\int_{\mathbb R^d} \psi_r(x-y)  {\mathop{}\!\mathrm{d}}\mu(x) {\mathop{}\!\mathrm{d}}\mu(y)  + \int_{\mathbb R^d} \int_{\mathbb R^d} \psi_a(x-y) {\mathop{}\!\mathrm{d}}\omega(x) {\mathop{}\!\mathrm{d}}\mu(y)$$ with respect to the $L^2$-Wasserstein metric, for $d=1$. A specific motivation to study $\mathcal E$ and its flow is given further below. Although the flow could be considered in the general framework of measure solutions $\mu$, we limit our analysis to absolutely continuous measures $\mu$ with bounded density of 