{"paper":{"title":"Asymptotic Behavior of Gradient Flows Driven by Nonlocal Power Repulsion and Attraction Potentials in One Dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniel Matthes, Jan-Christian H\\\"utter, Marco Di Francesco, Massimo Fornasier","submitted_at":"2014-01-10T14:13:50Z","abstract_excerpt":"We study the long time behavior of the Wasserstein gradient flow for an energy functional consisting of two components: particles are attracted to a fixed profile $\\omega$ by means of an interaction kernel $\\psi_a(z)=|z|^{q_a}$,and they repel each other by means of another kernel $\\psi_r(z)=|z|^{q_r}$. We focus on the case of one space dimension and assume that $1\\le q_r\\le q_a\\le 2$.\n  Our main result is that the flow converges to an equilibrium if either $q_r<q_a$ or $1\\le q_r=q_a\\le4/3$,and if the solution has the same (conserved) mass as the reference state $\\omega$. In the cases $q_r=1$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.2338","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}