{"paper":{"title":"Convergence and non-convergence in a nonlocal gradient flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DS"],"primary_cat":"math.CA","authors_text":"Robert L. Pego, Sangmin Park","submitted_at":"2023-08-08T14:22:59Z","abstract_excerpt":"We study the asymptotic convergence of solutions as $t\\rightarrow\\infty$ of $\\partial_t u=-f(u)+\\int f(u)$, a nonlocal differential equation that is formally a gradient flow in a constant-mass subspace of $L^2$ arising from simplified models of phase transitions. In case the solution takes finitely many values, we provide a new proof of stabilization that uses a {\\L}ojasiewicz-type gradient inequality near a degenerate curve of equilibria. Solutions with infinitely many values in general need not converge to equilibrium, however, which we demonstrate by providing counterexamples for piecewise "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.04281","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2308.04281/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}