{"paper":{"title":"Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"math.AP","authors_text":"Huaxiang L\\\"u, Michael R\\\"ockner, Xiangchan Zhu","submitted_at":"2025-04-23T13:16:04Z","abstract_excerpt":"We are concerned with the (stochastic) Lagrangian trajectories associated with Euler or Navier-Stokes equations. First, in the vanishing viscosity limit, we establish sharp non-uniqueness results for positive solutions to transport equations advected by weak solutions of the 3D Euler equations that exhibit kinetic energy dissipation with $C_{t,x}^{1/3-}$ regularity. As a corollary, in conjunction with the superposition principle, this yields the non-uniqueness of associated (deterministic) Lagrangian trajectories. Second, in dimension $d\\geq2$, for any $\\frac{1}{p}+\\frac{1}{r}>1$ or $p\\in(1,2)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.16687","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/2504.16687/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"}