{"paper":{"title":"Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Anuj Kumar, Elia Bru\\`e, Maria Colombo","submitted_at":"2024-05-02T18:46:58Z","abstract_excerpt":"Given a divergence-free vector field ${\\bf u} \\in L^\\infty_t W^{1,p}_x(\\mathbb R^d)$ and a nonnegative initial datum $\\rho_0 \\in L^r$, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of $L^\\infty_t L^r_x$ densities for $\\frac{1}{p} + \\frac{1}{r} \\leq 1$. This range was later improved in [BCDL21] to $\\frac{1}{p} + \\frac{d-1}{dr} \\leq 1$. We prove that this range is sharp by providing a counterexample to uniqueness when $\\frac{1}{p} + \\frac{d-1}{dr} > 1$.\n  To this end, we introduce a novel flow mechanism. It is not based on convex integration, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.01670","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.01670/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"}