{"paper":{"title":"Illposedness of incompressible fluids in supercritical Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaoyutao Luo","submitted_at":"2024-04-11T14:56:17Z","abstract_excerpt":"We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any $0 < s < \\frac{5}{2} $, we construct a $C^\\infty_c$ initial velocity field with arbitrarily small $H^{s}$ norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large $\\dot{H}^{s}$ norm inflation almost instantaneously. In the viscous case, the same $\\dot{H}^{s}$ norm inflation occurs in the 3D Navier-Stokes equation for $0< s < \\frac{1}{2} $, where $s = \\frac{1}{2}$ is scaling critical for this equation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.07813","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/2404.07813/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"}