{"paper":{"title":"Instantaneous blowup of incompressible flow with passive tracer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Solutions to incompressible flow with a passive tracer can blow up instantaneously in both velocity and tracer at a finite time while remaining smooth at all other times.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mimi Dai, Xiaotong \"Dawson\" Yang","submitted_at":"2026-04-13T17:40:23Z","abstract_excerpt":"We construct a family of solutions $(u,b)$ of the incompressible flow with a passive tracer for which both $\\|u(t)\\|_{L^\\infty}$ and $\\|b(t)\\|_{L^\\infty}$ blow up at time $T_*$. Away from $T_*$, the solutions remain smooth in both space and time. The argument adapts the inverse cascade mechanism from [CDP25] to the presence of an advected scalar, but the passive component creates a new compatibility constraint: the iteration must propagate the tracer while preserving the same principal velocity profiles from one stage to the next. We resolve it by introducing a simultaneous decomposition lemma"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct a family of solutions (u,b) of the incompressible flow with a passive tracer for which both ||u(t)||_{L^∞} and ||b(t)||_{L^∞} blow up at time T_*. Away from T_*, the solutions remain smooth in both space and time.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The iteration must propagate the tracer while preserving the same principal velocity profiles from one stage to the next; this compatibility constraint is resolved by introducing a simultaneous decomposition lemma for a symmetric tensor and a vector field.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Solutions to incompressible flow with passive tracer are constructed that blow up instantaneously in L^∞ for both velocity and tracer at finite time T_* while remaining smooth before.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Solutions to incompressible flow with a passive tracer can blow up instantaneously in both velocity and tracer at a finite time while remaining smooth at all other times.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8445163a4fdf689bdd87c505d3c07ba331bde01ec014d335c50635bae23e925b"},"source":{"id":"2604.11769","kind":"arxiv","version":2},"verdict":{"id":"72972e64-20f5-48ca-82bc-8f4021c015ea","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:00:03.315960Z","strongest_claim":"We construct a family of solutions (u,b) of the incompressible flow with a passive tracer for which both ||u(t)||_{L^∞} and ||b(t)||_{L^∞} blow up at time T_*. Away from T_*, the solutions remain smooth in both space and time.","one_line_summary":"Solutions to incompressible flow with passive tracer are constructed that blow up instantaneously in L^∞ for both velocity and tracer at finite time T_* while remaining smooth before.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The iteration must propagate the tracer while preserving the same principal velocity profiles from one stage to the next; this compatibility constraint is resolved by introducing a simultaneous decomposition lemma for a symmetric tensor and a vector field.","pith_extraction_headline":"Solutions to incompressible flow with a passive tracer can blow up instantaneously in both velocity and tracer at a finite time while remaining smooth at all other times."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11769/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"}