{"paper":{"title":"Instantaneous blowup and non-uniqueness of smooth solutions of MHD","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Smooth solutions to the incompressible MHD equations exist whose L^∞ norm blows up instantaneously at the critical rate.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mimi Dai","submitted_at":"2026-04-09T18:16:17Z","abstract_excerpt":"We construct a family of solutions $(u,B)$ of the incompressible magnetohydrodynamic (MHD) system, the $L^\\infty$ norm of which blows up instantaneously at the critical rate. The solutions remain smooth except at the blowup time. An inverse energy cascade mechanism and a convex integration scheme along a time sequence are the main ingredients of the construction, inspired by our recent work [CDP25] for the Navier-Stokes equations. The challenge of the construction for the MHD system stems from the coupling and the necessity of preserving the same ansatz of the principal solution at every itera"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct a family of solutions (u,B) of the incompressible magnetohydrodynamic (MHD) system, the L^∞ norm of which blows up instantaneously at the critical rate. The solutions remain smooth except at the blowup time.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"A new coupled geometric lemma exists that simultaneously decomposes a symmetric tensor and a skew-symmetric tensor while preserving the same principal solution ansatz at every iterative step of the convex integration scheme.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Family of smooth incompressible MHD solutions constructed with instantaneous critical-rate L^∞ blowup via inverse energy cascade and a new coupled geometric lemma in convex integration.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Smooth solutions to the incompressible MHD equations exist whose L^∞ norm blows up instantaneously at the critical rate.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1a4b41bebf3bc1aa6b9a24601cf682bce920aa82dfe1a98c2aca22715195a941"},"source":{"id":"2604.08684","kind":"arxiv","version":2},"verdict":{"id":"43a76004-29d5-4575-88eb-ed005d0cab64","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T17:09:48.574333Z","strongest_claim":"We construct a family of solutions (u,B) of the incompressible magnetohydrodynamic (MHD) system, the L^∞ norm of which blows up instantaneously at the critical rate. The solutions remain smooth except at the blowup time.","one_line_summary":"Family of smooth incompressible MHD solutions constructed with instantaneous critical-rate L^∞ blowup via inverse energy cascade and a new coupled geometric lemma in convex integration.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"A new coupled geometric lemma exists that simultaneously decomposes a symmetric tensor and a skew-symmetric tensor while preserving the same principal solution ansatz at every iterative step of the convex integration scheme.","pith_extraction_headline":"Smooth solutions to the incompressible MHD equations exist whose L^∞ norm blows up instantaneously at the critical rate."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.08684/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"}