{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"04ca33bf9d2f1fad29bad8d0794e7b90a8c3397146222ddf85bf4d518b5e4856","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["2290b0aa355fa47c12e8a6e077e1e4bbbd490ffa210b224725ab25138e73fd8c","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"204a9b211072546a2557fa97918707a571eb07f9d6a5fcb5f6244264b07954bb","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.fluid-singularities.leray-hopf-uniqueness","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"Weak solutions of the Navier-Stokes equations, which satisfy the equations only in an averaged sense, always exist, but it is unknown whether they are unique when no outside force acts. With a specially designed force, two different solutions from the same start are known.","posed_since":"1934","precise":"3D Navier-Stokes on $R^{3}$ with $f = 0$ and divergence-free $u_{0}$ in $L^{2}$: is the Leray-Hopf weak solution ($u\\ \\text{in}\\ L^{\\infty}_{t} L^{2}_{x}$ and $L^{2}_{t} H^{1}_{x}$, satisfying the energy inequality) unique? Albritton, Brue and Colombo proved non-uniqueness for a force $f\\ \\text{in}\\ L^{1}_{t} L^{2}_{x}$ built from an unstable self-similar profile. Answer: yes or no with proof.","problem_ref":"2290b0aa355fa47c12e8a6e077e1e4bbbd490ffa210b224725ab25138e73fd8c","references":"","settled_by":"A proof of uniqueness for all $L^{2}$ data, or two distinct Leray-Hopf solutions from the same data with $f = 0$.","status_note":"Non-uniqueness with forcing was shown by Albritton, Brue and Colombo (Annals of Mathematics, 2022); the unforced case remains open.","title":"Uniqueness of Leray-Hopf weak solutions without forcing","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"abceb24c84ccdd5553100a537f535f296764b7a6e2f6653092600c0b1a7ce3e6","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"6902d187720f85638012f7e703c7b7b718b6c347b7f28842cc08d52fe39df6d0","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"XXseCJUthly9NMUx6WX3g--5dyGoJWJUae5Xeqe5N2X9Tz1IHvSYvWV_UuQWI5ae7yCd4TMudZkMZdIpqPFmAQ"},"schema":"pubphys.envelope/1"},"record_hash":"abceb24c84ccdd5553100a537f535f296764b7a6e2f6653092600c0b1a7ce3e6","leaf_index":3416}