{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"7f979cce38e91fed3b310b96e523024a5689d074ea6d1faab1645aae5339a21a","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"e7f31fb110666e594fa4098b60cc8fdebbd017b7ab0e22013aece5c8736670d2","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"mathph.fluid-singularities.leray-hopf-uniqueness","kind":"well-posed","literature_status":"open","n":"1","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 in L^inf_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":null,"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":"56471002f5ad6f04f20e12861588794aa7901ecb598d7f960b48900e00132b59"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"2290b0aa355fa47c12e8a6e077e1e4bbbd490ffa210b224725ab25138e73fd8c","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"0576b03d81e614e68e330eaf2cc1d64294b83e3a72d340039230e0e4deb30783","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"KSjbNSMBfpsbkAwGgGZKZ3knfjrik3Ww2JaHCP-zjqIOxi-gTq68CLx7YZxuY57plhdMZrWlIu_wrbONsaggBg"},"schema":"pubphys.envelope/1"},"record_hash":"2290b0aa355fa47c12e8a6e077e1e4bbbd490ffa210b224725ab25138e73fd8c","leaf_index":1618}