{
  "id": "navier-stokes-singularity",
  "org": "OpenAI",
  "topic": "Partial differential equations",
  "published": "2026-09-08",
  "status": "formalized-provider-reported",
  "method": "Analytical proof and Lean artifact published",
  "math": "∂ₜu + (u·∇)u = −∇p + νΔu + f",
  "source": "https://openai.com/index/navier-stokes-solution/",
  "source_kind": "provider_announcement",
  "checked": "2026-10-29",
  "translations": {
    "en": {
      "title": "Can smooth fluid flow become singular?",
      "simple": "Picture a whirlpool spinning ever faster into an increasingly tiny region. Can the equations describing water or air become infinitely intense from a smooth start? OpenAI says its model constructed such an example.",
      "research": "OpenAI released an analytical and Lean-formalized construction of finite-time singularity for a three-dimensional Navier–Stokes flow, with smooth forcing and finite energy. This is framed as resolving the relevant forced case in the Clay Millennium formulation.",
      "academic": "The incompressible Navier–Stokes equations balance transport, pressure and viscosity. The announced construction starts with smooth finite-energy data, includes a smooth forcing term and develops singular behavior in finite time. The forcing assumptions and exact equivalence to Clay's formulations C/D matter. A Lean check is not identical to independent acceptance of all modeling and formulation choices.",
      "impact": "A genuine resolution would transform our understanding of regularity in nonlinear PDE. It does not imply that everyday fluid simulations universally fail.",
      "caution": "A Lean formalization is reported. The Clay Mathematics Institute has not been verified here as awarding or officially certifying the prize."
    },
    "nl": {
      "title": "Kan een gladde vloeistofstroming singulier worden?",
      "simple": "Stel je een draaikolk voor die steeds sneller draait in een steeds kleiner gebied. Kunnen stromingsvergelijkingen vanuit een glad begin onbeheersbaar worden?",
      "research": "OpenAI publiceerde een analytische constructie en Lean-formalisatie voor een singulariteit in eindige tijd, met gladde externe kracht en eindige energie.",
      "academic": "De Navier–Stokes-vergelijkingen combineren transport, druk en viscositeit. De exacte aannames over de kracht en de betekenis voor Clay-formuleringen C/D vereisen onafhankelijke controle.",
      "impact": "Mogelijke doorbraak in de theorie van niet-lineaire differentiaalvergelijkingen.",
      "caution": "Lean-formalisatie gemeld; geen hier gecontroleerde officiële Clay-certificering."
    },
    "de": {
      "title": "Kann glatte Strömung singulär werden?",
      "simple": "Man stelle sich einen Wirbel vor, der sich auf immer kleinerem Raum immer schneller dreht.",
      "research": "OpenAI veröffentlichte einen analytischen und in Lean formalisierten Singularitätsbeweis für Navier–Stokes mit glatter äußerer Kraft.",
      "academic": "Die Gleichungen koppeln Transport, Druck und Viskosität. Die Voraussetzungen der äußeren Kraft und ihre Bedeutung für die Clay-Formulierungen C/D müssen extern geprüft werden.",
      "impact": "Möglicher Wendepunkt für die Regularitätstheorie nichtlinearer PDEs.",
      "caution": "Kein hier bestätigter offizieller Clay-Preisentscheid."
    },
    "fr": {
      "title": "Un écoulement lisse peut-il devenir singulier ?",
      "simple": "Imaginez un tourbillon de plus en plus rapide concentré dans un volume minuscule.",
      "research": "OpenAI publie une construction analytique et Lean d'une singularité en temps fini pour Navier–Stokes avec force externe lisse.",
      "academic": "Transport, pression et viscosité interviennent dans ces équations. Les hypothèses sur la force et la correspondance aux formulations C/D du Clay Institute demandent un examen indépendant.",
      "impact": "Impact potentiel sur la régularité des équations différentielles non linéaires.",
      "caution": "Aucune certification officielle du prix Clay n'est confirmée ici."
    },
    "es": {
      "title": "¿Puede un flujo suave volverse singular?",
      "simple": "Imagina un remolino que gira cada vez más rápido en una región cada vez más pequeña.",
      "research": "OpenAI publicó una construcción analítica y en Lean de una singularidad en tiempo finito para Navier–Stokes con fuerza externa suave.",
      "academic": "Las ecuaciones combinan transporte, presión y viscosidad. Los supuestos de la fuerza y su relación con las formulaciones C/D del Clay Institute requieren revisión independiente.",
      "impact": "Posible avance en ecuaciones diferenciales no lineales.",
      "caution": "No consta aquí una certificación oficial del premio Clay."
    },
    "it": {
      "title": "Un flusso regolare può diventare singolare?",
      "simple": "Immagina un vortice sempre più veloce concentrato in uno spazio minuscolo.",
      "research": "OpenAI pubblica una costruzione analitica e in Lean di una singolarità in tempo finito per Navier–Stokes con forzante regolare.",
      "academic": "Le equazioni collegano trasporto, pressione e viscosità. Le ipotesi sulla forzante e le formulazioni C/D del Clay Institute vanno esaminate indipendentemente.",
      "impact": "Potenziale svolta nella teoria delle PDE non lineari.",
      "caution": "Nessuna certificazione ufficiale del premio Clay è verificata qui."
    }
  },
  "translation_method": "AI-assisted explanatory translations; not native mathematical peer review",
  "url": "https://appliedfrontier.nl/en/discoveries/navier-stokes-singularity/",
  "as_of": "2026-10-29"
}