{"query": "Navier-Stokes existence and smoothness", "count": 20, "results": [{"id": "card_question_navier_stokes", "title": "Navier-Stokes existence and smoothness", "shelf": "millennium", "surface": "secular", "snippet": "Navier-Stokes existence and smoothness. In three dimensions, smooth globally defined solutions of the Navier-Stokes equations exist for every smooth initial datum, or a counterexample does. Its chart ", "authority_tier": "reference", "source": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "readable": false, "generated": false}, {"id": "card_theory_fluid_mechanics", "title": "Fluid mechanics (Bernoulli, Reynolds, Navier–Stokes)", "shelf": "theories", "surface": "secular", "snippet": "Fluid mechanics (Bernoulli, Reynolds, Navier–Stokes) — an engine domain that can touch it: hydrology. Calibration: seals — continuity, Bernoulli and Reynolds number compute directly. CONTINUITY first:", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_bridge_theory_circulation_harvey__fluid_mechanics", "title": "Bridge: Circulation of the blood (Harvey's quantitative argument)  ↔  Fluid mechanics (Bernoulli, Reynolds, Navier–Stokes)", "shelf": "bridges", "surface": "secular", "snippet": "Circulation of the blood (Harvey's quantitative argument) and Fluid mechanics (Bernoulli, Reynolds, Navier–Stokes) are the same form in different domains. Harvey's argument was ARITHMETIC -- ejected v", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_bridge_theory_dimensional_analysis__fluid_mechanics", "title": "Bridge: Dimensional analysis & similarity (Buckingham Π)  ↔  Fluid mechanics (Bernoulli, Reynolds, Navier–Stokes)", "shelf": "bridges", "surface": "secular", "snippet": "Dimensional analysis & similarity (Buckingham Π) and Fluid mechanics (Bernoulli, Reynolds, Navier–Stokes) are the same form in different domains. match the dimensionless groups -- Reynolds, Mach, Frou", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "shelf": "codex", "surface": "secular", "snippet": "The seven Millennium Prize Problems on the one map of reality. The FLOOR is what is proven or observed: the joints where two of the questions meet at one established thing - the GUE statistics shared ", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor (operator seed)", "readable": false, "generated": false}, {"id": "card_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "shelf": "spine", "surface": "secular", "snippet": "63 sources located for the seven Millennium sticks (Riemann, Birch and Swinnerton-Dyer, Navier-Stokes, Yang-Mills, P versus NP, Hodge, Poincare), one reference card each: the bibliographic record, DOI", "authority_tier": "reference", "source": "The Millennium sources — a spine of located records", "readable": false, "generated": false}, {"id": "card_src_mill_forster_nelson_stephen_1977", "title": "D. Forster 1977 — Large-distance and long-time properties of a randomly stirred fluid", "shelf": "millennium", "surface": "secular", "snippet": "D. Forster, D. R. Nelson, M. J. Stephen (1977). Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16 (1977) 732–749. DOI 10.1103/PhysRevA.16.732. Canonical: https://doi", "authority_tier": "reference", "source": "D. Forster, D. R. Nelson, M. J. Stephen (1977), Phys. Rev. A 16 (1977) 732–749", "readable": false, "generated": false}, {"id": "card_joint_renormalization_group", "title": "The renormalization group, and the continuum limit as the question", "shelf": "codex", "surface": "secular", "snippet": "Wilson's lattice (1974) and Forster, Nelson and Stephen's randomly stirred fluid (1977) run the same renormalization group: a coupling that changes with scale, a finite system whose continuum limit is", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_arnold_geodesics", "title": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms", "shelf": "codex", "surface": "secular", "snippet": "Arnold (1966): the Euler equations of an ideal fluid are the geodesic equations of the group of volume-preserving diffeomorphisms with the kinetic-energy metric - the fluid as a point moving on an inf", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_arnold_1966", "title": "V. I. Arnold 1966 — Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits", "shelf": "millennium", "surface": "secular", "snippet": "V. I. Arnold (1966). Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16 (1966) 319–361. DOI 10.5", "authority_tier": "reference", "source": "V. I. Arnold (1966), Ann. Inst. Fourier 16 (1966) 319–361", "readable": false, "generated": false}, {"id": "card_src_mill_fefferman_2000", "title": "C. L. Fefferman 2000 — Existence and smoothness of the Navier-Stokes equation", "shelf": "millennium", "surface": "secular", "snippet": "C. L. Fefferman (2000). Existence and smoothness of the Navier-Stokes equation. Clay Mathematics Institute, the official problem description. Canonical: https://www.claymath.org/millennium/navier-stok", "authority_tier": "reference", "source": "C. L. Fefferman (2000), Clay Mathematics Institute, the official problem description", "readable": false, "generated": false}, {"id": "card_src_mill_fujita_kato_1964", "title": "H. Fujita 1964 — On the Navier-Stokes initial value problem. I", "shelf": "millennium", "surface": "secular", "snippet": "H. Fujita, T. Kato (1964). On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964) 269–315. DOI 10.1007/BF00276188. Canonical: https://doi.org/10.1007/BF00276188. No free c", "authority_tier": "reference", "source": "H. Fujita, T. Kato (1964), Arch. Rational Mech. Anal. 16 (1964) 269–315", "readable": false, "generated": false}, {"id": "card_src_mill_tao_2016", "title": "T. Tao 2016 — Finite time blowup for an averaged three-dimensional Navier-Stokes equation", "shelf": "millennium", "surface": "secular", "snippet": "T. Tao (2016). Finite time blowup for an averaged three-dimensional Navier-Stokes equation. J. Amer. Math. Soc. 29 (2016) 601–674. DOI 10.1090/jams/838. arXiv: 1402.0290. Canonical: https://doi.org/10", "authority_tier": "reference", "source": "T. Tao (2016), J. Amer. Math. Soc. 29 (2016) 601–674", "readable": false, "generated": false}, {"id": "card_src_mill_caffarelli_kohn_nirenberg_1982", "title": "L. Caffarelli 1982 — Partial regularity of suitable weak solutions of the Navier-Stokes equations", "shelf": "millennium", "surface": "secular", "snippet": "L. Caffarelli, R. Kohn, L. Nirenberg (1982). Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982) 771–831. DOI 10.1002/cpa.3160350604. Canonic", "authority_tier": "reference", "source": "L. Caffarelli, R. Kohn, L. Nirenberg (1982), Comm. Pure Appl. Math. 35 (1982) 771–831", "readable": false, "generated": false}, {"id": "card_src_mill_ladyzhenskaya_1959", "title": "O. A. Ladyzhenskaya 1959 — Solution 'in the large' of the nonstationary boundary value problem for the Navier-Stokes system with two space variables", "shelf": "millennium", "surface": "secular", "snippet": "O. A. Ladyzhenskaya (1959). Solution 'in the large' of the nonstationary boundary value problem for the Navier-Stokes system with two space variables. Comm. Pure Appl. Math. 12 (1959) 427–433. DOI 10.", "authority_tier": "reference", "source": "O. A. Ladyzhenskaya (1959), Comm. Pure Appl. Math. 12 (1959) 427–433", "readable": false, "generated": false}, {"id": "card_chart_ns_dispersion", "title": "Navier-Stokes — dissonance and dispersion", "shelf": "codex", "surface": "secular", "snippet": "Navier-Stokes — dissonance and dispersion. A chart on the one map: this is the tick stick stick_navier_stokes_dissonance_and_dispersion, placed at the node it charts. Basis: the stick seals the superc", "authority_tier": "engine_derived", "source": "Narrow Highway - a chart: a tick stick placed on the one map", "readable": false, "generated": false}, {"id": "card_src_mill_leray_1934", "title": "J. Leray 1934 — Sur le mouvement d'un liquide visqueux emplissant l'espace", "shelf": "millennium", "surface": "secular", "snippet": "J. Leray (1934). Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934) 193–248. DOI 10.1007/BF02547354. arXiv: 1604.02484 (an English translation). Canonical: https://doi.or", "authority_tier": "reference", "source": "J. Leray (1934), Acta Math. 63 (1934) 193–248", "readable": false, "generated": false}, {"id": "card_src_mill_beale_kato_majda_1984", "title": "J. T. Beale 1984 — Remarks on the breakdown of smooth solutions for the 3-D Euler equations", "shelf": "millennium", "surface": "secular", "snippet": "J. T. Beale, T. Kato, A. Majda (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94 (1984) 61–66. DOI 10.1007/BF01212349. Canonical: https://doi.org/1", "authority_tier": "reference", "source": "J. T. Beale, T. Kato, A. Majda (1984), Comm. Math. Phys. 94 (1984) 61–66", "readable": false, "generated": false}, {"id": "card_src_mill_orszag_1971", "title": "S. A. Orszag 1971 — Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations", "shelf": "millennium", "surface": "secular", "snippet": "S. A. Orszag (1971). Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representations. Stud. Appl. Math. 50 (1971) 293–327. DOI 10.1002/sapm1971504293. Can", "authority_tier": "reference", "source": "S. A. Orszag (1971), Stud. Appl. Math. 50 (1971) 293–327", "readable": false, "generated": false}, {"id": "card_chain_stokes_1845", "title": "Stokes 1845 — On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids", "shelf": "codex", "surface": "secular", "snippet": "G. G. Stokes (1845). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Philos. Soc. 8 (1845) 287–319. Cited by its rec", "authority_tier": "reference", "source": "G. G. Stokes (1845), Trans. Cambridge Philos. Soc. 8 (1845) 287–319", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_question_navier_stokes"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}