[ { "assumptions": [ "Initial velocity U_i(x) is square-summable on \\u03a0 and weakly divergence-free.", "Solutions satisfy energy dissipation inequality via W(t) nonincreasing.", "Turbulent solutions may be irregular on a closed set of measure zero." ], "authors": [ "Jean Leray", "Robert E. Terrell" ], "citation": "Leray, J.; Terrell, R.E. (2016). On the motion of a viscous liquid filling space. arXiv.", "claims": [ "For square-summable divergence-free initial data there exists at least one global turbulent solution.", "Turbulent solutions satisfy Navier's equations except possibly at a closed set of singular times of measure zero.", "Energy is nonincreasing along turbulent solutions (via W(t))." ], "conclusions": [ "Global weak solutions exist without assuming global smoothness.", "Uniqueness of turbulent solutions is not established." ], "contributions": [ "Introduces global weak (turbulent) solutions for 3D Navier-Stokes in R^3.", "Derives energy dissipation inequality and structural properties of singular times.", "Provides decay-type bounds on gradient quantities J(t) for large times." ], "future_work": [ "Construct explicit irregular (singular) solutions or prove global regularity.", "Establish uniqueness of turbulent solutions." ], "key_equations": [ "J^2(t) = \u222b\u222b\u222b_\\u03a0 U_{i,j}(x,t) U_{i,j}(x,t) dx", "w(t) <= w(\\u0398) exp( (1/(2\\u03bd)) \\u222b_\\u0398^t V^2(t') dt' )", "J^2(t) < ((1 + A)^2/2) W(0)/(\\u03bd t)" ], "limitations": [ "Does not prove regularity or uniqueness of weak solutions.", "Does not construct explicit blowup examples." ], "source_type": "paper", "summary": "English translation of Leray's 1934 paper introducing global-in-time weak (turbulent) solutions and the energy inequality framework. It formalizes existence of global weak solutions with possible singular times of measure zero, establishing foundational structure for Leray-Hopf theory.", "title": "On the motion of a viscous liquid filling space", "url": "https://arxiv.org/abs/1604.02484", "year": 2016 }, { "assumptions": [ "Divergence-free initial data with specified regularity classes (e.g., H^1, L^2*L^p).", "Viscosity normalized to \\u03bd = 1 for presentation.", "Strong solutions exist on maximal interval (0, T_0)." ], "authors": [ "Wojciech S. Ozanski", "Benjamin C. Pooley" ], "citation": "Ozanski, W.S.; Pooley, B.C. (2017). Leray's fundamental work on the Navier-Stokes equations: a modern review of \"Sur le mouvement d'un liquide visqueux emplissant l'espace\". arXiv.", "claims": [ "Global weak solutions exist and are strong except possibly on a compact set of singular times.", "Strong solutions exhibit quantified lower bounds on norm blowup as t -> T_0.", "Weak-strong uniqueness holds within the strong-solution lifespan." ], "conclusions": [ "Leray's framework already contains key modern ideas: compactness, mollification, weak solutions.", "Regularity/uniqueness of global weak solutions remains open." ], "contributions": [ "Provides modern proofs and clarifications of Leray's existence and weak-strong uniqueness results.", "Derives explicit lower bounds on blowup rates for strong solutions.", "Explains structure and dimension bounds for singular times." ], "future_work": [ "Sharper quantitative regularity criteria or improved singular-set bounds.", "Progress on uniqueness of weak solutions." ], "key_equations": [ "\\u2202_t u + (u * \\u2207)u - \\u03bd \\u0394u + \\u2207p = 0, \\u2207 * u = 0", "\\u2202_t u - \\u03bd \\u0394u + \\u2207p = F, \\u2207 * u = 0", "||u(t)||_\\u221e >= C / sqrt(T_0 - t), ||\\u2207u(t)|| >= C (T_0 - t)^(-1/4), ||u(t)||_p >= C (T_0 - t)^{-(1-3/p)/2}" ], "limitations": [ "Review article; no new resolution of the 3D global regularity problem.", "Relies on classical techniques and does not address constructive blowup." ], "source_type": "paper", "summary": "Modern, detailed reconstruction of Leray's 1934 arguments: local strong solutions, weak solutions, weak-strong uniqueness, and singular-time structure. Includes explicit blowup rate lower bounds for strong solutions and energy equality statements.", "title": "Leray's fundamental work on the Navier-Stokes equations: a modern review of \"Sur le mouvement d'un liquide visqueux emplissant l'espace\"", "url": "https://arxiv.org/abs/1708.09787", "year": 2017 }, { "assumptions": [ "Averaged bilinear operator \\u02dcB preserves cancellation <\\u02dcB(u,u),u> = 0.", "Solutions are smooth with Schwartz initial data for the averaged model.", "Viscosity normalized to 1." ], "authors": [ "Terence Tao" ], "citation": "Tao, T. (2014). Finite time blowup for an averaged three-dimensional Navier-Stokes equation. arXiv.", "claims": [ "There exist smooth solutions to the averaged equation that blow up in finite time.", "Any successful positive resolution of Navier-Stokes must use structure beyond energy identity and standard estimates." ], "conclusions": [ "Blowup in averaged model demonstrates limits of existing techniques.", "Regularity of actual Navier-Stokes remains open." ], "contributions": [ "Constructs finite-time blowup for averaged Navier-Stokes with energy identity.", "Formalizes a barrier: energy identity + harmonic analysis estimates are insufficient for global regularity.", "Outlines a program for adapting blowup mechanisms to true Navier-Stokes." ], "future_work": [ "Adapt averaged-model blowup scheme to true Navier-Stokes.", "Identify finer nonlinear structures needed for regularity proofs." ], "key_equations": [ "\\u2202_t u + (u * \\u2207)u = \\u0394u - \\u2207p, \\u2207 * u = 0", "B(u,v) = -(1/2) P[(u*\\u2207)v + (v*\\u2207)u]", " = 0", "\\u2202_t u = \\u0394u + \\u02dcB(u,u)" ], "limitations": [ "Blowup is for an averaged model, not the true Navier-Stokes equations.", "Does not provide blowup example for the real equations." ], "source_type": "paper", "summary": "Constructs finite-time blowup for an averaged Navier-Stokes nonlinearity preserving the energy identity. Shows that energy methods and harmonic-analysis estimates alone cannot resolve 3D regularity and proposes a program toward true Navier-Stokes blowup.", "title": "Finite time blowup for an averaged three-dimensional Navier-Stokes equation", "url": "https://arxiv.org/abs/1402.0290", "year": 2014 }, { "assumptions": [ "Periodic domain T^3 and zero-mean velocity fields.", "Weak solutions are in C^0_t L^2_x and divergence-free in distributions.", "Energy profile e(t) is prescribed smooth and nonnegative." ], "authors": [ "Tristan Buckmaster", "Vlad Vicol" ], "citation": "Buckmaster, T.; Vicol, V. (2018). Nonuniqueness of weak solutions to the Navier-Stokes equation. arXiv.", "claims": [ "There exist distinct weak solutions sharing the same initial data and energy profile.", "Some weak solutions can come to rest in finite time." ], "conclusions": [ "Weak solution nonuniqueness is possible within finite-energy class.", "Selection criteria for Euler via vanishing viscosity are insufficient." ], "contributions": [ "Establishes nonuniqueness of weak solutions with finite kinetic energy.", "Shows vanishing-viscosity limits can yield dissipative Euler solutions.", "Develops intermittent Beltrami flows for convex integration with viscosity." ], "future_work": [ "Extend nonuniqueness to Leray-Hopf class or stronger regularity.", "Understand implications for selection principles in Euler/Navier-Stokes." ], "key_equations": [ "\\u2202_t v + \\u2207*(v\\u2297 v) + \\u2207p - \\u03bd \\u0394 v = 0, \\u2207*v=0", "\\u222b\\u222b v * (\\u2202_t \\u03c6 + (v*\\u2207)\\u03c6 + \\u03bd\\u0394\\u03c6) dx dt = 0", "v(t) = e^{\\u03bd t\\u0394} v(0) + \\u222b_0^t e^{\\u03bd(t-s)\\u0394} P\\u2207*(v\\u2297 v)(s) ds" ], "limitations": [ "Does not show nonuniqueness for Leray-Hopf solutions (energy inequality).", "Does not address global regularity of strong solutions." ], "source_type": "paper", "summary": "Proves nonuniqueness of finite-energy weak solutions for 3D Navier-Stokes on the torus via convex integration and constructs vanishing-viscosity limits yielding dissipative Euler solutions. Establishes that weak solutions can match arbitrary smooth energy profiles.", "title": "Nonuniqueness of weak solutions to the Navier-Stokes equation", "url": "https://arxiv.org/abs/1709.10033", "year": 2018 }, { "assumptions": [ "Smooth solutions on (0,T) with divergence-free initial data.", "Leray-Hopf weak solutions satisfy energy inequality (1.8).", "Scale-critical spaces defined via 2/p + 3/q = 1." ], "authors": [ "Evan Miller" ], "citation": "Miller, E. (2020). Navier-Stokes regularity criteria in sum spaces. arXiv.", "claims": [ "If a sum-space norm of velocity (or vorticity, or middle strain eigenvalue) is finite, the solution is regular.", "Sum-space criteria subsume families of scale-critical criteria." ], "conclusions": [ "Sum-space regularity criteria unify multiple critical conditions.", "Endpoint and blowup-rate questions remain open." ], "contributions": [ "Introduces regularity criteria in sum spaces L^p_t L^q_x + L^2_t L^infty_x.", "Provides strain-eigenvalue and vorticity criteria in sum spaces.", "Proves new inclusion inequalities for sum spaces in mixed Lebesgue scales." ], "future_work": [ "Handle endpoint/critical cases and refine quantitative bounds.", "Extend to broader geometric or anisotropic criteria." ], "key_equations": [ "\\u2202_t u - \\u03bd\\u0394u + (u*\\u2207)u + \\u2207p = 0, \\u2207*u = 0", "\\u0394p = -\\u2211_{i,j} \\u2202_{x_i}u_j \\u2202_{x_j}u_i", "S_{ij} = 1/2(\\u2202_i u_j + \\u2202_j u_i)", "\\u2202_t \\u03c9 - \\u03bd\\u0394\\u03c9 + (u*\\u2207)\\u03c9 - S\\u03c9 = 0", "u_\\u03bb(x,t) = \\u03bbu(\\u03bb x, \\u03bb^2 t)" ], "limitations": [ "Conditional regularity only; does not resolve global regularity.", "Endpoint cases remain delicate and are not fully addressed." ], "source_type": "paper", "summary": "Develops regularity criteria in mixed Lebesgue sum spaces for velocity, vorticity, and strain eigenvalues. Provides unified criteria covering families of scale-critical conditions and proves new sum-space inclusions.", "title": "Navier-Stokes regularity criteria in sum spaces", "url": "https://arxiv.org/abs/2007.02023", "year": 2020 }, { "assumptions": [ "Leray weak solutions satisfy energy inequality.", "Regularity criteria stated for blowup time T_max < infinity.", "Scale-critical exponent relations are used throughout." ], "authors": [ "Evan Miller" ], "citation": "Miller, E. (2021). A survey of geometric constraints on the blowup of solutions of the Navier-Stokes equation. arXiv.", "claims": [ "Blowup would require fully three-dimensional geometric stretching features.", "Certain component-reduction criteria are equivalent via Helmholtz decomposition." ], "conclusions": [ "Geometric constraints are strong but insufficient to resolve global regularity.", "Substantially new ideas are likely required for a full resolution." ], "contributions": [ "Synthesizes component-reduction and geometric regularity criteria.", "Highlights equivalences between vorticity-component and strain-direction criteria.", "Discusses physical interpretations of geometric constraints." ], "future_work": [ "Develop mechanisms for depletion of nonlinearity.", "Bridge geometric criteria with quantitative blowup rates." ], "key_equations": [ "\\u2202_t u - \\u03bd\\u0394u + (u*\\u2207)u + \\u2207p = 0, \\u2207*u=0", "S_{ij} = 1/2(\\u2202_i u_j + \\u2202_j u_i)", "\\u03c9 = \\u2207 * u", "1/2 ||u(.,t)||_2^2 + \\u03bd \\u222b_0^t ||\\u2207u(.,\\u03c4)||_2^2 d\\u03c4 <= 1/2 ||u_0||_2^2", "\\u222b_0^{T_max} ||u(.,t)||_q^p dt = +infty, 2/p + 3/q = 1" ], "limitations": [ "Survey only; no new regularity theorem or blowup example.", "Many endpoint cases remain open." ], "source_type": "paper", "summary": "Survey of geometric regularity criteria constraining possible blowup, including vorticity direction, strain eigenvalues, and component-reduction results. Provides physical interpretations and links between different criteria.", "title": "A survey of geometric constraints on the blowup of solutions of the Navier-Stokes equation", "url": "https://arxiv.org/abs/2111.00040", "year": 2021 }, { "assumptions": [ "Focus on whole-space R^3 or local analysis away from boundaries.", "Quantitative results often assume suitable weak/Leray-Hopf solutions.", "Type I or scale-invariant bounds are used in some sections." ], "authors": [ "Tobias Barker", "Christophe Prange" ], "citation": "Barker, T.; Prange, C. (2022). From concentration to quantitative regularity: a short survey of recent developments for the Navier-Stokes equations. arXiv.", "claims": [ "Quantitative bounds for critical norms can imply mild criticality breaking.", "Concentration results are tied to quantitative regularity and dissipative scales." ], "conclusions": [ "Quantitative regularity is progressing but critical barriers remain.", "Explicit rates are still weak (e.g., triple-log blowup)." ], "contributions": [ "Synthesizes concentration (weak/strong) and quantitative regularity frameworks.", "Explains links between blowup rates, local smoothing, and criticality breaking.", "Collects recent quantitative estimates and identifies open problems." ], "future_work": [ "Improve quantitative blowup rates and concentration localization.", "Extend quantitative theory to broader classes of solutions." ], "key_equations": [ "\\u2202_t U - \\u0394U + U*\\u2207U + \\u2207P = 0, \\u2207*U=0", "lim sup_{t->T*} ||U(.,t)||_{L^3} (log log log(T*-t))^c = infinity", "E_r(t) = \\u222b_{B_0(r)} |U|^2 + \\u222b_0^t \\u222b_{B_0(r)} |\\u2207U|^2 + (1/r^2)\\u222b_0^t \\u222b_{B_0(r)} |p|^{3/2}", "sup_{t in (0,S(M))} t^{1/2} ||U(.,t)||_{L^infty(B_0(1/2))} <= 1" ], "limitations": [ "Survey only; many statements depend on specific assumptions (Type I, suitable solutions).", "No new resolution of global regularity." ], "source_type": "paper", "summary": "Survey of recent quantitative regularity and concentration results for 3D Navier-Stokes, including explicit blowup-rate bounds, local-in-space smoothing, and connections to criticality breaking.", "title": "From concentration to quantitative regularity: a short survey of recent developments for the Navier-Stokes equations", "url": "https://arxiv.org/abs/2211.16215", "year": 2022 }, { "assumptions": [ "Leray-Hopf weak solutions with u_0 in H^1 and divergence-free.", "Critical Besov/Lorentz norms control regularity; smallness in sum-space for borderline case.", "Viscosity normalized to 1." ], "authors": [ "Yiran Xu", "Ly Kim Ha", "Haina Li", "Zexi Wang" ], "citation": "Xu, Y.; Ha, L.K.; Li, H.; Wang, Z. (2024). New Regularity Criteria for Navier-Stokes and SQG Equations in Critical Spaces. arXiv.", "claims": [ "Leray-Hopf solutions are regular if critical Besov/Lorentz norms are finite as specified.", "Smallness of the L^infty_t \\u02d9B^{-1}_{\\u221e,\\u221e} component yields regularity in the sum-space case." ], "conclusions": [ "New critical-space criteria expand the landscape of conditional regularity.", "Endpoint/borderline cases can be handled with sum-space smallness." ], "contributions": [ "Establishes regularity criteria in critical Besov and Lorentz spaces for 3D Navier-Stokes.", "Provides sum-space criteria combining L_T^{2/(1-\\u03b1)} \\u02d9B^{-\\u03b1}_{\\u221e,\\u221e} and L^infty_t \\u02d9B^{-1}_{\\u221e,\\u221e}.", "Extends analogous criteria to supercritical SQG." ], "future_work": [ "Handle full endpoint cases without smallness conditions.", "Refine criteria with quantitative blowup rates." ], "key_equations": [ "\\u2202_t u - \\u0394u + u*\\u2207u + \\u2207p = 0, \\u2207*u=0, u|_{t=0}=u_0", "u_\\u03bb(t,x) = \\u03bbu(\\u03bb^2 t, \\u03bb x), p_\\u03bb(t,x) = \\u03bb^2 p(\\u03bb^2 t, \\u03bb x)", "u \\u2208 L^p(0,T;L^q), 2/p + 3/q = 1", "||\\u2207u||_{L^infty_t L^2_x} + ||\\u2207^2u||_{L^2_t L^2_x} <= ||\\u2207u_0||_{L^2} exp(C\\u222b_0^T ||u(s)||_{\\u02d9B^{-\\u03b1}_{\\u221e,\\u221e}}^{2/(1-\\u03b1)} ds)" ], "limitations": [ "Conditional results; do not resolve the global regularity problem.", "Require specific function-space bounds and, in one case, smallness assumptions." ], "source_type": "paper", "summary": "Derives new regularity criteria for 3D Navier-Stokes in critical Besov and Lorentz spaces and for supercritical SQG. Provides a priori estimates yielding regularity under scale-critical norms and sum-space smallness.", "title": "New Regularity Criteria for Navier-Stokes and SQG Equations in Critical Spaces", "url": "https://arxiv.org/abs/2403.12383", "year": 2024 }, { "assumptions": [ "Classical solutions (can be extended to weak notions via regularity theory).", "Uniform boundedness of critical L^infty_t L^3_x norm for quantitative regularity theorem.", "Finite blowup time T* for blowup criterion." ], "authors": [ "Terence Tao" ], "citation": "Tao, T. (2019). Quantitative bounds for critically bounded solutions to the Navier-Stokes equations. arXiv.", "claims": [ "If ||u||_{L^infty_t L^3_x} is bounded by A, then derivatives are bounded by exp(exp(exp(A^{O(1)}))).", "Any finite-time blowup must have L^3 norm diverging at least at triple-log rate." ], "conclusions": [ "Provides the first explicit quantitative blowup-rate bounds at the critical level.", "Bounds are extremely weak but fully explicit." ], "contributions": [ "Replaces compactness/unique continuation with quantitative Carleman-based bounds.", "Obtains explicit triple-exponential regularity bounds in terms of critical norm.", "Derives triple-logarithmic lower bound on L^3 blowup rate at T*." ], "future_work": [ "Improve quantitative rates and reduce exponential complexity.", "Extend to other critical Besov/Lorentz spaces." ], "key_equations": [ "\\u2202_t u + (u*\\u2207)u = \\u0394u - \\u2207p, \\u2207*u=0", "u_\\u03bb(t,x)=\\u03bbu(\\u03bb^2 t, \\u03bb x), p_\\u03bb(t,x)=\\u03bb^2 p(\\u03bb^2 t, \\u03bb x)", "lim sup_{t->T*} ||u(t)||_{L^3} (log log log(1/(T*-t)))^c = infinity" ], "limitations": [ "Triple-exponential bounds are far from sharp.", "Does not resolve global regularity; depends on critical norm control." ], "source_type": "paper", "summary": "Develops quantitative regularity bounds for solutions with bounded critical L^infty_t L^3_x norm using quantitative Carleman estimates, yielding triple-exponential regularity bounds and a triple-logarithmic blowup rate for the L^3 norm near singular time.", "title": "Quantitative bounds for critically bounded solutions to the Navier-Stokes equations", "url": "https://arxiv.org/abs/1908.04958", "year": 2019 } ]