[ { "assumptions": [ "Domain Omega is R^3 or bounded with smooth boundary", "Initial data u(0) in H^1_0,div(Omega)", "u is a Leray-Hopf weak solution on (0,T)", "u belongs to L^s(0,T; L^{r,\u221e}(Omega)) for a Prodi-Serrin pair" ], "authors": [ "Stefano Bosia", "Vittorino Pata", "James C. Robinson" ], "citation": "bosia_pata_robinson_2014", "claims": [ "If u in L^s_t L^{r,\u221e}_x with 3/r + 2/s = 1 and r>3 then the solution remains strong on [0,T]", "A small weak-Lp norm implies global-in-time regularity on [0,T]" ], "conclusions": [ "Weak-Lp (Lorentz) Prodi-Serrin conditions are sufficient for strong solution persistence and uniqueness" ], "contributions": [ "Proves regularity and uniqueness under weak-Lp Prodi-Serrin conditions", "Provides smallness criterion in L^{s,\u221e}(0,T; L^{r,\u221e}) for regularity", "Derives nonlinear term estimates in weak-Lp spaces" ], "future_work": [], "key_equations": [ "d_t u - nu Laplacian(u) + (u . grad)u = -grad p, div u = 0", "3/r + 2/s = 1 (Prodi-Serrin pair)", "d/dt ||grad u||_2^2 <= C_r * nu^(1-s) * ||u||_{L^{r,\u221e}}^s * ||grad u||_2^2" ], "limitations": [], "source_type": "paper", "summary": "Establishes Prodi-Serrin regularity criteria in weak-Lp (Lorentz) spaces for 3D Navier-Stokes. Provides short proofs that Leray-Hopf weak solutions with H1 initial data remain strong and unique on [0,T] when u lies in L^s_t L^{r,\u221e}_x or when a weak-Lp norm is sufficiently small, extending classical criteria to weak spaces relevant for critical-norm monitoring.", "title": "A Weak-Lp Prodi-Serrin Type Regularity Criterion for the Navier-Stokes Equations", "url": "https://re.public.polimi.it/bitstream/11311/862580/1/A%20Weak-Lp%20Prodi%E2%80%93Serrin%20Type%20Regularity%20Criterion%20for%20the%20Navier%E2%80%93Stokes%20Equations_11311-862580_Pata.pdf", "year": 2014 }, { "assumptions": [ "Cauchy problem on R^3", "Leray-Hopf weak solution with u0 in L^2 and div u0 = 0", "Additional integrability assumptions on u or p for regularity" ], "authors": [ "Chuong V. Tran", "Xinwei Yu" ], "citation": "tran_yu_2016", "claims": [ "Finite integrals of ||u||_Ls^r divided by (1+||u||_X)^k imply smoothness up to time T", "Pressure-based criteria with 2/r + 3/s <= 2 can ensure regularity" ], "conclusions": [ "Classical Prodi-Serrin criteria can be strengthened by incorporating scaling-invariant norms without losing regularity guarantees" ], "contributions": [ "Improves classical Prodi-Serrin criteria using scaling-invariant norms (L^3 and H^{1/2})", "Establishes new regularity conditions with weighted integrals involving (1+||u||) factors" ], "future_work": [], "key_equations": [ "u_t + (u . grad)u = -grad p + nu Laplacian(u), div u = 0", "||u(t)||_2^2 + 2 nu * integral_0^t ||grad u||_2^2 d\u03c4 <= ||u0||_2^2", "u in L^r(0,T; L^s(R^3)), 2/r + 3/s <= 1, s>3" ], "limitations": [], "source_type": "paper", "summary": "Develops new Prodi-Serrin-Ladyzhenskaya regularity criteria that improve classical L^r_t L^s_x conditions by incorporating scaling-invariant norms of u and p. The paper restates the Leray-Hopf framework, energy inequality, and pressure representation, then proves logarithmic and scaling-improved criteria for smoothness.", "title": "Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations", "url": "https://research-repository.st-andrews.ac.uk/bitstream/10023/10048/1/Tran_2016_Criteria_JMP_AAM.pdf", "year": 2016 }, { "assumptions": [ "Leray-Hopf solution and generalized local energy inequality", "Weak-Lp spatial norms and log-in-time integrability bounds" ], "authors": [ "Clayton Bjorland", "Alexis Vasseur" ], "citation": "bjorland_vasseur_2009", "claims": [ "Weak-space bounds with logarithmic time improvement suffice to ensure regularity" ], "conclusions": [ "Regularity criteria can be extended beyond classical Lebesgue spaces to weak-Lp with log-time control" ], "contributions": [ "Introduces weak-Lp plus log-in-time regularity criteria", "Adapts De Giorgi and blow-up techniques to weak-space settings" ], "future_work": [], "key_equations": [], "limitations": [], "source_type": "paper", "summary": "Extends Ladyzenskaja-Prodi-Serrin regularity by combining weak-Lp spatial norms with logarithmic time improvements. The work uses a De Giorgi-type approach and generalized energy inequalities to show regularity under log-improved, weak-space bounds.", "title": "Weak in Space, Log in Time Improvement of the Ladyzenskaja-Prodi-Serrin Criteria", "url": "https://ar5iv.labs.arxiv.org/html/0912.2969", "year": 2009 }, { "assumptions": [ "3D incompressible Navier-Stokes with viscosity set to 1", "Regularity analysis via sparseness classes Z_alpha for derivatives" ], "authors": [ "Zoran Grujic", "Liaosha Xu" ], "citation": "grujic_xu_2024", "claims": [ "Regularity class and a priori bound scales converge as derivative order increases", "Sparseness-based framework can asymptotically close the scaling gap" ], "conclusions": [ "Navier-Stokes regularity exhibits asymptotic criticality in a sparseness framework" ], "contributions": [ "Defines a hierarchy of sparseness classes for higher derivatives", "Shows scaling gap shrinks with derivative order (asymptotic criticality)" ], "future_work": [ "Extend techniques to hyper-dissipative Navier-Stokes models" ], "key_equations": [ "u_t + (u . grad)u = -grad p + Laplacian(u), div u = 0", "omega_t + (u . grad)omega = (omega . grad)u + Laplacian(omega)", "u_lambda(x,t) = lambda u(lambda x, lambda^2 t), p_lambda(x,t) = lambda^2 p(lambda x, lambda^2 t)", "m(S \u2229 B_r(x0)) / m(B_r(x0)) <= delta (sparseness definition)" ], "limitations": [], "source_type": "paper", "summary": "Introduces a framework based on sparseness of super-level sets of higher-order derivatives and shows that the scaling gap between regularity criteria and a priori bounds vanishes asymptotically as derivative order increases. This provides a pathway to asymptotic criticality in the Navier-Stokes regularity problem.", "title": "Asymptotic Criticality of the Navier-Stokes Regularity Problem", "url": "https://link.springer.com/content/pdf/10.1007/s00021-024-00888-x.pdf", "year": 2024 }, { "assumptions": [ "3D incompressible viscous flow with forcing", "Kida vortex and homogeneous isotropic turbulence datasets", "Sparseness framework applied to vorticity super-level sets" ], "authors": [ "Janet Rafner", "Zoran Grujic", "Christian Bach", "Jakob Andreas Baerentzen", "Bo Gervang", "Ruo Jia", "Scott Leinweber", "Marek Misztal", "Jacob Sherson" ], "citation": "rafner_et_al_2021", "claims": [ "Sparseness scale detects onset of dissipation in burst events", "Numerical evidence supports reduction of the scaling gap" ], "conclusions": [ "Sparseness framework is empirically consistent with dissipation-regime behavior", "Supports further mathematical work toward closing the scaling gap" ], "contributions": [ "Numerically validates sparseness-based regularity framework", "Shows sparseness scaling exponents slightly above criticality", "Analyzes JHTDB isotropic turbulence for geometric sparseness" ], "future_work": [], "key_equations": [ "d_t u + (u . grad)u = -grad p + nu Laplacian(u) + f, div u = 0", "d_t omega + (u . grad)omega = nu Laplacian(omega) + (omega . grad)u + curl f", "m(S \u2229 B_r(x0)) / m(B_r(x0)) <= delta" ], "limitations": [], "source_type": "paper", "summary": "Provides the first numerical test of the sparseness-based regularity framework. Using high-resolution Kida vortex simulations and JHTDB isotropic turbulence data, it measures the sparseness scale and finds scaling exponents consistent with dissipation onset, supplying numerical evidence that the scaling gap can be closed.", "title": "Geometry of turbulent dissipation and the Navier-Stokes regularity problem", "url": "https://www.nature.com/articles/s41598-021-87774-y.pdf", "year": 2021 }, { "assumptions": [ "Forced isotropic turbulence on a periodic box", "Pseudo-spectral DNS with phase-shift and spherical truncation de-aliasing", "Time series of 1024 snapshots at Re_lambda ~ 433" ], "authors": [ "Yi Li", "Eric Perlman", "Minping Wan", "Yunke Yang", "Charles Meneveau", "Randal Burns", "Shiyi Chen", "Alexander Szalay", "Gregory Eyink" ], "citation": "li_et_al_2008", "claims": [ "Database enables remote numerical experiments without downloading full datasets", "De-aliasing and spectral methods yield accurate spectra and gradients" ], "conclusions": [ "JHTDB infrastructure supports scalable turbulence analysis via web services" ], "contributions": [ "Introduces public turbulence database with remote analysis interface", "Documents processing functions (interpolation, differentiation) and validation tests", "Applies database to analyze Lagrangian intermittency models" ], "future_work": [], "key_equations": [], "limitations": [], "source_type": "paper", "summary": "Describes the JHTDB database architecture and a 27 TB DNS of forced isotropic turbulence on a periodic grid, including web-service access, in-database differentiation/interpolation, and validation calculations such as energy spectra. Provides details about the pseudo-spectral DNS, de-aliasing, and dataset parameters.", "title": "A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence", "url": "https://ar5iv.labs.arxiv.org/html/0804.1703", "year": 2008 }, { "assumptions": [], "authors": [ "G. S. Patterson", "Steven A. Orszag" ], "citation": "patterson_orszag_1971", "claims": [ "Aliasing interactions can be efficiently removed via spectral truncation" ], "conclusions": [ "De-aliasing improves reliability of pseudo-spectral DNS" ], "contributions": [ "Introduces 2/3 truncation de-aliasing in spectral turbulence calculations" ], "future_work": [], "key_equations": [], "limitations": [], "source_type": "paper", "summary": "Classic reference introducing efficient de-aliasing for spectral calculations of isotropic turbulence, establishing the 2/3 truncation rule that underpins pseudo-spectral DNS accuracy.", "title": "Spectral Calculations of Isotropic Turbulence: Efficient Removal of Aliasing Interactions", "url": "https://doi.org/10.1063/1.1693365", "year": 1971 }, { "assumptions": [], "authors": [ "Steven A. Orszag" ], "citation": "orszag_1971", "claims": [ "Filtering can eliminate aliasing in numerical schemes" ], "conclusions": [ "Provides de-aliasing rationale used in turbulence DNS" ], "contributions": [ "Formalizes high-wavenumber filtering to remove aliasing" ], "future_work": [], "key_equations": [], "limitations": [], "source_type": "paper", "summary": "Introduces filtering of high-wavenumber components to eliminate aliasing errors, a foundational reference for de-aliasing practice in spectral and finite-difference turbulence simulations.", "title": "On the Elimination of Aliasing in Finite-Difference Schemes by Filtering High-Wavenumber Components", "url": "https://doi.org/10.1175/1520-0469(1971)028<1074:OTEOAI>2.0.CO;2", "year": 1971 }, { "assumptions": [], "authors": [ "Nishant Parashar", "Sawan S. Sinha", "Balaji Srinivasan" ], "citation": "jhtdb_dataset_2024", "claims": [], "conclusions": [], "contributions": [ "Provides a citable DOI for a JHTDB isotropic turbulence dataset" ], "future_work": [], "key_equations": [], "limitations": [], "source_type": "paper", "summary": "Dataset record for the John Hopkins University turbulence database (JHTD), providing a DOI for a high-resolution isotropic turbulence dataset derived from JHTDB.", "title": "John Hopkins University turbulence database (JHTD) dataset record", "url": "https://doi.org/10.57702/efm5g5uy", "year": 2024 } ]