Quantitative Protocol for Distinguishing Finite-Time Singularity from Global Regularity in Numerical Time Series of 3D Euler and Navier–Stokes Norms
Year: 2026
Type: preprint
Abstract: We present a rigorous, reproducible protocol for the quantitative analysis of numerical time series of maximum vorticity and velocity gradient arising in simulations of the three-dimensional incompressible Euler and Navier–Stokes equations. The protocol is designed to discriminate between two competing hypotheses: finite-time blow-up at some T<∞T < \inftyT<∞ and global regularity (unbounded but finite growth for all finite ttt). The protocol comprises: (i) an explicit partition of candidate models into a finite-singularity class (S) and a global-growth class (G); (ii) mandatory extrapolating (rolling-origin) cross-validation; (iii) analytic filters derived from the Beale–Kato–Majda criterion and, for Navier–Stokes, from Leray-type lower bounds; (iv) strict requirements on dynamic range and grid resolution; (v) consistency checks on the characteristic length scale of the high-vorticity region; and (vi) the use of joint confidence regions rather than one-dimensional intervals for highly correlated parameters. The protocol is validated on an extensive synthetic test bench (216 realisations). Pure power-law singularities with dynamic range R≳100R \gtrsim 100R≳100 are recovered with high reliability, while false-positive declarations of finite-time blow-up remain rare (approximately 8 %). The method is deliberately conservative: on short dynamic ranges or in the presence of logarithmic corrections and slowly drifting exponents it frequently returns an “undecided” verdict. The protocol does not constitute a proof of singularity or of global regularity. Its purpose is to raise the evidentiary standard for numerical claims and to isolate those scenarios that merit further analytic or computer-assisted investigation. For the classical Clay problem (unforced Navier–Stokes with smooth finite-energy data) the protocol presently functions primarily as a false-positive filter.
Author: Glushkov, Oleg
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225312723