Reformulation and Interpretation of the Regularity Criterion for 3D NSE Based on Finitely Many Observations

TL;DR

This paper refines a regularity criterion for 3D Navier-Stokes equations, showing that finite measurements of the flow can determine regularity, with explicit control of derivatives based on data.

Contribution

It introduces a new interpolation approach that removes mollification, providing explicit derivative control from finite observational data.

Findings

The criterion is necessary and sufficient for regularity.

The new interpolation method simplifies analysis and improves data utilization.

Finite measurements suffice to determine flow regularity.

Abstract

We revisit and sharpen a recent observable regularity criterion for the three-dimensional Navier-Stokes equations on the periodic cube by requiring only finitely many measurements of the flow on a given time interval. Two data models are treated: (i) modal observations (a finite set of low Fourier modes), and (ii) nodal observations, i.e. values of the velocity field sampled at finitely many points on a uniform grid. The key upgrade is a piecewise linear interpolant built on a fixed five-tetrahedra subdivision of each grid cube, which removes the mollification step used previously and yields an explicit control of the derivatives of the interpolation operator purely in terms of the measured data. The criterion is also shown to be both necessary and sufficient for regularity.