Can a finite number of measurements determine whether a fluid flow is smooth? The Navier-Stokes regularity problem is infinite-dimensional — the solution lives in a function space, and singularities could hide at arbitrarily fine scales. The standard criteria for regularity (Prodi-Serrin, Beale-Kato-Majda) require global integral bounds over the full velocity field.
This paper reformulates and sharpens observable regularity criteria using piecewise linear interpolation on finitely many measured data points. The insight is that the observability conditions — bounds on what the finite measurements see — are sufficient to exclude singularity formation, provided the measurement resolution satisfies a precise relationship with the viscosity and other physical parameters.
The reformulation turns a function-analytic question into an interpolation question: given N point measurements, is the piecewise linear reconstruction smooth enough to guarantee the full solution is smooth? The answer depends on whether the interpolation error is controlled, which in turn depends on the spacing and placement of measurement points relative to the smallest dynamically relevant scales.
This is a strengthening because it connects regularity — a property of the abstract solution — to observability, a property of what can actually be measured. The finite measurement network encodes enough information about the infinite-dimensional flow to exclude the pathological behaviors that regularity criteria are designed to detect. If the measurements are sufficiently dense relative to the flow's energy, no singularity can hide between them.
The sharpened criteria quantify exactly how many observations suffice, moving the regularity question from “is the solution smooth?” to “are these N measurements consistent with smoothness?” — a question that is finite and checkable.
Finitely many point measurements can encode full regularity of an infinite-dimensional flow, because the physical dynamics constrain the inter-measurement behavior tightly enough to exclude hidden singularities.