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In this paper we present two results: (1) A data assimilation algorithm forthe 3D Navier-Stokes equation (3D NSE) using nodal data, and, as a consequence(2) a novel regularity criterion for the 3D NSE based on finitely manyobservations of the velocity. The data assimilation algorithm we employutilizes nudging, a method based on a Newtonian relaxation scheme motivated byfeedback-control. The observations, which may be either modal, nodal or volumeelements, are drawn from a weak solution of the 3D NSE and are collected almosteverywhere in time over a finite grid and our results, including the regularitycriterion, hold for data of any of the aforementioned forms. The regularitycriterion we propose follows from our data assimilation algorithm and is henceintimately connected to the notion of determining functionals (modes, nodes andvolume elements). To the best of our knowledge, all existing regularitycriteria require knowing the solution of the 3D NSE almost everywhere in space.Our regularity criterion is fundamentally different from any preexistingregularity criterion as it is based on finitely many observations (modes, nodesand volume elements). We further prove that the regularity criterion we proposeis both a necessary and sufficient condition for regularity. Thus our resultcan be viewed as a natural generalization of the notion of determining modes,nodes and volume elements as well as the asymptotic tracking property of thenudging algorithm for the 2D NSE to the 3D setting.

A Novel Regularity Criterion For The three-dimensional Navier-Stokes Equations Based On Finitely many observations