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

TL;DR

This paper introduces a new regularity criterion for the 3D Navier-Stokes equations based on finitely many observations, using a data assimilation approach with nudging, which is a significant departure from existing criteria requiring full solution knowledge.

Contribution

It presents a novel regularity criterion for 3D NSE based on limited observational data, extending the concept of determining functionals to a finite observational framework.

Findings

Proposes a data assimilation algorithm using nodal data for 3D NSE.

Establishes a necessary and sufficient regularity criterion based on finite observations.

Connects the criterion to determining functionals and the nudging algorithm.

Abstract

In this paper we present two results: (1) A data assimilation algorithm for the 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 many observations of the velocity. The data assimilation algorithm we employ utilizes nudging, a method based on a Newtonian relaxation scheme motivated by feedback-control. The observations, which may be either modal, nodal or volume elements, are drawn from a weak solution of the 3D NSE and are collected almost everywhere in time over a finite grid and our results, including the regularity criterion, hold for data of any of the aforementioned forms. The regularity criterion we propose follows from our data assimilation algorithm and is hence intimately connected to the notion of determining functionals (modes, nodes and volume elements). To the best of our knowledge, all existing regularity criteria require knowing the solution of the 3D NSE almost everywhere in space. Our regularity criterion is fundamentally different from any preexisting regularity criterion as it is based on finitely many observations (modes, nodes and volume elements). We further prove that the regularity criterion we propose is both a necessary and sufficient condition for regularity. Thus our result can be viewed as a natural generalization of the notion of determining modes, nodes and volume elements as well as the asymptotic tracking property of the nudging algorithm for the 2D NSE to the 3D setting.