A Novel Regularity Criterion For The three-dimensional Navier-Stokes Equations Based On Finitely many observations
Date
2022-11-28Type of Work
17 pagesText
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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.
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