Abridged Continuous Data Assimilation for the 2D Navier–Stokes Equations Utilizing Measurements of Only One Component of the Velocity Field

We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier–Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal...

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Bibliographic Details
Published inJournal of mathematical fluid mechanics Vol. 18; no. 1; pp. 1 - 23
Main Authors Farhat, Aseel, Lunasin, Evelyn, Titi, Edriss S.
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.03.2016
Subjects
Classical and Continuum Physics
Fluid- and Aerodynamics
Mathematical Methods in Physics
Physics
Physics and Astronomy
signal synchronization
76B75
Navier–Stokes equations
volume elements and nodes
downscaling
coarse mesh measurements of only one component of the velocity field
feedback control
continuous data assimilation
35Q30
34D06
37C50
93C20
nudging
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Summary:We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier–Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate asymptotically in time, to the unique exact unknown reference solution, of the 2D Navier–Stokes equations, associated with the observed (finite dimensional projection of) velocity.
ISSN:1422-6928
1422-6952
DOI:10.1007/s00021-015-0225-6