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On the reconstruction of unknown driving forces from low-mode observations in the 2D Navier–Stokes equations

Part of: Partial differential equations Equations of mathematical physics and other areas of application Controllability, observability, and system structure Incompressible inviscid fluids Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  01 April 2024

Vincent R. Martinez Open the ORCID record for Vincent R. Martinez [Opens in a new window]
Vincent R. Martinez*
Affiliation:
Department of Mathematics & Statistics, CUNY Hunter College, New York, NY, USA Department of Mathematics, New York, NY, CUNY Graduate Center, USA (vrmartinez@hunter.cuny.edu)
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Abstract

This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which makes use of a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier–Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decreases to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned.

Keywords

inferring unknown external forceparameter estimation and inverse problemsmodel errorsystem identificationdata assimilationfeedback controlsynchronizationNavier–Stokes equationsconvergencesensitivity

MSC classification

Primary: 35Q30: Navier-Stokes equations
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters 93B30: System identification 35R30: Inverse problems 76B75: Flow control and optimization
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 24
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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