Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

Adam Larios; Yuan Pei

doi:10.3934/eect.2020031

Article Contents

2020, Volume 9, Issue 3: 733-751. Doi: 10.3934/eect.2020031

This issue Previous Article Remarks on the damped nonlinear Schrödinger equation Next Article Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay

Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

Adam Larios^1, , and
Yuan Pei^2,

1.
Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 68588, USA
2.
Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA

^* Corresponding author: Adam Larios
^* Corresponding author: Adam Larios

Received: June 2019

Revised: October 2019

Early access: December 2019

Published: September 2020

The first author is supported by NSF grant number DMS-1716801.

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Abstract

We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that for any admissible initial data, the $ L^2 $ and $ H^1 $ norms of error are bounded by a constant times a power of the Voigt-regularization parameter $ \alpha>0 $, plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as $ \alpha $ goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the $ H^2 $ norm.

Keywords:

Continuous data assimilation,
Azouani-Olson-Titi algorithm,
nudging,
two-dimensional Navier-Stokes-Voigt equations,
observable data,
Voigt regularization.

Mathematics Subject Classification: Primary: 35Q30, 37C50; Secondary: 93C20, 76B75, 34D06.

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