September  2020, 9(3): 733-751. doi: 10.3934/eect.2020031

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

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

Received  June 2019 Revised  October 2019 Published  September 2020 Early access  December 2019

Fund Project: The first author is supported by NSF grant number DMS-1716801

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.

Citation: Adam Larios, Yuan Pei. Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data. Evolution Equations and Control Theory, 2020, 9 (3) : 733-751. doi: 10.3934/eect.2020031
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