Approximate continuous data assimilation of the 2D NavierStokes equations via the Voigtregularization with observable data
Abstract
We propose a data assimilation algorithm for the 2D NavierStokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D NavierStokesVoigt equations. Adapting the AOT algorithm to regularized versions of NavierStokes 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 wellposed. 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 Voigtregularization parameter $\alpha>0$, plus a term which decays exponentially fast in time. In particular, the largetime 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1810.10616
 arXiv:
 arXiv:1810.10616
 Bibcode:
 2018arXiv181010616L
 Keywords:

 Mathematics  Analysis of PDEs;
 35A01;
 35B65;
 35K40;
 35K61;
 35Q35;
 35Q30;
 35Q93;
 76D03