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

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 \begin{document}$ L^2 $\end{document} and \begin{document}$ H^1 $\end{document} norms of error are bounded by a constant times a power of the Voigt-regularization parameter \begin{document}$ \alpha>0 $\end{document} , plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as \begin{document}$ \alpha $\end{document} goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the \begin{document}$ H^2 $\end{document} norm.