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.

中文翻译：

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通过带有可观测数据的Voigt正则化对2D Navier-Stokes方程进行近似连续数据同化

我们基于Azouani，Olson和Titi（AOT）算法，为2D Navier-Stokes方程提出了一种数据同化算法，但将其应用于2D Navier-Stokes-Voigt方程。以前已经完成了使AOT算法适应Navier-Stokes正规化版本的工作，但是这项工作的创新之处在于用观测数据而不是来自正规化系统的数据来驱动同化方程。我们首先证明该新系统在全球范围内状况良好。此外，我们证明，对于任何可接受的初始数据，错误的$ L ^ 2 $和$ H ^ 1 $范数都由常数乘以Voigt正则化参数$ \ alpha> 0 $的幂加上一个项来定界它随时间快速衰减。特别地，随着$ \ alpha $变为零，长时间误差代数变为零。