Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data
Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data
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DOI：
10.3934/eect.2020031
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发表时间：
2018-10
影响因子：
1.5
通讯作者：
Adam Larios;Yuan Pei
中科院分区：
文献类型：
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作者：
Adam Larios;Yuan Pei
我们提出了一种基于Azouani， Olson, and Titi （AOT）算法的二维Navier-Stokes方程的数据同化算法，并将其应用于二维Navier-Stokes- voigt方程。将AOT算法应用于正则化版本的Navier-Stokes之前已经做过，但这项工作的创新之处在于用观测数据驱动同化方程，而不是来自正则化系统的数据。我们首先证明了这个新系统是全局适定的。此外，我们证明了对于任何可容许的初始数据，误差范数$L^2$和$H^1$被一个常数乘以voigt正则化参数$\alpha> $的幂，加上一个随时间呈指数级快速衰减的项所限定。特别地，当$\alpha$趋于零时，大时间误差在代数上趋于零。假设初始数据和强迫更平滑，我们也证明了$H^2$范数的类似结果。
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.