Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations

Animikh Biswas; Ciprian Foias; Cecilia F. Mondaini; Edriss S. Titi

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arXiv · 1711.04067

Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations

Animikh Biswas·Ciprian Foias·Cecilia F. Mondaini·Edriss S. Titi

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Abstract

Based on a previously introduced downscaling data assimilation algorithm, which employs a nudging term to synchronize the coarse mesh spatial scales, we construct a determining map for recovering the full trajectories from their corresponding coarse mesh spatial trajectories, and investigate its properties. This map is then used to develop a downscaling data assimilation scheme for statistical solutions of the two-dimensional Navier-Stokes equations, where the coarse mesh spatial statistics of the system is obtained from discrete spatial measurements. As a corollary, we deduce that statistical solutions for the Navier-Stokes equations are determined by their coarse mesh spatial distributions. Notably, we present our results in the context of the Navier-Stokes equations; however, the tools are general enough to be implemented for other dissipative evolution equations.

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Animikh Biswas, Ciprian Foias, Cecilia F. Mondaini, Edriss S. Titi. 2017-11-11. Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations. https://arxiv.org/abs/1711.04067

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