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

  • Author(s): Biswas, A
  • Foias, C
  • Mondaini, CF
  • Titi, ES
  • et al.
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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