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

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

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 2, pp. 295-326.

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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