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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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Cecilia F. Mondaini, Edriss S. Titi, Animikh Biswas, Ciprian Foias, Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, pp. 295–326DOI 10.1016/J.ANIHPC.2018.05.004