We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the 2D Navier-Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.

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Abstract. An abstract framework for the theory of statistical solutions is developed for general evolution equations, extending the theory initially developed for the three-dimensional incompressible Navier-Stokes equations. The motivation for this concept is to model the evolution of uncertainties on the initial conditions for systems which have global solutions that are not known to be unique. Both concepts of statistical solution in trajectory space and in phase space are given, and the corresponding results of existence of statistical solution for the associated initial value problems are proved. The wide applicability of the theory is illustrated with the very incompressible Navier-Stokes equations, a reaction-diffusion equation, and a nonlinear wave equation, all displaying the property of global existence of weak solutions without a known result of global uniqueness.

A general framework for the theory of statistical solutions on trajectory spaces is constructed for a wide range of equations involving incompressible viscous flows. This framework is constructed with a general Hausdorff topological space as the phase space of the system, and with the corresponding set of trajectories belonging to the space of continuous paths in that phase space. A trajectory statistical solution is a Borel probability measure defined on the space of continuous paths and carried by a certain subset which is interpreted, in the applications, as the set of solutions of a given problem. The main hypotheses for the existence of a trajectory statistical solution concern the topology of that subset of "solutions", along with conditions that characterize those solutions within a certain larger subset (a condition related to the assumption of strong continuity at the origin for the Leray-Hopf weak solutions in the case of the Navier-Stokes and related equations). The aim here is to raise the current theory of statistical solutions to an abstract level that applies to other evolution equations with properties similar to those of the three-dimensional Navier-Stokes equations. The applicability of the theory is illustrated with the Bénard problem of convection in fluids.

Abstract Our aim is to approximate a reference velocity field solving the two-dimensional Navier–Stokes equations (NSE) in the absence of its initial condition by utilizing spatially discrete measurements of that field, available at a coarse scale, and continuous in time. The approximation is obtained via numerically discretizing a downscaling data assimilation algorithm. Time discretization is based on semiimplicit and fully implicit Euler schemes, while spatial discretization (which can be done at an arbitrary scale regardless of the spatial resolution of the measurements) is based on a spectral Galerkin method. The two fully discrete algorithms are shown to be unconditionally stable, with respect to the size of the time step, the number of time steps and the number of Galerkin modes. Moreover, explicit, uniform-in-time error estimates between the approximation and the reference solution are obtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensional NSE, subject to the no-slip Dirichlet or periodic boundary conditions, are used in this work as a paradigm. The complete analysis that is presented here can be extended to other two- and three-dimensional dissipative systems under the assumption of global existence and uniqueness.