Continuous data assimilation for the three-dimensional Navier-Stokes-α model

Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present here a continuous data assimilation algorithm for three-dimensional viscous hydrodynamic models. However, to validate the convergence of this algorithm our proofs require the existence of uniform global bounds on the gradients of the solutions of the underlying system in terms of certain combinations of the physical parameters (such as kinematic viscosity, the size of the domain and the forcing term). Therefore our proofs cannot be applied to the three-dimensional Navier-Stokes equations; instead we demonstrate the implementation of this algorithm, for instance, in the context of the three-dimensional Navier-Stokes-alpha equations. This algorithm consists of introducing a nudging process through a general type of approximation interpolation operator (which is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of unknown solutions of the Navier-Stokes-a equations corresponding to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, which is obtained from this collected data, converges to the unknown reference solution over time. These conditions are given in terms of the physical parameters.