Data assimilation for the Navier–Stokes- α equations

The well-posedness of the data assimilation problem for the Navier–Stokes- α equations on a bounded three-dimensional domain is investigated. The data assimilation procedures under consideration are the adjoint method of variational data assimilation (4D-Var) and the method of continuous data assimilation. Concerning the adjoint method the existence of optimal initial conditions with respect to an observation-dependent cost functional is proven, the optimizers are characterized by a first-order necessary condition involving the adjoint linearized Navier–Stokes- α equations and conditions for the uniqueness of the initial conditions are given. Well-posedness of the continuous data assimilation problem is proven and convergence rates in terms of observational resolution are provided.