The 3DVAR filter for the Navier-Stokes equation: Accuracy and stability in the limit of high-frequency observations

We study the problem of accuracy and stability of filters for the Navier-Stokes equation in continuous time. In particular, we formally derive continuous filters from discrete filters in the frequent observations limit and focus on the 3DVAR case in which the covariance is assumed to be constant. The diffusion limit of continuous observations gives rise to a Stochastic Partial Differential Equation (SPDE) describing the time evolution of the estimated posterior filtering mean. Accuracy of this filter is established in the mean square sense.