Higher-order synchronization for a data assimilation algorithm for the 2D Navier-Stokes equations

We consider the two-dimensional (2D) Navier-Stokes equations (NSE) with space periodic boundary conditions and an algorithm for continuous data assimilation developed by Azouani et al. (2014). The algorithm is based on the observation that existence of finite determining parameters for nonlinear dissipative systems can be exploited as a feedback control mechanism for a companion system into which observables, e.g, modes, nodes, or volume elements, are input directly for the purpose of assimilation. It has been shown that in the case of the 2D NSE, the approximating solution induced by the algorithm synchronizes with the exact solution of the 2D NSE in the topology of the Sobolev space, H1, provided that the number of observed modes, nodes, or volume elements is sufficiently large in terms of the Grashof number. In this article, we adapt a technique, introduced by Grujić and Kukavica (1998) and Kukavica (1999) to obtain good estimates of the analyticity radius for the 2D NSE, and show that one can in fact obtain synchronization in the analytic Gevrey class in the case of modal observables, given sufficiently many, but fixed number of such observables. For these types of observables, we additionally show that synchronization in the uniform norm, L∞, can be achieved by assuming the same number of modal observables (in terms of the order of the Grashof number) as is required for the H1 synchronization.