Super-Exponential Convergence Rate of a Nonlinear Continuous Data Assimilation Algorithm: The 2D Navier–Stokes Equation Paradigm

We study a nonlinear-nudging modification of the Azouani–Olson–Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier–Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well posed and, moreover, that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that once the error has decreased below a certain order one threshold, the convergence becomes double exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally.