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

• DOI: 10.1007/s00332-024-10014-w
• Publication Date: 2024-02-24T13:02:03Z
• Authors: Elizabeth Carlson; Adam Larios; Edriss S. Titi
• Keywords: Mathematics - Analysis of PDEs; 4901 Applied Mathematics; 518; 49 Mathematical Sciences; FOS: Mathematics; 34D06, 35K61, 35Q93, 93C20; Analysis of PDEs (math.AP)

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.<br/>33 pages, 7 figures<br/>