Sensitivity Analysis for the 2D Navier-Stokes Equations with Applications to Continuous Data Assimilation

TL;DR

This paper provides a rigorous proof of the well-posedness of sensitivity equations for the 2D Navier-Stokes equations and related data assimilation algorithms, ensuring stable parameter recovery as the system evolves.

Contribution

It offers the first rigorous proof of global existence and uniqueness of solutions to the sensitivity equations for 2D Navier-Stokes, including convergence of difference quotients.

Findings

Proved well-posedness of sensitivity equations for 2D Navier-Stokes.

Established convergence of difference quotients to sensitivity solutions.

Demonstrated stability of parameter recovery algorithms over time.

Abstract

We rigorously prove the well-posedness of the formal sensitivity equations with respect to the Reynolds number corresponding to the 2D incompressible Navier-Stokes equations. Moreover, we do so by showing a sequence of difference quotients converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the related data assimilation equations, which utilize the continuous data assimilation algorithm proposed by Azouani, Olson, and Titi. As a result, this method of proof provides uniform bounds on difference quotients, demonstrating parameter recovery algorithms that change parameters as the system evolves will not blow-up. We also note that this appears to be the first such rigorous proof of global existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.