# Parameter Recovery and Sensitivity Analysis for the 2D Navier-Stokes Equations Via Continuous Data Assimilation

## TL;DR

This paper analyzes a continuous data assimilation method for the 2D Navier-Stokes equations, focusing on parameter recovery, sensitivity analysis, and error bounds related to the Reynolds number, with rigorous proofs of solution uniqueness.

## Contribution

It introduces an algorithm to recover the true solution and Reynolds number using discrete velocity data, and provides the first rigorous proof of sensitivity equations' existence and uniqueness.

## Key findings

- Error bounds between true and assimilated solutions due to Reynolds number discrepancy
- Algorithm successfully recovers true solution and Reynolds number from discrete data
- Proved existence and uniqueness of solutions to sensitivity equations for 2D Navier-Stokes

## Abstract

We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the assimilated solution due to discrepancy between an approximate Reynolds number and the physical Reynolds number. Additionally, we develop an algorithm that can be run in tandem with the AOT algorithm to recover both the true solution and the Reynolds number (or equivalently the true viscosity) using only spatially discrete velocity measurements. The algorithm we propose involves changing the viscosity mid-simulation. Therefore, we also examine the sensitivity of the equations with respect to the Reynolds number. We prove that a sequence of difference quotients with respect to the Reynolds number converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the assimilated equations. We also note that this appears to be the first such rigorous proof of 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.