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

Abstract

We rigorously prove the well-posedness of the formal sensitivity equations with respect to the viscosity 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 be well behaved. Furthermore, our analysis can be extended to analyze the sensitivity of the 2D Euler equations to a viscous regularization. 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 viscosity.