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We study a continuous data assimilation algorithm proposed by Azouani, Olson,and Titi (AOT) in the context of an unknown Reynolds number. We determine thelarge-time error between the true solution of the 2D Navier-Stokes equationsand the assimilated solution due to discrepancy between an approximate Reynoldsnumber and the physical Reynolds number. Additionally, we develop an algorithmthat can be run in tandem with the AOT algorithm to recover both the truesolution and the Reynolds number (or equivalently the true viscosity) usingonly spatially discrete velocity measurements. The algorithm we proposeinvolves changing the viscosity mid-simulation. Therefore, we also examine thesensitivity of the equations with respect to the Reynolds number. We prove thata sequence of difference quotients with respect to the Reynolds numberconverges to the unique solution of the sensitivity equations for both the 2DNavier-Stokes equations and the assimilated equations. We also note that thisappears to be the first such rigorous proof of existence and uniqueness tostrong or weak solutions to the sensitivity equations for the 2D Navier-Stokesequations (in the natural case of zero initial data), and that they can beobtained as a limit of difference quotients with respect to the Reynoldsnumber.

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