  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.
