  The Bayesian approach to inverse problems is of paramount importance in
quantifying uncertainty about the input to and the state of a system of
interest given noisy observations. Herein we consider the forward problem of
the forced 2D Navier Stokes equation. The inverse problem is inference of the
forcing, and possibly the initial condition, given noisy observations of the
velocity field. We place a prior on the forcing which is in the form of a
spatially correlated temporally white Gaussian process, and formulate the
inverse problem for the posterior distribution. Given appropriate spatial
regularity conditions, we show that the solution is a continuous function of
the forcing. Hence, for appropriately chosen spatial regularity in the prior,
the posterior distribution on the forcing is absolutely continuous with respect
to the prior and is hence well-defined. Furthermore, the posterior distribution
is a continuous function of the data. We complement this theoretical result
with numerical simulation of the posterior distribution.
