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
to make inference concerning 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 and
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, it may then be shown that the posterior distribution is a
continuous function of the data. We complement these theoretical results
with numerical simulations showing the feasibility of computing the
posterior distribution, and illustrating its properties.