  We study Bayesian data assimilation (filtering) for time-evolution PDEs, for
which the underlying forward problem may be very unstable or ill-posed. Such
PDEs, which include the Navier-Stokes equations of fluid dynamics, are
characterized by a high sensitivity of solutions to perturbations of the
initial data, a lack of rigorous global well-posedness results as well as
possible non-convergence of numerical approximations. Under very mild and
readily verifiable general hypotheses on the forward solution operator of such
PDEs, we prove that the posterior measure expressing the solution of the
Bayesian filtering problem is stable with respect to perturbations of the noisy
measurements, and we provide quantitative estimates on the convergence of
approximate Bayesian filtering distributions computed from numerical
approximations. For the Navier-Stokes equations, our results imply uniform
stability of the filtering problem even at arbitrarily small viscosity, when
the underlying forward problem may become ill-posed, as well as the compactness
of numerical approximants in a suitable metric on time-parametrized probability
measures.
