  We consider the martingale problem associated to the Navier-Stokes in
dimension 2 or 3. Existence is well known and it has been recently shown that
markovian transition semi group associated to these equations can be
constructed. We study the Kolmogorov operator associated to these equations. It
can be defined formally as a differential operator on an infinite dimensional
Hilbert space. It can be also defined in an abstract way as the infinitesimal
generator of the transition semi group. We explicit cores for these abstract
operators and identify them with the concrete differential operators on these
cores. In dimension 2, the core is explicit and we can use a classical argument
to prove uniqueness for the martingale problem. In dimension 3, we are only
able to exhibit a core which is defined abstractly and does not allow to prove
uniqueness for the martingale problem. Instead, we exhibit a core for a
modified Kolmogorov operator which enables us to prove uniqueness for the
martingale problem up to the time the solutions are regular.
