  The purpose of this paper is to establish the Donsker-Varadhan type large
deviations principle (LDP) for the two-dimensional stochastic Navier-Stokes
system. The main novelty is that the noise is assumed to be highly degenerate
in the Fourier space. The proof is carried out by using a criterion for the LDP
developed in arXiv:1410.6188 in a discrete-time setting and extended in
arXiv:1505.03686 to the continuous-time. One of the main conditions of that
criterion is the uniform Feller property for the Feynman-Kac semigroup, which
we verify by using Malliavin calculus.
