We study the two-dimensional incompressible Navier-Stokes equation on the torus, driven by Gaussian noise that is white in time and colored in space. We consider the case where the magnitude of the random forcing √ ǫ and its correlation scale δ(ǫ) are both small. We prove a large deviations principle for the solutions, as well as for the family of invariant measures, as ǫ and δ(ǫ) are simultaneously sent to 0, under a suitable scaling.
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