Ergodic and mixing properties of the 2D Navier–Stokes equations with a degenerate multiplicative Gaussian noise

In this paper, we establish the ergodic and mixing properties of stochastic 2D Navier–Stokes equations driven by a highly degenerate multiplicative Gaussian noise. The noise can appear in as few as four directions, and its intensity depends on the solution. The case of additive Gaussian noise was previously treated by Hairer and Mattingly (Ann. of Math. (2) 164 (2006) 993–1032). To derive the ergodic and mixing properties in the present setting, we employ Malliavin calculus to establish the asymptotically strong Feller property. The primary challenge lies in proving the “invertibility” of the Malliavin matrix, which differs fundamentally from the additive case.