Invariant Measures in Two-Dimensional Fluid Dynamics

The present thesis is an integrated compilation of the two main works that defined my PhD research at the University of Pavia, University of Milano-Bicocca, and University of York, where I completed a four-month Erasmus Traineeship Programme. The two principal works are presented as the main chapters of this thesis. They both address the two-dimensional stochastic Navier-Stokes Equations (sNSEs) for homogeneous incompressible fluids and the study of invariant probability measures. The first work, supervised by Professor Enrico Priola from the University of Pavia, introduces a novel a priori estimate for the sNSEs, set in a bounded domain and with additive noise, which leads to an intriguing application regarding the uniqueness and ergodic properties of its invariant measure. The second work, co-authored with Professor Zdzislaw Brzezniak from the University of York, explores the inviscid limit for the hyperviscous sNSEs, set in R^2 with additive noise. This investigation results in the proof of the existence, along with some moment estimates, of an invariant measure for the deterministic Euler Equations.