Some examples of absolute continuity of measures in stochastic fluid dynamics

A non linear Itô equation in a Hilbert space is studied by means of Girsanov theorem. We consider a non linearity of polynomial growth in suitable norms, including that of quadratic type which appears in the Kuramoto–Sivashinsky equation and in the Navier– Stokes equation. We prove that Girsanov theorem holds for the 1-dimensional stochastic Kuramoto–Sivashinsky equation and for a modification of the 2and 3-dimensional stochastic Navier–Stokes equation; this modification consists in substituting the Laplacian −∆ with (−∆), where α > d 2 + 1 (d = 2, 3). In this way, we prove existence and uniqueness of solutions for these stochastic equations. Moreover, the asymptotic behaviour for t → ∞ is characterized.