Stochastic Fluids - Theoretical Models and Machine Learning Applications

The thesis deals with various problems arising in the study of fluid dynamics. Its topics can be divided in two groups: problems presenting rougher terms than usual (either in the form of very irregular noise or in form of distributional terms) and problems arising in the context of climate studies. To the former group belong Chapters 1 and 3, respectively proving well-posedness of 2D Euler equations with transport noise of finite p-variation, p greater or equal than 2, and enhanced dissipation for an advection-diffusion equation when the transport term belongs to a distributional space. In the latter group, Chapters 2 proves existence of an invariant measure for a multi-layer quasi-geostrophic system, while in Chapter 4 a new methodology is developed to detect and locate Mediterranean cyclones based on statistical learning techniques.