The mathematical analysis of fast rotating fluids is nowadays a classical topic. Recent results have dealt with non-homogeneous flows, and especially flows exhibiting density variations. Nonetheless, very little is known (even on the physical side) when the density is a perturbation of a non-constant state, and when irregularities of the domain (topography effects) are taken into account.
The aim of this PhD thesis is to investigate some stability and instability issues in the mathematical theory of fast rotating fluids. The goal is to investigate the interplay of stratification and fast rotation in the dynamics of geophysical flows, focusing on the case of the compressible Navier-Stokes-Fourier system and following recent results on multi-scale analysis by Feireisl and Novotný.
Multi-scale asymptotic limit for the Navier-Stokes-Fourier system with strong Coriolis force will be considered, taking into account the presence of centrifugal force. Various scaling choices will be discussed, leading to qualitatively different target systems: the hope is to extend the known results also to other ranges of the parameters. Finally, special attention will be devoted to the isotropic scaling, a setting which allows one to consider non-constant target density profiles (stratification).