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 first goal was 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.
We considered the multi-scale asymptotic limit for the Navier-Stokes-Fourier system with strong Coriolis force, taking into account the presence of centrifugal force and the gravity. We discussed various scaling choices, leading to qualitatively different target systems: we extended the known results also to other ranges of the parameters.
Now, in the second part of the project, we will focus on the study of hyperbolic problems and, in particular, of the incompressible Euler equations with density variation and, in addition, the Coriolis force.
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