Special Session 7: 

Density-dependent incompressible fluids in fast rotation

Francesco Fanelli
Universite' de Lyon
In this talk we consider a class of singular perturbation problems for non-homogeneous flows whose dynamics is influenced by the Earth rotation. We specilize on the $2$-D density-dependent incompressible Navier-Stokes equations with Coriolis force: our goal is to characterize the asymptotic dynamics of weak solutions to this model, in the limit when the rotation becomes faster and faster. We present two kinds of results, deeply different from each other from a qualitative viewpoint. If the initial density is a small perturbation of a constant state, we prove that the limit dynamics is essentially described by a homogeneous Navier-Stokes system with an additional forcing term, which can be seen as a remainder of density variations. %and which is a remainder of the action of the Coriolis force. If, instead, the initial density is a small perturbation of a truly variable reference state, we show that the final equations become linear, and moreover one can identify only a mean motion, described in terms of the limit vorticity and the limit density fluctuation function; this issue can be interpreted as a sort of turbulent behaviour of the limit flow. \medbreak This talk is based on a joint work with \emph{Isabelle Gallagher}.