Incompressible and fast rotation limit for barotropic Navier–Stokes equations at large Mach numbers

Abstract

In the present paper we study the incompressible and fast rotation limit for the barotropic Navier-Stokes equations with Coriolis force, in the case when the Mach number $\\rm Ma$ is large with respect to the Rossby number $\\rm Ro$: namely, we focus on the regime ${\\rm Ro}\\ll{\\rm Ma}$. For this, we follow a recent approach by Danchin and Mucha in \\cite{D-M} and take also a large bulk viscosity coefficient. We prove that the limit dynamics is described by an incompressible Navier-Stokes type equation, recasted in the vorticy formulation, where however an additional unknown, linked to density oscillations around a fixed constant reference state, comes into play. The proof of the convergence is based on a compensated compactness argument and on the derivation of sharp decay estimates for solutions to a heat equation with fast diffusion in time.