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

Abstract

We study the incompressible and fast rotation limit for the barotropic Navier–Stokes equations with Coriolis force, in the case when the Mach number Ma is large with respect to the Rossby number Ro: namely, we focus on the regime Ro≪Ma. For this, we follow a recent approach in Danchin and Mucha (2017) 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 vorticity 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.