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

TL;DR

This paper investigates the combined limit of incompressibility and rapid rotation in the barotropic Navier-Stokes equations at high Mach numbers, revealing a modified incompressible Navier-Stokes model with an extra density oscillation component.

Contribution

It extends previous work by analyzing the simultaneous incompressible and fast rotation limit at large Mach numbers, incorporating a large bulk viscosity and deriving a new limit system.

Findings

Limit dynamics described by an incompressible Navier-Stokes type equation with an additional density oscillation term.

Proved convergence using compensated compactness and decay estimates for a heat equation with fast diffusion.

Established the regime where the combined effects lead to a modified incompressible flow model.

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.