Anisotropy and stratification effects in the dynamics of fast rotating compressible fluids

TL;DR

This paper develops new methods to analyze the dynamics of fast rotating compressible fluids with anisotropic viscosity and stratification, establishing well-posedness and approximations in geophysical flow regimes.

Contribution

It introduces a maximal regularity approach for local well-posedness that handles anisotropic viscous stress and stratification, extending beyond isotropic models.

Findings

Established local well-posedness for finite-energy solutions.

Derived error estimates for quasi-geostrophic approximations.

Applicable to a broad class of pressure laws.

Abstract

The primary goal of this paper is to develop robust methods to handle two ubiquitous features appearing in the modeling of geophysical flows: (i) the anisotropy of the viscous stress tensor, (ii) stratification effects. We focus on the barotropic Navier-Stokes equations with Coriolis and gravitational forces. Two results are the main contributions of the paper. Firstly, we establish a local well-posedness result for finite-energy solutions, via a maximal regularity approach. This method allows us to circumvent the use of the effective viscous flux, which plays a key role in the weak solutions theories of Lions-Feireisl and Hoff, but seems to be restricted to isotropic viscous stress tensors. Moreover, our approach is sturdy enough to take into account non constant reference density states; this is crucial when dealing with stratification effects. Secondly, we study the structure of the solutions to the previous model in the regime when the Rossby, Mach and Froude numbers are of the same order of magnitude. We prove an error estimate on the relative entropy between actual solutions and their approximation by a large-scale quasi-geostrophic flow supplemented with Ekman boundary layers. Our analysis holds for a large class of barotropic pressure laws.