An asymptotic limit of a Navier-Stokes system with capillary effects

TL;DR

This paper proves the asymptotic limit of a 2D compressible Navier-Stokes system with capillary effects, including rotation and density-dependent viscosity, converging to a viscous quasi-geostrophic equation.

Contribution

It establishes the singular limit for a complex Navier-Stokes model with capillarity, rotation, and variable viscosity, using the modulated energy method.

Findings

Proved the combined incompressible and vanishing capillarity limit.

Derived the viscous quasi-geostrophic equation as the limiting model.

Validated the limit on the two-dimensional torus with well-prepared initial data.

Abstract

A combined incompressible and vanishing capillarity limit in the barotropic compressible Navier-Stokes equations for smooth solutions is proved. The equations are considered on the two-dimensional torus with well prepared initial data. The momentum equation contains a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity. The limiting model is the viscous quasi-geostrophic equation for the "rotated" velocity potential. The proof of the singular limit is based on the modulated energy method with a careful choice of the correction terms.