From compressible to incompressible inhomogeneous flows in the case of large data

TL;DR

This paper rigorously derives the inhomogeneous incompressible Navier-Stokes equations from the compressible version in the large volume viscosity limit, demonstrating convergence of solutions in 2D with large density variations.

Contribution

It provides a mathematical proof of the convergence from compressible to incompressible flows under large volume viscosity, including existence results for the compressible system.

Findings

Solutions of CNS exist for large times with regularity.

CNS solutions converge to INS solutions as volume viscosity increases.

The analysis handles large density variations in 2D.

Abstract

This paper is concerned with the mathematical derivation of the inhomoge-neous incompressible Navier-Stokes equations (INS) from the compressible Navier-Stokes equations (CNS) in the large volume viscosity limit. We first prove a result of large time existence of regular solutions for (CNS). Next, as a consequence, we establish that the solutions of (CNS) converge to those of (INS) when the volume viscosity tends to infinity. Analysis is performed in the two dimensional torus, for general initial data. In particular, we are able to handle large variations of density.