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

Abstract

We are concerned with the mathematical derivation of the inhomogeneous 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 ${\mathbb{T}}^2$ for general initial data. Compared to prior works, the main breakthrough is that we are able to handle large variations of density.