Global weak solutions and incompressible limit to the isentropic compressible Navier-Stokes equations in 2D bounded domains with ripped density and large initial data

TL;DR

This paper proves the existence of global weak solutions and their incompressible limit for the 2D isentropic compressible Navier-Stokes equations in bounded domains with complex boundary conditions and large initial data.

Contribution

It extends previous results to bounded convex domains with Navier-slip boundary conditions, introducing new estimates for curved boundaries.

Findings

Established global weak solutions in 2D bounded domains.

Proved the incompressible limit under complex boundary conditions.

Developed new estimates based on effective viscous flux.

Abstract

This paper is a continuation of our previous work (arXiv:2507.03505), where the global existence and incompressible limit of weak solutions to the isentropic compressible Navier-Stokes equations in the half-plane with ripped density and large initial data were established. We extend such results to the case of two-dimensional bounded convex domains under a Navier-slip boundary condition. To overcome difficulties in the presence of a curved boundary, some new estimates based on the effective viscous flux and a Desjardins-type logarithmic interpolation inequality play decisive roles.