Global weak solutions and incompressible limit to the isentropic compressible Navier-Stokes equations in the half-plane with ripped density and large initial data

TL;DR

This paper establishes the global existence of weak solutions for the isentropic compressible Navier-Stokes equations with ripped density in a half-plane, and demonstrates their convergence to incompressible solutions as viscosity increases, accommodating large initial data and vacuum.

Contribution

It proves the existence and incompressible limit of weak solutions with large initial data and vacuum in a half-plane setting, using novel analytical techniques.

Findings

Global weak solutions exist under large initial data.

Solutions converge to incompressible Navier-Stokes solutions as viscosity increases.

The method employs a Desjardins-type inequality and effective viscous flux techniques.

Abstract

We prove the global existence of weak solutions to the isentropic compressible Navier-Stokes equations with ripped density in the half-plane under a slip boundary condition provided the bulk viscosity coefficient is properly large. Moreover, we show that such solutions converge globally in time to a weak solution of the inhomogeneous incompressible Navier-Stokes equations as the bulk viscosity coefficient tends to infinity. In particular, the large initial data and an initial patch of density as well as a vacuum are allowed. Our method relies on a Desjardins-type logarithmic interpolation inequality and some new techniques based on the effective viscous flux.