Global weak solutions and incompressible limit to the isentropic compressible Navier-Stokes equations in the half-plane with ripped density and large initial data
Shuai Wang, Guochun Wu, Xin Zhong
TL;DR
The paper addresses global existence of weak solutions and the incompressible limit for the isentropic compressible Navier–Stokes equations in a half-plane with ripped density and slip boundary. The authors develop a two-tier approach that yields uniform a priori control, notably via a Desjardins-type logarithmic interpolation inequality and the effective viscous flux, to handle large bulk viscosity and vacuum. They prove the existence of a global weak solution with controlled density for large initial energy when $\lambda$ is sufficiently large, and show that, as $\lambda\to\infty$, a subsequence converges to a weak solution of the inhomogeneous incompressible Navier–Stokes equations with $\operatorname{div}\mathbf{v}=0$ on compact sets. This work extends prior incompressible-limit results to a boundary-rich half-space, accommodating vacuum and nonzero initial momentum, and clarifies the boundary’s role in the limit behavior.
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.