Compressible flows with a density-dependent viscosity coefficient

TL;DR

This paper establishes the global existence of weak solutions for 2-D compressible Navier-Stokes equations with a density-dependent viscosity, highlighting the role of viscosity in vacuum behavior and system singularities.

Contribution

It proves global weak solutions with density-dependent viscosity and analyzes vacuum persistence and singularity formation related to viscosity conditions.

Findings

Vacuum domains persist over time if initially present.

Constant viscosity leads to singularities at vacuum.

Solutions exist globally under small energy norms.

Abstract

We prove the global existence of weak solutions for the 2-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient ($\lambda=\lambda(\rho)$). Initial data and solutions are small in energy-norm with nonnegative densities having arbitrarily large sup-norm. Then, we show that if there is a vacuum domain at the initial time, then the vacuum domain will retain for all time, and vanishes as time goes to infinity. At last, we show that the condition of $\mu=$constant will induce a singularity of the system at vacuum. Thus, the viscosity coefficient $\mu$ plays a key role in the Navier-Stokes equations.