Compressible Flows with a Density-Dependent Viscosity Coefficient

Abstract

We prove the global existence of weak solutions for the two-dimensional compressible Navier–Stokes equations with a density-dependent viscosity coefficient ($\lambda=\lambda(\rho)$). Initial data and solutions are small in energy-norm with nonnegative density 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 $\mu=\text{constant}$ will induce some singularities of the system at vacuum. Thus, the viscosity coefficient $\mu$ plays a key role in the Navier–Stokes equations.