Compressible flows with a density-dependent viscosity coefficient

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.

32 pages

References in corpus (2)

• Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations
• Global Behavior of Spherically Symmetric Navier-Stokes Equations with Density-Dependent Viscosity