Global solutions of compressible barotropic Navier–Stokes equations with a density-dependent viscosity coefficient

Abstract

In this paper, we consider the global existence and uniqueness of the classical (weak) solution for the 2D or 3D compressible Navier–Stokes equations with a density-dependent viscosity coefficient (λ = λ(ρ)). Initial data and solutions are only small in the energy-norm. We also give a description of the large time behavior of the solution. Then, we study the propagation of singularities in solutions. We obtain that if there is a vacuum domain initially, then the vacuum domain will exist for all time, and vanishes as time goes to infinity.