2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226443We prove 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 at initially, then the vacuum domain will exists for all time, and vanishes as time goes to infinity.25 pagesAnalysis of PDEsFluid Dynamics35Q30 (Primary) 35R35, 76N10 (Secondary)text