Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity

摘要

The Navier-Stokes equations for compressible barotropic fluid in 1D with the mass force under zero velocity boundary conditions are studied. We prove the uniform upper and lower bounds for the density ρ as well as the uniform in time L~2(Ω)-estimates for ρ_x and u_x (u is the velocity). Moreover, a collection of the decay rate estimates for ρ-ρ_∞ (with ρ_∞ being the stationary density) and u in L~2(Ω)-norm and H~1(Ω)- norm as time t → ∞ are established. The results are given for general state function p(ρ) (but mainly monotone) and viscosity coefficient μ(ρ) of arbitrarily fast (or slow) growth as well as for the large data.