On Global Classical Solutions to 1D Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

Abstract: For the initial boundary value problem of compressible barotropic Navier-Stokes equations in one-dimensional bounded domains with general density-dependent viscosity and large external force, we prove that there exists a unique global classical solution with large initial data containing vacuum. Furthermore, we show that the density is bounded from above independently of time which in particular yields the large time behavior of the solution as time tends to infinity: the density and the velocity converge to the steady states in $L^p$ and in $W^{1,p}$ ($1\le p<+\infty$) respectively. Moreover, the decay rate in time of the solution is shown to be exponential. Finally, we also prove that the spatial gradient of the density will blow up as time tends to infinity when vacuum states appear initially even at one point.