Decay Rates for Solutions to the Compressible Navier-Stokes Equations with Vacuum Free Boundary and Large Initial Data

. In this paper, we are concerned with the vacuum-free boundary problem of the one-dimensional compressible Navier-Stokes equations with constant viscosity. We derive the decay estimates on the L2 and L infinity norms of the velocity and the pointwise estimates in terms of the derivatives of the solution that further imply a clear description of the large-time behavior of the free interfaces. Moreover, we improve the estimates in [19, Luo-Xin-Yang, SIAM J. Math. Anal., 31: 1175-1191, 2000] on the behaviors of the solution near the interfaces in Eulerian coordinates. In particular, we obtain the pointwise lower and upper bounds of the density function near the interfaces with time-decay rates. The proof is based on various weighted estimates associated with the degeneracy near the interfaces and the decay properties of the solution with respect to the time variable. It is worth noting that no smallness assumptions upon the initial data are required in our analysis.