In this paper, we consider the large time behavior of the weak solution to the free boundary problem for one-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum. Under appropriate smallness conditions on the initial data (initial energy), we give the optimal decay rate of the density function along with the behavior of it near the interfaces is studied. In the meanwhile, we obtain also sharper decay rates for the norms in terms of the velocity function. The proof is based on the standard line method. The key is to establish some new global-in-time weighted estimates (both in time and space) uniformly up to the vacuum boundary, which ensures the uniform convergence of the approximate solutions.
This is a joint work with Guangyi Hong.