Regularity of 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier–Stokes equations with the viscosity depending on density and with free boundary. The viscosity coefficient μ is proportional to ρθ with 0<θ<1, where ρ is the density. The existence and uniqueness of global weak solutions in H1([0,1]) have been established in [S. Jiang, Z. Xin, P. Zhang, Global weak solutions to 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity, Methods Appl. Anal. 12 (2005) 239–252]. We will establish the regularity of global solution under certain assumptions imposed on the initial data by deriving some new a priori estimates.