A remark on 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 the free boundary. The viscosity coefficient μ is proportional to ρ θ with θ > 0 , where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H 1 ( [ 0 , 1 ] ) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier–Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier–Stokes equations, preprint] by driving some new a priori estimates.