Existence Of Classical Solutions And Asymptotic Behavior For 1D Isentropic, Compressible Navier-Stokes Equations With Free Boundary Value Problem

The free boundary for one-dimensional compressible isentropic Navier-Stokes equations is studied in this paper, namely ??G [0, a(?)],?> 0, where ?= ?(?,?), u=u(?,?) and P(?) denote the density, the velocity and the pressure of the fluid, respectively, the viscosity coefficient ?=?(?)=???+1, with ? a positive constant.Here the initial values are given by the free boundary condition is imposed byIt is proved that the global existence of smooth solutions is constructed when 0< ?<? and the asymptotic behavior, the decay rate of solutions is also obtained. The key to the proof is that the positive upper and lower bound of the density p is obtained by using some appropriate energy functionals. Moreover, the regularity of solutions is established by using a series of priori estimates.