The Free Boundary For One-dimensional Density-dependent Viscosity Nonisentropic Navier-Stokes Equations

In this article,the free boundary for one-dimensional compressible nonisen-tropic Navier-Stokes equations is studied,namely where p,v,θ denote respectively the density,the velocity and the tempcrature, P=P（ρ,θ）denote the pressure of the fluid respectively. We study the ideal gas model in this paper,where P=R_ρθ.Furthermore κ=1 is the thermal conductivity coefficient,μ（ρ）=ρ^α+1 is the viscosity coefficient,and E=e+v²/2 is the total energy of the fluids with e=Cθ the specific internal energy. The physical quantities to satisfythe second law of thermodynamics define the region Ω=｛（ζ,τ）|a（τ）≤ζ≤b)（τ）,τ>0｝. Assuming the system（0.4）is statisfied with the initial conditions and the bound-ary conditions and where a（τ）,b（τ）are free boundaries defined by a′（τ）=v（a（τ）,τ）,a（0）=a, b’（τ）= v（b（τ）,τ）,b（0）= b, and a< b.In this paper, we have got the existence of global classical solutions for μ（ρ）= ρ^α+1 with α∈ （0,+∞）, The difference from others is that the positive upper and lower bound of the density p is obtained by using some appropriate energy functionals, so it reduces the restriction to a enoughly. Moreover, the regularity of solutions is established by using a series of priori estimates and we complete the proof of this paper.