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,? denote respectively the density,the velocity and the tempcrature, P=P??,??denote the pressure of the fluid respectively. We study the ideal gas model in this paper,where P=R_??.Furthermore ?=1 is the thermal conductivity coefficient,????=?^?+1 is the viscosity coefficient,and E=e+v²/2 is the total energy of the fluids with e=C? the specific internal energy. The physical quantities to satisfythe second law of thermodynamics define the region ?=???,??|a??????b)???,?>0?. Assuming the system?0.4?is statisfied with the initial conditions and the bound-ary conditions and where a???,b???are free boundaries defined by a????=v?a???,??,a?0?=a, b'???= v?b???,??,b?0?= b, and a< b.In this paper, we have got the existence of global classical solutions for ????= ?^?+1 with ?? ?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.