The problem of free boundary for non-isentropic Navier-Stokes equations with density-dependent viscosity in one-dimension

In this article, we obtain the existence of global classical solutions for non-isentropic compressible Navier-Stokes equations with density-dependent viscosity in one-dimension. Here, let the viscosity coefficient be $\mu(\rho)=\rho^{\alpha}+1$, where $\rho$ denotes the density of fluids and $\alpha\in(0,+\infty)$ is a constant. The key point is that the positive upper and lower bounds of the density $\rho$ are obtained by using some appropriate energy functionals, so it reduces the restriction to $\alpha$ enough. Moreover, we get the regularity of solutions by using a series of priori estimates and obtain the existence of classical solutions.