Global stability of combination of viscous contact wave with rarefaction wave for compressible Navier-Stokes equations with temperature-dependent viscosity

We study the nonlinear stability of a composite wave pattern, which is a combination of a viscous contact wave with a rarefaction wave, to the Cauchy problem of one-dimensional compressible Navier-Stokes equations for a viscous and heat conducting ideal polytropic gas with large initial perturbation when the transport coefficients depend on both temperature and density. Our main idea is to use the "smallness mechanism" induced by the structures of the equations under consideration and the smallness of the strengths of the two elementary waves to control the possible growth of the solutions caused by the nonlinearities of the equations, the interactions between the solutions themselves and the wave pattern, and the interactions of waves between different families. The main ingredient in the analysis is to derive the uniform positive lower and upper bounds on the specific volume and the temperature.