Global stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system with temperature-dependent transport coefficients and large data

Abstract

We study the Cauchy problem for the one-dimensional full compressible Navier--Stokes equations with viscosity $\mu(\theta)=\tilde\mu\theta^\alpha$ and heat conductivity $\kappa(\theta)=\tilde\kappa\theta^\beta$. For each fixed $\beta\ge0$, we prove the global stability of the combination of a viscous contact wave with rarefaction waves, provided that the viscosity exponent $\alpha$ and the total wave strength are sufficiently small. The initial perturbation may be large in $H^1$, and the specific volume and temperature are assumed to have positive initial lower bounds. The solution remains uniformly bounded in $H^1$ relative to the wave, and converges uniformly to that wave as time tends to infinity. We show that both the specific volume and the temperature admit time-uniform lower and upper bound. Estimates for the logarithmic volume derivative and the temperature gradient then close the $H^1$ estimates without requiring a second derivative of the initial specific volume.