Initial and initial boundary value problems for the Navier-Stokes equations with temperature-dependent transport coefficients and large data

We consider initial and initial boundary value problems for the one-dimensional compressible Navier–Stokes equations of an ideal polytropic gas in unbounded domains. The viscosity and heat conductivity are proportional to ${\theta }^{\alpha }$ and ${\theta }^{\beta }$, respectively. For every fixed $\beta \ge 0$ and sufficiently small $\alpha \ge 0$, depending on the initial data, we prove the global existence and asymptotic stability of a strong solution for arbitrarily large $H^1$ initial data. Dividing the momentum equation by the viscosity allows us to estimate the specific-volume derivative without an initial $H^2$ assumption.