The Outer Pressure Problem for the Compressible Navier--Stokes System with Temperature-Dependent Transport Coefficients

We study global strong solutions and their large-time behavior for the one-dimensional outer pressure problem of a viscous, heat-conducting ideal polytropic gas. The viscosity and heat conductivity satisfy $\mu =\tilde{\mu }{\theta }^{\alpha }$ and $\kappa =\tilde{\kappa }{\theta }^{\beta }$. For each prescribed nonnegative pressure and each fixed $\beta \ge 0$, we allow large $H^1$ initial data with positive specific volume and temperature, provided that $\alpha \ge 0$ is sufficiently small. When the limiting outer pressure is positive, the solution has uniform bounds and converges to a stationary state. When the limiting outer pressure is zero and the stated weighted pressure conditions hold, the normalized solution converges to an expanding state at an algebraic rate in physical time.