Global strong solutions to the 1D full compressible Navier--Stokes equations with temperature-dependent coefficients and vacuum

Abstract

We study the initial--boundary value problem for the one-dimensional full compressible Navier--Stokes system whose viscosity and heat conductivity are given by $μ(θ)=μθ^α$ and $κ(θ)=κθ^β$, respectively. The initial density is nonnegative and is allowed to vanish. For $α>2$ and $β>2α+4$, we prove the existence of a global strong solution for arbitrarily large compatible $H^2$ data, provided that the prescribed boundary temperature is sufficiently high relative to their size. No velocity or temperature perturbation is required to shrink as the boundary temperature tends to infinity. A shifted initial-layer weight and an exact symmetrization of the two material-derivative equations separate the fast thermal mode from the momentum mode. The resulting estimates are uniform with respect to the positive lower bound used in the approximation and therefore survive the limit to genuine vacuum. The proof further combines a relative-entropy identity centered at the conserved mean density, the physical effective viscous flux, a Zlotnik-type estimate along particle trajectories, and finite-time transport estimates for the density derivatives. The density remains uniformly bounded and the temperature stays uniformly comparable with the prescribed boundary temperature. After the initial layer we establish vacuum-compatible exponential relaxation: the density converges to its conserved mean in every finite $L^p$, the velocity and temperature perturbation decay in $H^1$, and the weighted material-derivative energy decays at rate $c\barθ^{1-α}$.