📝 SummarXiv

Abstract

We investigate the time-asymptotic stability of solutions to the one-dimensional Navier-Stokes-Fourier system in the half space, focusing on the outflow and impermeable wall problems. When the asymptotic profile determined by the prescribed constant states at the boundary and at the far field is a viscous shock, we show that the solution converges asymptotically to the viscous shock profile, up to a dynamical shift, provided the initial perturbation and the shock amplitude are sufficiently small. In order to obtain our results, we employ the method of a-contraction with shifts. Although the impermeable wall problem is technically simpler to analyze in Lagrangian mass coordinates, the outflow problem leads to a free boundary in that framework. Therefore, we use Eulerian coordinates to provide a unified approach to both problems. This is the first result on the time-asymptotic stability of viscous shocks for initial-boundary value problems of the Navier-Stokes-Fourier system for the outflow and impermeable wall cases.