Title: Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems Show Chinese title

Title: 粘性激波的稳定性：纳维-斯托克斯-傅里叶系统的流出和不可渗透壁问题

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 prescribed boundary and far-field conditions form an outgoing viscous shock, we prove that the solution converges to the viscous shock profile, up to a dynamical shift, provided that 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. Show Chinese abstract

Abstract: 我们研究了一维Navier-Stokes-Fourier系统在半空间中的解的时间渐近稳定性，重点关注流出和不可渗透壁问题。 当规定的边界条件和远场条件形成一个向外的粘性激波时，我们证明了在初始扰动和激波幅度足够小的情况下，解会收敛到粘性激波轮廓，最多有一个动态位移。 为了获得我们的结果，我们采用了带有位移的a-压缩方法。 尽管不可渗透壁问题在拉格朗日质量坐标中技术上更容易分析，但流出问题会导致该框架中的自由边界。 因此，我们使用欧拉坐标来为这两个问题提供一种统一的方法。 这是关于Navier-Stokes-Fourier系统流出和不可渗透壁情况下初始边界值问题的粘性激波时间渐近稳定性的第一个结果。