This paper is concerned with the time-asymptotic behavior of strong solutions to an initial-boundary value problem of the compressible Navier-Stokes-Korteweg system on the half line $\mathbb{R}^+$. The asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary. Moreover, we prove that if the initial data around the shifted viscous shock profile and the strength of the shifted viscous shock profile are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. The analysis is based on the elementary $L^2$-energy method and the key point is to deal with the boundary estimates.

中文翻译：

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具有边界效应的一维可压缩 Navier-Stokes-Korteweg 系统粘性冲击剖面的渐近稳定性

本文涉及可压缩 Navier-Stokes-Korteweg 系统在半线 $\mathbb{R}^+$ 上的初边值问题的强解的时间渐近行为。问题的渐近剖面显示为偏移的粘性冲击剖面，它适当地远离边界。此外，我们证明，如果围绕移动粘性冲击剖面的初始数据和移动粘性冲击剖面的强度足够小，那么该问题具有独特的全局强解，随着时间的推移趋于移动粘性冲击剖面无限。分析基于基本的$L^2$-energy方法，重点是处理边界估计。