In this study, we are concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes Korteweg equations of a compressible fluid in the half space. We assume that the space-asymptotic states and the boundary data satisfy some conditions so that the time-asymptotic state of this solution is a rarefaction wave. Then we show that the rarefaction wave is non-linearly stable, as time goes to infinity, provided that the strength of the wave is weak and the initial perturbation is small. The proof is mainly based on $L^{2}$-energy method and some time-decay estimates in $L^{p}$-norm for the smoothed rarefaction wave.

在这项研究中，我们关注半空间中可压缩流体的 Navier-Stokes Korteweg 方程的流出问题解的稀疏波的渐近稳定性。我们假设空间渐近状态和边界数据满足一些条件，因此该解的时间渐近状态是稀疏波。然后我们证明稀疏波是非线性稳定的，当时间趋于无穷时，假设波的强度很弱并且初始扰动很小。证明主要基于$L^{2}$ -energy 方法和$L^{p}$ -norm 中的一些时间衰减估计，用于平滑稀疏波。