We shall consider the one-dimensional Navier–Stokes–Korteweg (NSK) equations that are used to model compressible fluids with internal capillarity. Formally, the NSK equations converge, as the viscosity and capillarity vanish, to the corresponding Euler equations, and we do justify this for the case that the Euler equations have a rarefaction wave with one-side vacuum state. To prove this result, we first construct a sequence of approximate solutions to the NSK equations and then show that the sequence converges, as both viscosity and capillarity tend simultaneously to zero, to this rarefaction wave. The uniform convergence rates are also obtained. The key ingredients of our proof are the re-scaling technique and energy estimates.

中文翻译：

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可压缩流体的 Navier-Stokes-Korteweg 方程对真空稀疏波的零粘度-毛细管极限

我们将考虑用于模拟具有内部毛细作用的可压缩流体的一维 Navier-Stokes-Korteweg (NSK) 方程。形式上，NSK 方程随着粘度和毛细作用的消失收敛到相应的欧拉方程，我们确实证明了欧拉方程具有一侧真空状态的稀疏波的情况。为了证明这个结果，我们首先构造了 NSK 方程的一系列近似解，然后表明该序列收敛到这个稀疏波，因为粘度和毛细管力同时趋于零。还获得了统一的收敛速度。我们证明的关键要素是重新缩放技术和能量估计。