SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 6155-6179, January 2020.
We consider a nonlocal version of the quasi-static Navier--Stokes--Korteweg equations with a nonmonotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapor state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn--Hilliard equation. Crucial for the analysis is the convolution-like structure of the nonlocal capillarity term that allows us to equip the system with a generalized convex free energy. Moreover, the capillarity term accounts not only for the energetic interaction within the fluid but also for the interaction with a solid wall boundary.

中文翻译：

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多孔介质中可压缩液-汽流的非局部Navier-Stokes-Korteweg方程的均质化

SIAM数学分析杂志，第52卷，第6期，第6155-6179页，2020年1月。
我们考虑具有非单调压力定律的拟静态Navier-Stokes-Korteweg方程的非局部版本。该系统控制可采取液体或蒸汽状态的可压缩粘性流体的低雷诺数动力学。对于由直径与它们的相互距离成正比的孔洞穿孔的多孔区域，将分析均质极限。我们用多变压力定律扩展了可压缩单相流的结果，并证明有效运动受Cahn-Hilliard方程的非局部形式控制。对于分析至关重要的是非局域毛细作用项的类似卷积的结构，它使我们能够为系统配备广义的凸自由能。此外，