AI 中文总结
研究含大量小孔区域中三维Navier-Stokes-Cahn-Hilliard系统均匀化，考虑粘度和迁移率依赖相场变量，通过建立渐近区域，在不同毛细强度条件下得出极限系统，首次对演化NSCH流在亚临界孔缩放时进行严格均匀化分析。
AI 中文摘要
本文研究了在包含大量固体障碍物（称为孔）的区域中三维Navier-Stokes-Cahn-Hilliard（NSCH）系统的均匀化。每个孔的直径为\(\varepsilon^{\alpha}(\alpha>3)\)量级，\(\varepsilon>0\)表示孔间分离的小长度尺度。粘度和迁移率都依赖于相场变量。我们建立了两种不同的渐近区域：当\(\varepsilon\to0\)时，若毛细强度\(\lambda_\varepsilon\to\lambda>0\)，极限系统与原NSCH系统一致；若\(\lambda_\varepsilon\to0\)，缩放后的速度、相场和化学势收敛到Stokes-Cahn-Hilliard（SCH）系统的弱解。据我们所知，这项工作是首次对具有依赖相的粘度和迁移率的演化NSCH流在亚临界孔缩放情况下进行严格的均匀化分析。
英文摘要
This paper investigates the homogenization of the 3D Navier--Stokes--Cahn--Hilliard (NSCH) system in domains containing a large number of solid obstacles (named holes). Each hole has diameter of order $\varepsilon^α(α>3)$, where $\varepsilon > 0$ denotes the small length scale for inter-hole separation. Both viscosity and mobility depend on the phase-field variable. We establish two distinct asymptotic regimes: if the capillary strength $λ_\varepsilon\to λ>0$ as $\varepsilon\to 0$, the limit system coincides with the original NSCH system; if $λ_\varepsilon\to 0$ as $\varepsilon\to 0$, the scaled velocity, phase field and chemical potential converge to a weak solution to a Stokes--Cahn--Hilliard (SCH) system. To the best of our knowledge, this work constitutes the first rigorous homogenization analysis for evolutionary NSCH flows with phase-dependent viscosity and mobility under the subcritical hole scaling.
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