穿孔区域中Navier--Stokes--Cahn--Hilliard系统的临界均匀化
Critical homogenization of the Navier--Stokes--Cahn--Hilliard system in perforated domains
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中文总结 AI 辅助
研究穿孔域中NSCH系统的临界均匀化，在Stokes容量尺度下得到带Brinkman阻力的极限方程，并处理相依赖粘度与校正子的相互作用，同时分析毛细系数消失的极限。
中文摘要 AI 辅助
我们考虑在由无滑移障碍物穿孔的三维不可压缩Navier--Stokes--Cahn--Hilliard（NSCH）系统，障碍物直径为\\(O(\varepsilon^3)\\)，间距为\\(\varepsilon\\)量级。粘度和迁移率都可能依赖于相变量。在此临界Stokes容量尺度下，极限速度\\(\boldsymbol u\\)和相场\\(\phi\\)满足带有额外Brinkman阻力\\(\nu(\phi)\mathbf B\boldsymbol u\\)的NSCH系统，其中\\(\mathbf B\\)由参考障碍物的外部Stokes容量决定。主要的分析难点在于相依赖粘度与临界Stokes校正子的阶一能量集中之间的相互作用。变分化学势恒等式在标量延拓后，给出相在\\(L^2(0,T;H^1(\Omega))\\)中的强收敛，从而相依赖粘度在同一拓扑中强收敛。结合逐胞Hardy乘子估计，这种紧性可以传递通过集中的校正子层。所得的加权容量陈述适用于具有\\(L^2(0,T;H^1(\Omega))\\)正则性的均匀正且有界系数，并同时识别有效Brinkman力和相应的粘性耗散下界。我们还研究了消失的毛细系数\\(\lambda_\varepsilon\to0\\)。在自然速度缩放后，极限是耦合到无平流Cahn--Hilliard方程的非定常Stokes--Brinkman方程。
英文摘要
We consider the three-dimensional incompressible Navier--Stokes--Cahn--Hilliard (NSCH) system in domains perforated by no-slip obstacles of diameter of order \(\varepsilon^3\) separated by distances of order \(\varepsilon\). Both viscosity and mobility may depend on the phase variable. At this critical Stokes-capacity scale, the limiting velocity \(\boldsymbol u\) and phase field \(ϕ\) satisfy an NSCH system with the additional Brinkman resistance \(ν(ϕ)\mathbf B\boldsymbol u\), where \(\mathbf B\) is determined by the exterior Stokes capacity of the reference obstacle. The principal analytical difficulty is the interaction between the phase-dependent viscosity and the order-one energy concentration of the critical Stokes correctors. The variational chemical-potential identity yields, after scalar extension, strong convergence of the phase in \(L^2(0,T;H^1(Ω))\), and hence strong convergence of the phase-dependent viscosity in the same topology. Combined with a cellwise Hardy multiplier estimate, this compactness can be transferred through the concentrated corrector layer. The resulting weighted-capacity statement applies to uniformly positive and bounded coefficient sequences converging strongly in \(L^2(0,T;H^1(Ω))\) and simultaneously identifies the effective Brinkman force and the corresponding viscous dissipation lower bound. We also investigate a vanishing capillary coefficient \(λ_\varepsilon\to0\). After the natural velocity scaling, the limit is an unsteady Stokes--Brinkman equation coupled to an unadvected Cahn--Hilliard equation.
发表机构
- School of Mathematics, Nanjing University（南京大学数学系）
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