发表机构
Institute of Applied Physics and Computational Mathematics; Nanjing University（北京应用物理与计算数学研究所; 南京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立了三维非均匀热黏性流体的定量Darcy极限，涵盖环面和有界域，给出了误差界并构造了微观弱解。
AI 中文摘要
本文针对具有温度依赖性黏度和准静态热反馈的三维非均匀不可压缩流体，建立了定量的Darcy极限。区域为平坦环面或具有$C^{3,\beta}$边界的有界域，其中$0<\beta<1$。孔洞尺寸为$\varepsilon^\alpha$，间距为$\varepsilon$阶，且$1<\alpha<3$。归一化前，流体和固体的热导率分别为$\sigma_\varepsilon^2\kappa_f$和$\sigma_\varepsilon^2\kappa_s$，其中$\sigma_\varepsilon^2\sim\varepsilon^{3-\alpha}$。该缩放保持了极限中的热反馈。对于足够正则的参考解，并在显式吸收条件下，平方密度、速度和温度误差由加权初始差异以及环面上的$\varepsilon^{\alpha-1}+\varepsilon^{3-\alpha}$和有界域中的$\varepsilon^{\alpha-1}+\varepsilon^{(3-\alpha)/2}$界定。一个与界面无关的温度估计控制了黏度差异，加权Stokes残差保留了系数和单元压力交换子。有界比较使用了无散壁面修正和逐单元散度修复。我们还构造了固定参数的微观弱解。局部强有效解在两种设置中都需要单独的收缩条件，在有界情形下还需要更强的边界正则性。
英文摘要
In this paper, we establish quantitative Darcy limits for a three-dimensional inhomogeneous incompressible fluid with temperature-dependent viscosity and quasi-static thermal feedback. The domain is either the flat torus or a bounded domain with $C^{3,β}$ boundary, $0<β<1$. The holes have size $\varepsilon^α$ and separation of order $\varepsilon$, with $1<α<3$. Before normalization, the fluid and solid conductivities are $σ_\varepsilon^2κ_f$ and $σ_\varepsilon^2κ_s$, where $σ_\varepsilon^2\sim\varepsilon^{3-α}$. This scaling preserves the thermal feedback in the limit. For a sufficiently regular reference solution and under an explicit absorption condition, the squared density, velocity and temperature errors are bounded by the weighted initial discrepancy together with $\varepsilon^{α-1}+\varepsilon^{3-α}$ on the torus and $\varepsilon^{α-1}+\varepsilon^{(3-α)/2}$ in a bounded domain. An interface-independent temperature estimate controls the viscosity discrepancy, and a weighted Stokes residual retains the coefficient and cell-pressure commutators. The bounded comparison uses a solenoidal wall correction and cellwise divergence repair. We also construct fixed-parameter microscopic weak solutions. The local strong effective solutions require a separate contraction condition in both settings and stronger boundary regularity in the bounded case.
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