三维穿孔域中非均匀热黏性流体的临界定量均匀化
Quantitative critical homogenization of a three-dimensional non-homogeneous thermoviscous fluid in perforated domains
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中文总结 AI 辅助
针对三维穿孔域中的非均匀热黏性流体，在临界Stokes-容量尺度下证明了耦合系统的定量均匀化稳定性估计，给出了误差阶$O(\eps)$的收敛结果，并建立了有效系统的局部正则可解性。
中文摘要 AI 辅助
本文研究了一个在临界Stokes-容量尺度下，位于有界周期穿孔域中的三维非均匀不可压缩流体。密度由流动输运，而黏度依赖于一个由均匀椭圆传输问题控制的准静态温度，该问题具有齐次Neumann边界条件和零空间均值。在临界状态下，均匀化的动量方程包含一个Brinkman阻力项。在已有的定性临界Brinkman极限之外，我们证明了耦合密度-速度-温度系统的定量稳定性估计。对于每个固定的穿孔几何，全局有限能量弱解存在。给定一个正则的有效解，每个微观弱解的平方相对误差由初始失配加上一个$O(\eps^{2})$余项界定。对于良好准备的数据，该估计在$L^\infty(0,T;L^2)$中产生密度和未修正速度的$O(\eps)$阶收敛，在$L^2(0,T;H^1)$中产生温度和修正速度的$O(\eps)$阶收敛。证明结合了一个螺线管限制算子、对偶能量范数中的$O(\eps)$单元容量残差、$O(\eps)$的$L^{6/5}$校正子梯度估计以及固定域温度稳定性。临界边界层保持一阶黏性能量，在极限中由Brinkman耗散表示。最后，对于满足有限层次初始边界相容条件的光滑规定力，建立了有效系统的局部正则可解性。
英文摘要
In this paper, we consider a three-dimensional non-homogeneous incompressible fluid in a bounded periodically perforated domain at the critical Stokes-capacity scale. The density obeys a transport equation. The viscosity depends on a quasi-static temperature perturbation, which solves a uniformly elliptic transmission problem with insulated outer boundary conditions and zero spatial mean. The limiting momentum equation contains a Brinkman resistance term generated by the critical perforation regime. For each fixed perforated domain, we construct global finite-energy weak solutions and derive a quantitative relative-energy stability estimate with respect to regular solutions of the homogenized system. As long as a regular effective solution exists, the squared relative error between any microscopic weak solution and the effective solution is bounded by the initial mismatch plus $O(\eps^{2})$. For well-prepared data, the density and uncorrected velocity converge at order $O(\eps)$ in $L^\infty(0,T;L^2)$, the temperature at the same order in both $L^\infty((0,T)\timesΩ)$ and $L^2(0,T;H^1)$, and the corrected velocity in $L^2(0,T;H^1)$. The key estimates are an $O(\eps)$ cell-capacity residual in the dual energy norm, an $O(\eps)$ corrector-gradient bound in $L^{6/5}$, and fixed-domain thermal stability. The critical boundary layers retain order-one viscous energy, represented in the limit by the Brinkman dissipation, so the uncorrected velocity gradient does not converge strongly to the effective gradient whenever the effective velocity is nonzero. Finally, we prove local existence of regular effective solutions for smooth data satisfying a finite set of boundary compatibility conditions, which closes the stability argument.
发表机构
- School of Mathematics, Nanjing University（南京大学数学系）
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