Quantitative critical homogenization of a three-dimensional non-homogeneous thermoviscous fluid in perforated domains
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Pan, Jiaojiao, Wang, Luqi
Institution: Nanjing University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214293526