Critical homogenization of the Navier--Stokes--Cahn--Hilliard system in perforated domains
Abstract
We consider the three-dimensional incompressible Navier--Stokes--Cahn--Hilliard (NSCH) system in domains perforated by no-slip obstacles of diameter of order ε3 separated by distances of order ε. Both viscosity and mobility may depend on the phase variable. At this critical Stokes-capacity scale, the limiting velocity u and phase field ϕ satisfy an NSCH system with the additional Brinkman resistance ν(ϕ)Bu, where 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 L2(0,T;H1(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 L2(0,T;H1(Ω)) and simultaneously identifies the effective Brinkman force and the corresponding viscous dissipation lower bound. We also investigate a vanishing capillary coefficient λε→0. After the natural velocity scaling, the limit is an unsteady Stokes--Brinkman equation coupled to an unadvected Cahn--Hilliard equation.
Related
- Homogenization of an optimal control problem for a Navier-Stokes-Cahn-Hilliard system in a 3D perforated domain
- Quantitative critical homogenization of a three-dimensional non-homogeneous thermoviscous fluid in perforated domains
- Bulk-surface spectral homogenization with indefinite mass in periodic perforated domains
- Surface-to-Volume Homogenization of NonLinear Steklov Eigencouples with NonHomogeneous Orlicz Growth in Perforated Domains
- Homogenization of degenerate parabolic equations with nonlinear Robin reactions in perforated domains
BibTeX
@misc{pan2026critical,
title = {{Critical homogenization of the Navier--Stokes--Cahn--Hilliard system in perforated domains}},
author = {Jiaojiao Pan and Luqi Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25119},
eprint = {2609.25119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25119},
}
RIS
TY - UNPB
TI - Critical homogenization of the Navier--Stokes--Cahn--Hilliard system in perforated domains
AU - Jiaojiao Pan
AU - Luqi Wang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.25119
UR - https://arxiv.org/abs/2609.25119
AB - We consider the three-dimensional incompressible Navier--Stokes--Cahn--Hilliard (NSCH) system in domains perforated by no-slip obstacles of diameter of order ε^3 separated by distances of order ε. Both viscosity and mobility may depend on the phase variable. At this critical Stokes-capacity scale, the limiting velocity u and phase field ϕ satisfy an NSCH system with the additional Brinkman resistance ν(ϕ)Bu, where 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_ε→0. After the natural velocity scaling, the limit is an unsteady Stokes--Brinkman equation coupled to an unadvected Cahn--Hilliard equation.
ER -