Quantitative thermoviscous Darcy limits in periodic and bounded domains
Abstract
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 C3,β boundary, 0<β<1. The holes have size εα and separation of order ε, with 1<α<3. Before normalization, the fluid and solid conductivities are σε2κf and σε2κs, where σε2∼ε3−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 εα−1+ε3−α on the torus and εα−1+ε(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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BibTeX
@misc{pan2026quantitative,
title = {{Quantitative thermoviscous Darcy limits in periodic and bounded domains}},
author = {Jiaojiao Pan and Luqi Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09528},
eprint = {2610.09528},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09528},
}
RIS
TY - UNPB
TI - Quantitative thermoviscous Darcy limits in periodic and bounded domains
AU - Jiaojiao Pan
AU - Luqi Wang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09528
UR - https://arxiv.org/abs/2610.09528
AB - 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 ε^α and separation of order ε, with 1<α<3. Before normalization, the fluid and solid conductivities are σ_ε^2κ_f and σ_ε^2κ_s, where σ_ε^2∼ε^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 ε^α−1+ε^3−α on the torus and ε^α−1+ε^(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.
ER -