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 $C^{3,\beta }$ boundary, $0<\beta <1$. The holes have size ${\epsilon }^{\alpha }$ and separation of order $\epsilon$, with $1<\alpha <3$. Before normalization, the fluid and solid conductivities are ${\sigma }_{\epsilon }^2{\kappa }_f$ and ${\sigma }_{\epsilon }^2{\kappa }_s$, where ${\sigma }_{\epsilon }^2\sim {\epsilon }^{3-\alpha }$. 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 ${\epsilon }^{\alpha -1}+{\epsilon }^{3-\alpha }$ on the torus and ${\epsilon }^{\alpha -1}+{\epsilon }^{(3-\alpha )/2}$ 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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