<p>In this paper, we consider the homogenization of stationary and evolutionary incompressible viscous non-Newtonian flows of Carreau–Yasuda type in domains perforated with a large number of periodically distributed small holes in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>, where the mutual distance between the holes is measured by a small parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and the size of the holes is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon ^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ε</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha \in (1, 3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The Darcy’s law is recovered in the limit, thus generalizing the results from Bourgeat and Mikelić (Nonlinear Anal Theory Methods 26(7):1221–1253, 1996. <a href="https://doi.org/10.1016/0362-546X(94)00285-P">https://doi.org/10.1016/0362-546X(94)00285-P</a>) and Lu and Qian (J Differ Equ 411: 619–639, 2024. <a href="https://doi.org/10.1016/j.jde.2024.08.021">https://doi.org/10.1016/j.jde.2024.08.021</a>) for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Instead of using their restriction operator to derive the estimates of the pressure extension by duality, we use the Bogovskiĭ type operator in perforated domains [constructed in Diening et al. (ESAIM Control Optim Calc Var 23:851–868, 2017. <a href="https://doi.org/10.1051/cocv/2016016">https://doi.org/10.1051/cocv/2016016</a>)] to deduce the uniform estimates of the pressure directly. Moreover, quantitative convergence rates are given.</p>

Qualitative/quantitative homogenization of some non-Newtonian flows in perforated domains

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2026年4月28日

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Richard M. Höfer,
Yong Lu,
Florian Oschmann

摘要

In this paper, we consider the homogenization of stationary and evolutionary incompressible viscous non-Newtonian flows of Carreau–Yasuda type in domains perforated with a large number of periodically distributed small holes in $\mathbb {R}^{3}$ $R^{3}$ , where the mutual distance between the holes is measured by a small parameter $\varepsilon >0$ $ε > 0$ and the size of the holes is $\varepsilon ^{\alpha }$ $ε^{α}$ with $\alpha \in (1, 3)$ $α \in (1, 3)$ . The Darcy’s law is recovered in the limit, thus generalizing the results from Bourgeat and Mikelić (Nonlinear Anal Theory Methods 26(7):1221–1253, 1996. https://doi.org/10.1016/0362-546X(94)00285-P) and Lu and Qian (J Differ Equ 411: 619–639, 2024. https://doi.org/10.1016/j.jde.2024.08.021) for $\alpha =1$ $α = 1$ . Instead of using their restriction operator to derive the estimates of the pressure extension by duality, we use the Bogovskiĭ type operator in perforated domains [constructed in Diening et al. (ESAIM Control Optim Calc Var 23:851–868, 2017. https://doi.org/10.1051/cocv/2016016)] to deduce the uniform estimates of the pressure directly. Moreover, quantitative convergence rates are given.