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Large-scale
regularity for the stationary Navier–Stokes equations over
non-Lipschitz boundaries
Mitsuo Higaki, Christophe Prange and Jinping Zhuge |
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Vol. 17 (2024), No. 1, 171–242
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Abstract |
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We address the large-scale regularity theory for the stationary Navier–Stokes equations in highly oscillating bumpy John domains. These domains are very rough, possibly with fractals or cusps, at the microscopic scale, but are amenable to the mathematical analysis of the Navier–Stokes equations. We prove a large-scale Calderón–Zygmund estimate, a large-scale Lipschitz estimate, and large-scale higher-order regularity estimates, namely, and estimates. These nice regularity results are inherited only at mesoscopic scales, and clearly fail in general at the microscopic scales. We emphasize that the large-scale regularity is obtained by using first-order boundary layers constructed via a new argument. The large-scale regularity relies on the construction of second-order boundary layers, which allows for certain boundary data with linear growth at spatial infinity. To the best of our knowledge, our work is the first to carry out such an analysis. In the wake of many works in quantitative homogenization, our results strongly advocate in favor of considering the boundary regularity of the solutions to fluid equations as a multiscale problem, with improved regularity at or above a certain scale. |
Keywords
Navier–Stokes equations, large-scale regularity, boundary
layers, John domains, Green's functions, homogenization
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Mathematical Subject Classification
Primary: 35Q30, 35B27, 76D03, 76D10, 76M50
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Milestones
Received: 24 September 2021
Revised: 27 March 2022
Accepted: 16 June 2022
Published: 5 February 2024
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Authors |
| Mitsuo Higaki | |
| Department of Mathematics Graduate School of Science Kobe University Kobe Japan |
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| Christophe Prange | |
| Laboratoire de Mathématiques
AGM UMR CNRS 8088 Cergy Paris Université Cergy-Pontoise France |
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| Jinping Zhuge | |
| Department of Mathematics University of Chicago Chicago, IL United States |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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