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Large-scale regularity for the stationary Navier–Stokes equations over non-Lipschitz boundaries

Mitsuo Higaki, Christophe Prange and Jinping Zhuge

Vol. 17 (2024), No. 1, 171–242

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, C1,γ and C2,γ 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 C1,γ regularity is obtained by using first-order boundary layers constructed via a new argument. The large-scale C2,γ 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.

Navier–Stokes equations, large-scale regularity, boundary layers, John domains, Green's functions, homogenization
Mathematical Subject Classification
Primary: 35Q30, 35B27, 76D03, 76D10, 76M50
Received: 24 September 2021
Revised: 27 March 2022
Accepted: 16 June 2022
Published: 5 February 2024
Mitsuo Higaki
Department of Mathematics
Graduate School of Science
Kobe University
Christophe Prange
Laboratoire de Mathématiques AGM
Cergy Paris Université
Jinping Zhuge
Department of Mathematics
University of Chicago
Chicago, IL
United States

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