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Partial regularity of Leray–Hopf weak solutions to the incompressible Navier–Stokes equations with hyperdissipation

Wojciech S. Ożański

Vol. 16 (2023), No. 3, 747–783
Abstract

We show that if u is a Leray–Hopf weak solution to the incompressible Navier–Stokes equations with hyperdissipation α (1, 5 4) then there exists a set S 3 such that u remains bounded outside of S at each blow-up time, the Hausdorff dimension of S is bounded above by 5 4α and its box-counting dimension is bounded by 1 3(16α2 + 16α + 5). Our approach is inspired by the ideas of Katz and Pavlović (Geom. Funct. Anal. 12:2 (2002), 355–379).

Keywords
Navier–Stokes, hyperdissipation, partial regularity, Leray–Hopf weak solutions, box-counting dimension, Hausdorff dimension
Mathematical Subject Classification
Primary: 35Q30, 35Q35, 35R11, 76D03, 76D05
Secondary: 35B44, 35B65
Milestones
Received: 10 August 2020
Revised: 27 August 2021
Accepted: 6 October 2021
Published: 25 May 2023
Authors
Wojciech S. Ożański
Department of Mathematics
Florida State University
Tallahassee, FL
United States

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