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

Ożański, Wojciech S.

doi:10.2140/apde.2023.16.747

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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

DOI: 10.2140/apde.2023.16.747

Abstract

We show that if $u$ is a Leray–Hopf weak solution to the incompressible Navier–Stokes equations with hyperdissipation $α \in (1, \frac{5}{4})$ then there exists a set $S \subset ℝ^{3}$ such that $u$ remains bounded outside of $S$ at each blow-up time, the Hausdorff dimension of $S$ is bounded above by $5 - 4 α$ and its box-counting dimension is bounded by $\frac{1}{3} (- 16 α^{2} + 16 α + 5)$ . 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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Vol. 16, No. 3, 2023