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

TL;DR

This paper investigates the partial regularity of Leray-Hopf weak solutions to the incompressible Navier-Stokes equations with hyperdissipation, establishing bounds on the singular set’s Hausdorff and box-counting dimensions.

Contribution

It provides new bounds on the size of the singular set for solutions with hyperdissipation parameter between 1 and 5/4, extending partial regularity results.

Findings

Hausdorff dimension of singular set ≤ 5 - 4α

Box-counting dimension ≤ (-16α^2 + 16α + 5)/3

Solutions are bounded outside a set of controlled fractal dimension

Abstract

We show that if $u$ is a Leray-Hopf weak solution to the incompressible Navier--Stokes equations with hyperdissipation $\alpha \in (1,5/4)$ then there exists a set $S\subset \mathbb{R}^3$ such that $u$ remains bounded outside of $S$ at each blow-up time, the Hausdorff dimension of $S$ is bounded above by $ 5-4\alpha $ and its box-counting dimension is bounded by $(-16\alpha^2 + 16\alpha +5)/3$. Our approach is inspired by the ideas of Katz & Pavlovi\'c (Geom. Funct. Anal., 2002).