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

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).