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 & Pavlović (Geom. Funct. Anal., 2002).

37 pages, 3 figures

References in corpus (3)

• The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system
• Improved bounds for box dimensions of potential singular points to the Navier--Stokes equations
• Local regularity of weak solutions of the hypodissipative Navier-Stokes equations

Cited by in corpus (1)

• Local regularity of weak solutions of the hypodissipative Navier-Stokes equations