Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1

TL;DR

This paper demonstrates the existence of weak solutions to the Navier-Stokes equations that are smooth except on a fractal set of singular times with Hausdorff dimension less than 1, showing non-uniqueness in such cases.

Contribution

It introduces a class of weak solutions with fractal singular sets in time, expanding understanding of solution behaviors and non-uniqueness in Navier-Stokes equations.

Findings

Existence of solutions with fractal singular sets in time

Solutions have bounded kinetic energy and integrable vorticity

Singular sets have Hausdorff dimension less than 1

Abstract

We prove non-uniqueness for a class of weak solutions to the Navier-Stokes equations which have bounded kinetic energy, integrable vorticity, and are smooth outside a fractal set of singular times with Hausdorff dimension strictly less than 1.