Non-uniqueness of weak solutions for a logarithmically supercritical hyperdissipative Navier-Stokes system

TL;DR

This paper demonstrates the existence of multiple finite-energy solutions for a supercritical hyperdissipative Navier-Stokes system, extending prior results using convex integration and introducing impulsed Beltrami flows.

Contribution

It extends non-uniqueness results to a logarithmically supercritical setting by developing impulsed Beltrami flows, advancing the understanding of solution behavior near critical thresholds.

Findings

Proves non-uniqueness of solutions in supercritical Navier-Stokes setting

Introduces impulsed Beltrami flows as a new tool

Extends previous non-uniqueness results to a closer supercritical regime

Abstract

The existence of non-unique solutions of finite kinetic energy for the three dimensional Navier-Stokes equations is proved in the slightly supercritical hyper-dissipative setting introduced by Tao. The result is based on the convex integration techniques of Buckmaster and Vicol and extends Luo and Titi result in the slightly supercritical setting. To be able to be closer to the threshold identified by Tao, we introduce the impulsed Beltrami flows, a variant of the intermittent Beltrami flows of Buckmaster and Vicol.