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

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.