  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.
