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

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 [20]. The result is based on the convex integration techniques of Buckmaster and Vicol [3], and extends Luo and Titi [16] in the slightly supercritical setting. To reach the threshold identified by Tao, we introduce the impulsed Beltrami flows, a variant of the intermittent Beltrami flows of Buckmaster and Vicol.