Almost sure existence of global weak solutions for super-critical   Navier-Stokes equations
  In this paper we show that after suitable data randomization there exists a large set of super-critical periodic initial data, in $H^{-\alpha}({\mathbb T}^d)$ for some $\alpha(d) > 0$, for both 2d and 3d Navier-Stokes equations for which global energy bounds are proved. As a consequence, we obtain almost sure super-critical global weak solutions. We also show that in 2d these global weak solutions are unique. 
Andrea R. Nahmod, Nata\v{s}a Pavlovi\'c and Gigliola Staffilani
math.AP