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.
Categories: math.AP