Almost Sure Existence of Global Weak Solutions for Supercritical Navier--Stokes Equations

Abstract

In this paper we show that after suitable data randomization there exists a large set of supercritical periodic initial data, in $H^{-\alpha}(\boldsymbol{T}^d)$ for some $\alpha(d) > 0$, for both two- and three-dimensional Navier--Stokes equations for which global energy bounds hold. As a consequence, we obtain almost sure large data supercritical global weak solutions. We also show that in two dimensions these global weak solutions are unique.