Almost sure existence of global weak solutions for super-critical Navier-Stokes equations

TL;DR

This paper demonstrates that, through data randomization, there is a high probability of global weak solutions existing for super-critical Navier-Stokes equations in both 2D and 3D, with uniqueness in 2D.

Contribution

It introduces a probabilistic approach to establish the existence of global weak solutions for super-critical Navier-Stokes equations, including uniqueness in 2D.

Findings

Existence of global weak solutions for super-critical data after randomization.

Global energy bounds are established for these solutions.

Uniqueness of solutions in 2D cases.

Abstract

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.