Almost sure existence of Navier-Stokes Equations with randomized data in the whole space

TL;DR

This paper demonstrates the almost sure existence of global weak solutions to the supercritical Navier-Stokes equations in the whole space with randomized initial data, using probabilistic methods to handle a broad class of initial conditions.

Contribution

It introduces a probabilistic approach to establish the almost sure existence of weak solutions for supercritical Navier-Stokes equations with randomized initial data in the whole space.

Findings

Global weak solutions exist almost surely for a large set of initial data.

The method applies to initial data in $H^{-s}( ^d)$ for some $s>0$.

Probabilistic techniques enable handling supercritical regimes.

Abstract

This paper considers the supercritical Navier-Stokes equations posed in the whole space $\R^d$, with suitably randomized initial data, in the weak solution setting. The global weak solutions are constructed for a large set of initial data in $H^{-s}(\R^d)$ for some $s>0$ via a probabilistic argument, and this in turn implies the almost sure existence.