Almost Sure Existence of Global Weak Solutions to the 3D Incompressible Navier-Stokes Equation

TL;DR

This paper proves that for a large set of initial data in certain negative Sobolev spaces, there almost surely exist global weak solutions to the 3D incompressible Navier-Stokes equations, extending previous results to larger data sets.

Contribution

It extends the almost sure existence of global weak solutions to larger initial data sets in negative Sobolev spaces for the 3D Navier-Stokes equations, improving prior restrictions on the parameter alpha.

Findings

Almost sure existence of global weak solutions for large data.

Energy of solutions remains finite for all time.

Improves previous restrictions on initial data regularity.

Abstract

In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $\dot{H}^{-\alpha}(\mathbb{R}^{3})$ or $\dot{H}^{-\alpha}(\mathbb{T}^{3})$ with $0<\alpha\leq 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $t\geq0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlovi\'{c} and Staffilani on (SIMA, [1])in which $\alpha$ is restricted to $0<\alpha<\frac{1}{4}$.