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

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 $0<\alpha \le 1\mathrm{/}2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $t\ge 0$. 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 $0<\alpha <\frac{1}{4}$.