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 $0<\alpha<\frac{ 1}{ 4}$.
Categories: math.AP