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 $\dot{H}^{-α}(\mathbb{R}^{3})$ or $\dot{H}^{-α}(\mathbb{T}^{3})$ with $0<α\leq 1/2$. 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, Pavlović and Staffilani on (SIMA, [1])in which $α$ is restricted to $0<α<\frac{1}{4}$.

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