Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations

In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus ${\mathbb{T}}^d$ with $d\ge 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-\mathrm{\Delta }{)}^{\text{al}}$ with $\text{al}\in (\frac{2}{3},1]$. After an appropriate randomization on the initial data, we obtain the almost sure existence of global weak solutions for initial data being in $\text{Dot}{H}^s({\mathbb{T}}^d)$ with $s\in (1-2\text{al},0)$.