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 $T^d$ with $d\ge 2$. The generalized Navier-Stokes equations are obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-{(-\Delta )}^{\alpha }$ with $\alpha \in (\frac{2}{3},1]$. After an appropriate randomization on the initial data, we obtain the almost sure existence of global weak solutions for initial data being in ${\dot{H}}^s\left(T^d\right)$ with $s\in (1-2\alpha ,0)$.