Almost sure global weak solutions and optimal decay for the incompressible generalized Navier-Stokes equations

In this paper, we consider the initial value problem of the incompressible generalized Navier-Stokes equations with initial data being in negative order Sobolev spaces, in the whole space ${\mathbb{R}}^d$ with $d\ge 2$. The generalized Navier-Stokes equations studied here is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-\mathrm{\Delta }{)}^{\text{al}}$ with $\text{al}\in (\frac{1}{2},\frac{d+2}{4}]$. After an appropriate randomization on the initial data, we obtain the almost sure existence and optimal decay rate of global weak solutions when the initial data belongs to $\text{Dot}{H}^s({\mathbb{R}}^d)$ with $s\in (-\text{al}+(1-\text{al}{)}_{+},0)$. Moreover, we show that the weak solutions are unique when $\text{al}=\frac{d+2}{4}$ with $d\ge 2$.