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 $d \geq 2$. 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 $-(-\Delta)^\al$ with $\al \in \left( \frac{1}{2},\frac{d+2}{4} \right]$. 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 $\Dot{H}^s(\mathbb{R}^d)$ with $s\in (-\al+(1-\al)_+,0)$. Moreover, we show that the weak solutions are unique when $\al=\frac{d+2}{4}$ with $d \geq 2$.