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

Abstract

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 $-(-Δ)^\al$ with $\al \in \left( \frac{2}{3},1 \right]$. After an appropriate randomization on the initial data, we obtain the almost sure existence of global weak solutions for initial data being in $\Dot{H}^s(\mathbb{T}^d)$ with $s\in (1-2\al,0)$.