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

TL;DR

This paper proves the almost sure existence of global weak solutions for a class of generalized Navier-Stokes equations with fractional Laplacian, using randomization of initial data in a specific Sobolev space.

Contribution

It establishes the almost sure existence of solutions for fractional Navier-Stokes equations with randomized initial data in a certain Sobolev space, extending previous deterministic results.

Findings

Global weak solutions exist almost surely for initial data in specified Sobolev spaces.

Randomization technique is effective in proving existence results.

Results apply to equations with fractional Laplacian of order between 2/3 and 1.

Abstract

In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-\Delta)^\al$ with $\al \in \left( \frac{2}{3},1 \right]$. 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 $s\in (1-2\al,0)$.