Global existence of uniformly locally energy solutions for the incompressible fractional Navier-Stokes equations

TL;DR

This paper establishes the existence of local and global energy solutions for the fractional Navier-Stokes equations with initial data in a local square-integrable space, extending the understanding of solution behavior for fractional powers of the Laplacian.

Contribution

It introduces local Leray solutions for fractional Navier-Stokes equations with initial data in locally square-integrable spaces and proves their local and global existence for certain fractional orders.

Findings

Existence of local Leray solutions for $s \\in [3/4,1)$.

Global-in-time solutions for initial data vanishing at infinity when $s \\in [5/6,1)$.

Singularities are confined to a bounded region for solutions starting from vanishing initial data.

Abstract

In this paper, we introduce the concept of local Leray solutions starting from a locally square-integrable initial data to the fractional Navier-Stokes equations with $s\in [3/4,1)$. Furthermore, we prove its local in time existence when $s\in (3/4, 1)$. In particular, if the locally square-integrable initial data vanishs at infinity, we show that the fractional Navier-Stokes equations admit a global-in-time local Leray solution when $s\in [5/6, 1)$. For such local Leray solutions starting from locally square-integrable initial data vanishing at infinity, the singularity only occurs in $B_R(0)$ for some $R$.