  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 $R$.
