Global existence of a uniformly local energy solution for the incompressible fractional Navier–Stokes equations
We introduce the concept of local Leray solutions starting from locally square-integrable initial data to the fractional Navier–Stokes equations with $s\in [3/4,1)$. Furthermore, we prove their local-in-time existence when $s\in (3/4, 1)$. In particular, if a locally square-integrable initial datum vanishes 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, a singularity only occurs in $B_R(0)$ for some $R$.