Global regularity for the hyperdissipative Navier-Stokes equation below the critical order

We consider solutions of the Navier-Stokes equation with fractional dissipation of order $\alpha \ge 1$. We show that for any divergence-free initial datum $u_0$ such that $\mathrm{\mid }\mathrm{\mid }{u}_0\mathrm{\mid }{\mathrm{\mid }}_{H^{\delta }}\le M$, where $M$ is arbitrarily large and $\delta$ is arbitrarily small, there exists an explicit $ϵ=ϵ(M,\delta )>0$ such that the Navier-Stokes equations with fractional order $\alpha$ has a unique smooth solution for $\alpha \in (\frac{5}{4}-\epsilon, \frac{5}{4}]$. This is related to a new stability result on smooth solutions of the Navier-Stokes equations with fractional dissipation showing that the set of initial data and fractional orders giving rise to smooth solutions is open in $H^{5\mathrm{/}4}\times (\frac{3}{4},\frac{5}{4}]$.