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 $\alpha\geq 1$. We show that for any divergence-free initial datum $u_0$ such that $||u_0||_{H^{\delta}} \leq M$, where $M$ is arbitrarily large and $\delta$ is arbitrarily small, there exists an explicit $\epsilon=\epsilon(M, \delta)>0$ 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 $H^{5/4} \times (\frac 34, \frac{5}{4}]$.