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

Abstract

We consider solutions of the Navier-Stokes equation with fractional dissipation of order $α\geq 1$. We show that for any divergence-free initial datum $u_0$ such that $||u_0||_{H^δ} \leq M$, where $M$ is arbitrarily large and $δ$ is arbitrarily small, there exists an explicit $ε=ε(M, δ)>0$ such that the Navier-Stokes equations with fractional order $α$ has a unique smooth solution for $α\in (\frac{5}{4}-ε, \frac{5}{4}]$. 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 $H^{5/4} \times (\frac 34, \frac{5}{4}]$.