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 $\hat{I}\pm \ge 1$. We show that for any divergence-free initial datum $u_0$ such that $||u_0|{|}_{H^{\hat{I}}´}\le M$, where $M$ is arbitrarily large and $\hat{I}´$ is arbitrarily small, there exists an explicit $\hat{I}\mu =\hat{I}\mu (M,\hat{I}´)>0$ such that the Navier-Stokes equations with fractional order $\hat{I}\pm$ has a unique smooth solution for $\hat{I}\pm \in (\frac{5}{4}-\hat{I}\mu ,\frac{5}{4}]$. 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 $H^{5/4}\times (\frac{3}{4},\frac{5}{4}]$.