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≥1. We show that for any divergence-free initial datum u0 such that ‖u0‖Hδ≤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 have a unique global smooth solution for α∈(54−ε,54]. 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 H5/4×(34,54].