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 >= 1. We show that for any divergence-free initial datum u(0) such that parallel to u(0)parallel to(H delta) <= M, where M is arbitrarily large and S is arbitrarily small, there exists an explicit epsilon = epsilon(M, delta) > 0 such that the Navier-Stokes equations with fractional order a have a unique global smooth solution for alpha is an element of(5/4-epsilon, 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) x (3/4, 5/4] . (C) 2020 Elsevier Inc. All rights reserved.