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

TL;DR

This paper proves global regularity for the hyperdissipative Navier-Stokes equations with fractional dissipation near the critical order, showing smooth solutions exist under broad initial data conditions.

Contribution

It establishes the existence and uniqueness of smooth solutions for fractional dissipation orders close to the critical value, extending understanding of stability in these equations.

Findings

Unique smooth solutions for fractional order  in (, 5/4]

Stability of smooth solutions under small perturbations in initial data and dissipation order

Open set of initial data leading to regular solutions in specified fractional order range

Abstract

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 $H^{5/4} \times (\frac 34, \frac{5}{4}]$.