Partial regularity for the steady hyperdissipative fractional Navier-Stokes equations

TL;DR

This paper extends partial regularity results for steady fractional Navier-Stokes equations to hyperdissipative cases in five dimensions, employing blowup techniques and adapting existing methods to this new setting.

Contribution

It introduces a partial regularity theory for hyperdissipative steady fractional Navier-Stokes equations, expanding the understanding of regularity in higher-dimensional and more dissipative contexts.

Findings

Established partial regularity for hyperdissipative steady fractional Navier-Stokes equations.

Adapted blowup procedures from classical methods to the fractional and hyperdissipative setting.

Extended regularity results to five-dimensional flows with external forces.

Abstract

We extend the Caffarelli-Kohn-Nirenberg type partial regularity theory for the steady $5$-dimensional fractional Navier-Stokes equations with external force to the hyperdissipative setting. In our argument we use the methods of Colombo-De Lellis-Massaccesi to apply a blowup procedure adapted from work of Ladyzhenskaya-Seregin.