Remarks on sparseness and regularity of Navier-Stokes solutions

TL;DR

This paper presents a simple proof that sparse solutions to the Navier-Stokes equations remain regular and analyzes claims that sparseness estimates can help resolve the regularity problem.

Contribution

It offers an alternative proof for regularity of sparse solutions and critically examines prior claims about sparseness reducing the scaling gap.

Findings

Sparse Navier-Stokes solutions do not develop singularities.

Alternative proof based on sparseness avoids analyticity methods.

Analysis of prior claims questions the effectiveness of sparseness estimates in regularity.

Abstract

The goal of this paper is twofold. First, we give a simple proof that sufficiently sparse Navier--Stokes solutions do not develop singularities. This provides an alternative to the approach of \cite{Grujic2013}, which is based on analyticity and the `harmonic measure maximum principle'. Second, we analyze the claims in \cite{algebraicreduction,grujic2019asymptotic} that \emph{a priori} estimates on the sparseness of the vorticity and higher velocity derivatives reduce the 'scaling gap' in the regularity problem.