Remarks on sparseness and regularity of Navier–Stokes solutions

Abstract

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 (Grujić 2013 Nonlinearity 26 289–96), which is based on analyticity and the ‘harmonic measure maximum principle’. Second, we analyse the claims in (Bradshaw et al 2019 Arch. Ration. Mech. Anal. 231 1983–2005; Grujić and Xu 2019 arXiv:1911.00974) that a priori estimates on the sparseness of the vorticity and higher velocity derivatives reduce the ‘scaling gap’ in the regularity problem.