Remarks on sparseness and regularity of Navier-Stokes solutions

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 (Grujic 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; Grujic 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.