Note on Leray's criterion for Navier-Stokes singularity

Leray's criterion for Navier-Stokes singularity states that if a solution to theNavier-Stokes equations becomes singular att=T*, then it is necessary that thevelocity norm||u||Ls(R3), fors >3, grows unbounded in the manner||u||Ls(R3)>= Cs/(T*-t)(s-3)/2sast -> T*. HereCsmay depend ons. This note examinesLeray's necessary condition for singularity in some detail. Optimality of theexponent(s-3)/2sis confirmed and constraints onCsare derived and discussed.Furthermore, assuming singular growth of||u||Ls(R3)exactly respecting Leray'sscaling, a logarithmic constraint on the divergence of||u||L3(R3)is deduced.(c) 2022 Published by Elsevier Ltd.

Automated summary

This article discusses Leray's criterion for singularity in the Navier-Stokes equations, examines the necessary conditions and constraints for singularity, and derives the logarithmic constraint on divergence related to Leray's scaling.