Late formation of singularities in solutions to the Navier–Stokes equations

Abstract

We study how late the first singularity can form in solutions of the Navier–Stokes equations and estimate the size of the potentially dangerous time interval, where it can possibly appear. According to Leray (1934), its size is estimated as when normalized by the local existence time, for a general blowup of the enstrophy at Here is the Reynolds number defined with initial energy E(0) and enstrophy Q(0). Applying dynamic scaling transformations, we give a general estimate parameterized by the behaviour of the scaled enstrophy. In particular, we show that the size is reduced to for a class of type II blow up of the form On the basis on the structure theorem of Leray (1934), we note that the self-similar and asymptotically self-similar blowup are ruled out for any singularities of weak solutions. We also apply the dynamic scaling to weak solutions with more than one singularities to show that the size is estimated as for the type II blowup above.