On Local Type I Singularities of the Navier–Stokes Equations and Liouville Theorems

Abstract

We prove that suitable weak solutions of the Navier–Stokes equations exhibit Type I singularities if and only if there exists a non-trivial mild bounded ancient solution satisfying a Type I decay condition. The main novelty is in the reverse direction, which is based on the idea of zooming out on a regular solution to generate a singularity. By similar methods, we prove a Liouville theorem for ancient solutions of the Navier–Stokes equations bounded in \(L^3\) along a backward sequence of times.