Multi-scale self-similar finite-time blowups of the Constantin-Lax-Majda model for the 3D Euler equations

Abstract

We construct a new class of asymptotically self-similar finite-time blowups that have two collapsing spatial scales for the 1D Constantin-Lax-Majda model. The larger spatial scale measures the decreasing distance between the bulk of the solution and the eventual blowup point, while the smaller scale measures the shrinking size of the bulk of the solution. Similar multi-scale blowup phenomena have recently been discovered for many higher dimensional equations. Our study may provide some understanding of the common mechanism behind these multi-scale blowups.