Multiscale Self-Similar Finite-Time Blowups of the Constantin–Lax–Majda Model for the Three-Dimensional Euler Equations
De Huang, Xiang Qin, Xiuyuan Wang · SIAM Journal on Mathematical Analysis · 2025
Abstract. We construct a new class of asymptotically self-similar finite-time blowups that have two collapsing spatial scales for the one-dimensional 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. Our construction is based on an asymptotic analysis applied to the explicit solution formula of the model, and a complex analysis is performed to provide an understanding of multiscale blowups from a pole dynamics perspective. Similar multiscale blowup phenomena have recently been discovered for many higher-dimensional equations. Our study may provide some understanding of the common mechanism behind these multiscale blowups.