# On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equation

## TL;DR

This paper proves finite time self-similar blowup for the De Gregorio model and its generalization, introducing a new analytical framework that employs energy methods and stability analysis of approximate profiles.

## Contribution

It introduces a novel method to establish finite time blowup in the De Gregorio model using stability of self-similar profiles and energy estimates, applicable to a range of parameters.

## Key findings

- Proved finite time blowup for the De Gregorio model with smooth initial data.
- Established self-similar blowup results for the generalized De Gregorio model.
- Developed a new analytical framework based on stability of approximate profiles.

## Abstract

We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ for H\"older continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations.