On self-similar finite-time blowups of the De Gregorio model on the real line

TL;DR

This paper demonstrates the existence of infinitely many self-similar solutions to the De Gregorio model on the real line, which blow up in finite time, including solutions linked to numerical singularity solutions.

Contribution

It introduces a classification of self-similar solutions into basic and general classes, revealing their structure and connection to eigenfunctions of a compact operator.

Findings

Existence of infinitely many self-similar solutions

Basic class solutions are eigenfunctions of a self-adjoint operator

Leading eigenfunction matches numerically obtained finite-time singularity

Abstract

We show that the De Gregorio model on the real line admits infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions fall into two classes: the basic class and the general class. The basic class consists of countably infinite solutions that are eigenfunctions of a self-adjoint compact operator. In particular, the leading eigenfunction coincides with the finite-time singularity solution of the De Gregorio model recently obtained by numerical approaches. The general class consists of more complicated solutions that can be obtained by solving nonlinear eigenvalue problems associated with the same compact operator.