Self-similar finite-time blowups with singular profiles of the generalized Constantin-Lax-Majda model: theoretical and numerical investigations

TL;DR

This paper explores new types of finite-time self-similar blowups in the generalized Constantin-Lax-Majda model, revealing distinct behaviors depending on the parameter and establishing rigorous results for specific cases.

Contribution

It introduces novel self-similar blowup scenarios for the model, including explicit profiles and rigorous proofs for the case when the parameter is zero.

Findings

For $a>0$, one-scale self-similar blowups with new regular profiles.

For $a extless 0$, a two-scale blowup scenario with explicit singular profiles.

Rigorous proof of convergence to singular profiles for $a=0$.

Abstract

We investigate novel scenarios of self-similar finite-time blowups of the generalized Constantin-Lax-Majda model with a parameter $a$, which are induced by a new setting where the smooth initial data satisfy certain derivative degeneracy condition. In this setting, our numerical study reveals distinct self-similar blowup behaviors depending on the sign of $a$. For $a>0$, we observe one-scale self-similar blowups with regular profiles that have not been found in previous studies. In contrast, for $a\le 0$, we discover a novel two-scale self-similar blowup scenario where the outer profile converges to a singular function at the blowup time while the inner profile remains regular on a much smaller scale. Correspondingly, an $a$-parameterized family of singular self-similar profiles with explicit expressions are constructed for $a<0$ and shown to match nicely with the limiting profiles obtained in numerical simulation. In particular, for the specific case of $a=0$, we rigorously prove the convergence of the outer profile to an explicit singular function in self-similar coordinates. Furthermore, we demonstrate the two-scale nature of the blowup in this scenario by showing that the local inner profile behavior around the singularity point of the outer profile is governed by a traveling wave on a smaller scale. To support this observation, we rigorously establish the existence of such traveling wave solutions via a fixed-point method.