Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation

TL;DR

This paper numerically investigates novel self-similar finite-time blowup scenarios with singular profiles in the 1D Hou-Luo model and 2D Boussinesq equations, revealing a two-stage blowup process with asymptotic self-similarity.

Contribution

It introduces new self-similar blowup scenarios with singular profiles and a two-stage process in these fluid models, supported by numerical evidence.

Findings

Solutions develop unbounded singular profiles at blowup.

Blowup occurs in two stages: local $L^{ abla}$ blowup followed by $L^p$ blowup.

Both stages exhibit asymptotic self-similarity.

Abstract

We present novel self-similar finite-time blowup scenarios for the 1D Hou--Luo model. We numerically demonstrate that solutions that initially satisfy certain derivative degeneracy condition can develop asymptotically self-similar finite-time blowups with singular self-similar profiles that are unbounded at some point. Moreover, this blowup phenomenon exhibits a two-stage feature: the solution first undergoes a local $L^{\infty}$ blowup at some time $\tilde{T}$, then continues in the weak sense beyond $\tilde{T}$ and develops a local $L^p$ blowup at a later time $T>\tilde{T}$ for some $p>0$. A further numerical investigation indicates that both stages are asymptotically self-similar. Finally, we extend our numerical study to the 2D Boussinesq equations and discover similar self-similar finite-time blowups with singular profiles that also exhibit a two-stage feature.