Solar models and McKean's breakdown theorem for the $\mu$CH and $\mu$DP equations

TL;DR

This paper proves that for the $mbda$CH and $mbda$DP equations on the circle, solutions break down in finite time if the initial momentum changes sign, extending McKean's approach with a new variable change technique.

Contribution

It introduces a novel change of variables to analyze breakdown in $mbda$CH and $mbda$DP equations, providing a new perspective on finite-time singularity formation.

Findings

Finite-time breakdown occurs if initial momentum changes sign.

The new variable change links the PDE to planar systems with conserved angular momentum.

Insights are provided for related continuum mechanics PDEs like the De Gregorio equation.

Abstract

We study the breakdown for $\mu$CH and $\mu$DP equations on the circle, given by $$m_t + u m_{\theta} + \lambda u_{\theta} m = 0,$$ for $m = \mu(u) - u_{\theta\theta}$, where $\mu$ is the mean and $\lambda=2$ or $\lambda=3$ respectively. It is already known that if the initial momentum $m_0$ never changes sign, then smooth solutions exist globally. We prove the converse: if the initial momentum changes sign, then $C^2$ solutions $u$ must break down in finite time. The technique is similar to that of McKean, who proved the same for the Camassa-Holm equation, but we introduce a new perspective involving a change of variables to treat the equation as a family of planar systems with central force for which the conserved angular momentum is precisely the conserved vorticity. We also demonstrate how this perspective can apply to give some insights for other PDEs of continuum mechanics, such as the Okamoto-Sakajo-Wunsch equation (and in particular the De Gregorio equation).