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We study the breakdown for $\mu$CH and $\mu$DP equations on the circle, givenby $$m_t + u m_{\theta} + \lambda u_{\theta} m = 0,$$ 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 $m_0$ neverchanges sign, then smooth solutions exist globally. We prove the converse: ifthe initial momentum changes sign, then $C^2$ solutions $u$ must break down infinite time. The technique is similar to that of McKean, who proved the samefor the Camassa-Holm equation, but we introduce a new perspective involving achange of variables to treat the equation as a family of planar systems withcentral force for which the conserved angular momentum is precisely theconserved vorticity. We also demonstrate how this perspective can apply to givesome insights for other PDEs of continuum mechanics, such as theOkamoto-Sakajo-Wunsch equation (and in particular the De Gregorio equation).

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