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

Abstract

We study the breakdown for µCH and µDP equations on the circle, given by mt + um θ + λu θ m = 0, for m = µ(u) − u θθ , where µ is the mean and λ = 2 or λ = 3 respectively. It is already known that if the initial momentum m0 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 momentum. We also demonstrate how this perspective can apply for general PDEs of continuum mechanics, such as the Okamoto-Sakajo-Wunsch equation (and in particular the De Gregorio equation).

Singularity formation in the incompressible Euler equation in finite and infinite time

“…H 1 , H ´1, etc). Some of the systems described in this framework are the n-dimensional top [153], the Camassa-Holm equation [135], the Euler-α mean motion model [111,184], the Burgers equation and compressible fluids [171,125], the Hunter-Saxton equation [144,172], among others. See [8].…”