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Solar models for Euler-Arnold equations

Banff International Research Station for Mathematical Innovation and Discovery

Many one-dimensional Euler-Arnold equations can be recast in the form of a central-force problem $\Gamma_{tt}(t,x) = -F(t,x) \Gamma(t,x)$, where $\Gamma$ is a vector in $\mathbb{R}^2$ and $F$ is a nonlocal function possibly depending on $\Gamma$ and $\Gamma_t$. Angular momentum of this system is precisely the conserved momentum for the Euler-Arnold equation. In particular this picture works for the Camassa-Holm equation, the Hunter-Saxton equation, and the Okamoto-Sakajo-Wunsch family of equations. In the solar model, breakdown comes from a particle hitting the origin in finite time, which is only possible with zero angular momentum. Results due to McKean (for Camassa-Holm), Lenells (for Hunter-Saxton), and Bauer-Kolev-Preston/Washabaugh (for the Wunsch equation) show that breakdown of smooth solutions occurs exactly when momentum changes from positive to negative. I will discuss some conjectures and numerical evidence for the generalization of this picture to other equations such as the $\mu$-Camassa-Holm equation or the DeGregorio equation.

Mathematics; Calculus of variations and optimal control; optimization; Differential geometry

video/mp4

Author affiliation: Brooklyn College

2019-06-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70608

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/