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arXiv · 1212.4203

On the Euler-Poincar\'e equation with non-zero dispersion

Dong Li·Xinwei Yu·Zhichun Zhai
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Abstract

We consider the Euler-Poincar\'e equation on $\mathbb R^d$, $d\ge 2$. For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu \cite{Chae Liu}. Our analysis exhibits some new concentration mechanism and hidden monotonicity formula associated with the Euler-Poincar\'e flow. In particular we show the abundance of blowups emanating from smooth initial data with certain sign properties. No size restrictions are imposed on the data. We also showcase a class of initial data for which the corresponding solution exists globally in time.

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Dong Li, Xinwei Yu, Zhichun Zhai. 2012-12-18. On the Euler-Poincar\'e equation with non-zero dispersion. https://doi.org/10.1007/s00205-013-0662-4

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