Finite time singularity of the Euler-Poincare equation

Xinwei Yu

We consider the Euler-Poincare equation in $\mathbb{R}^d$ with $d\ge 2$. For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. Our analysis exhibits some new concentration mechanism and hidden monotonicity formula associated with the Euler-Poincare flow. 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. This is joint work with Dong Li and Zhichun Zhai.