On the Euler-Poincaré Equation with Non-Zero Dispersion

Abstract

We consider the Euler-Poincaré equation on ℝd, d ≧ 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 (Commun Math Phys 314:671-687, 2012). Our analysis exhibits some new concentration mechanisms and hidden monotonicity formulas associated with the Euler-Poincaré flow. In particular we show an 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. © 2013 Springer-Verlag Berlin Heidelberg.