On the Euler-Poincaré equation with non-zero dispersion

Abstract

We consider the Euler-Poincaré 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-Poincaré 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.

18 pages

References in corpus (2)

• Soliton Dynamics in Computational Anatomy
• Blow-up, zero $\alpha$ limit and the Liouville type theorem for the Euler-Poincaré equations

Cited by in corpus (7)

• Non-uniform continuous dependence on initial data of solutions to the Euler-Poincaré system
• On the modified multi-component Camassa-Holm system in higher dimensions
• On the continuity of the solution map of the Euler-Poincaré equations in Besov spaces
• On a parabolic sine-Gordon model
• Blow-up phenomenon, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations
• Non-uniform dependence for higher dimensional Camassa-Holm equations in Besov spaces
• Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces