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

TL;DR

This paper proves finite-time blowup for solutions to the Euler-Poincaré equation on  for a broad class of smooth initial data, revealing new concentration mechanisms and conditions for global existence.

Contribution

It establishes finite-time blowup for the Euler-Poincare9 equation with non-zero dispersion for a wide range of initial data, resolving an open problem.

Findings

Finite-time blowup occurs for many smooth initial conditions.

New concentration mechanisms and monotonicity formulas are identified.

Some initial data classes lead to global solutions without blowup.

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.