Global existence and blow-up for the Euler-Poincar\'{e} equations with a class of initial data

TL;DR

This paper studies the Euler-Poincaré equations, transforming them into a Camassa-Holm type equation for specific initial data, and establishes conditions for global existence and blow-up of solutions.

Contribution

It introduces a new class of initial data that simplifies the Euler-Poincaré equations to a known form, enabling analysis of global existence and blow-up phenomena.

Findings

Global existence results for certain initial data

New blow-up conditions under different assumptions

Transformation of Euler-Poincaré equations to Camassa-Holm type

Abstract

In this paper we investigate the Cauchy problem of d-dimensional Euler-Poincar\'{e} equations. By choosing a class of new and special initial data, we can transform this d-dimensional Euler-Poincar\'{e} equations into the Camassa-Holm type equation in the real line. We first obtain some global existence results and then present a new blow-up result to the system under some different assumptions on this special class of initial data.