Blow-up phenomenon, ill-posedness and peakon solutions for the periodic Euler-Poincar\'e equations

TL;DR

This paper studies the periodic Euler-Poincaré equations, demonstrating blow-up phenomena, ill-posedness in critical spaces, and the existence of peakon solutions, thereby advancing understanding of their mathematical properties.

Contribution

It introduces a new blow-up result, proves ill-posedness in critical Besov spaces, and confirms the existence of peakon solutions for the system.

Findings

Blow-up occurs for certain smooth initial data.

The system is ill-posed in critical Besov spaces.

Peakons are verified as distributional solutions.

Abstract

In this paper we mainly investigate the initial value problem of the periodic Euler-Poincar\'e equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant properties of the system. Then, we prove that the periodic Euler-Poincar\'e equations is ill-posed in critical Besov spaces by a contradiction argument. Finally, we verify the system possesses a class of peakon solutions in the sense of distributions.