# Non-uniform continuous dependence on initial data of solutions to the Euler-Poincar\'{e} system

## TL;DR

This paper demonstrates that solutions to the Euler-Poincaré system do not depend uniformly continuously on initial data in certain Sobolev spaces, highlighting limitations in the stability of solutions.

## Contribution

It provides a rigorous proof of non-uniform continuous dependence of solutions on initial data for the Euler-Poincaré system in Sobolev spaces.

## Key findings

- Data-to-solution map is not uniformly continuous in $H^s$ for $s>1+d/2$
- Constructs approximate solutions to analyze dependence
- Shows limitations in stability of solutions to the Euler-Poincaré system

## Abstract

In this paper, we investigate the continuous dependence on initial data of solutions to the Euler-Poincar\'{e} system. By constructing a sequence approximate solutions and calculating the error terms, we show that the data-to-solution map is not uniformly continuous in Sobolev space $H^s(\mathbb{R}^d)$ for $s>1+\frac d2$.