On the continuity of the solution map of the Euler-Poincar\'e equations in Besov spaces

TL;DR

This paper proves that the solution map for the Euler-Poincaré equations is nowhere uniformly continuous in certain Besov spaces, using Fourier localization and translation techniques, improving previous results on non-uniform continuity.

Contribution

It establishes the non-uniform continuity of the data-to-solution map for Euler-Poincaré equations in Besov spaces with broader conditions, extending earlier findings.

Findings

Solution map is nowhere uniformly continuous in specified Besov spaces.

Constructs perturbation functions via Fourier localization and translation.

Improves previous results on non-uniform continuity near the origin.

Abstract

By constructing a series of perturbation functions through localization in the Fourier domain and translation, we show that the data-to-solution map for the Euler-Poincar\'e equations is nowhere uniformly continuous in $B^s_{p,r}(\mathbb{R} ^d)$ with $s>\max\{1+\frac d2,\frac32\}$ and $(p,r)\in (1,\infty)\times [1,\infty)$. This improves our previous result which shows the data-to-solution map for the Euler-Poincar\'e equations is non-uniformly continuous on a bounded subset of $B^s_{p,r}(\mathbb{R} ^d)$ near the origin.