On the continuity of the solution map of the Euler-Poincaré equations in Besov spaces

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-Poincaré equations is nowhere uniformly continuous in $B_{p,r}^s({\mathbb{R}}^d)$ with $s>\max \{1+\frac{d}{2},\frac{3}{2}\}$ and $(p,r)\in (1,\mathrm{\infty })\times [1,\mathrm{\infty })$. This improves our previous result which shows the data-to-solution map for the Euler-Poincaré equations is non-uniformly continuous on a bounded subset of $B_{p,r}^s({\mathbb{R}}^d)$ near the origin.