The failure of Hölder regularity of solutions for the Euler-Poincaré equations in Besov spaces

In this paper, we investigate the continuity of solution to the Euler-Poincaré equations. We show that the continuity of the solution cannot be improved to the Hölder continuity. That is, the solution of the Euler-Poincaré equations with initial data $u_0\in B_{p,r}^s$ belongs to $\mathcal{C}([0,T];B_{p,r}^s({\mathbb{R}}^d))$ but not to ${\mathcal{C}}^{\alpha }([0,T];B_{p,r}^s({\mathbb{R}}^d))$ with any $\alpha \in (0,1)$.