Well-posedness and no-uniform dependence for the Euler-Poincaré equations in Triebel-Lizorkin spaces

Abstract

In this paper, we study the Cauchy problem of the Euler-Poincaré equations in ${\mathbb{R}}^d$ with initial data belonging to the Triebel-Lizorkin spaces. We prove the local-in-time unique existence of solutions to the Euler-Poincaré equations in $F_{p,r}^s({\mathbb{R}}^d)$. Furthermore, we obtain that the data-to-solution of this equation is continuous but not uniformly continuous in these spaces.

17pages