Well-posedness and no-uniform dependence for the Euler-Poincar\'{e} equations in Triebel-Lizorkin spaces

Abstract

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