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

In this paper, we study the Cauchy problem of the Euler-Poincar\'{e} equations in 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 Fˢp,r(ᵈ). Furthermore, we obtain that the data-to-solution of this equation is continuous but not uniformly continuous in these spaces.