  In this paper we study the Euler-Poincar\'{e} equations in $\Bbb R^N$. We
prove local existence of weak solutions in $W^{2,p}(\Bbb R^N),$ $p>N$, and
local existence of unique classical solutions in $H^k (\Bbb R^N)$, $k>N/2+3$,
as well as a blow-up criterion. For the zero dispersion equation($\alpha=0$) we
prove a finite time blow-up of the classical solution. We also prove that as
the dispersion parameter vanishes, the weak solution converges to a solution of
the zero dispersion equation with sharp rate as $\alpha\to0$, provided that the
limiting solution belongs to $C([0, T);H^k(\Bbb R^N))$ with $k>N/2 +3$. For the
{\em stationary weak solutions} of the Euler-Poincar\'{e} equations we prove a
Liouville type theorem. Namely, for $\alpha>0$ any weak solution $\mathbf{u}\in
H^1(\Bbb R^N)$ is $\mathbf{u}=0$; for $\alpha=0$ any weak solution
$\mathbf{u}\in L^2(\Bbb R^N)$ is $\mathbf{u}=0$.
