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In this paper we study the Euler-Poincar\'{e} equations in $\Bbb R^N$. Weprove local existence of weak solutions in $W^{2,p}(\Bbb R^N),$ $p>N$, andlocal 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($\alpha=0$) weprove a finite time blow-up of the classical solution. We also prove that asthe dispersion parameter vanishes, the weak solution converges to a solution ofthe zero dispersion equation with sharp rate as $\alpha\to0$, provided that thelimiting 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 aLiouville type theorem. Namely, for $\alpha>0$ any weak solution $\mathbf{u}\inH^1(\Bbb R^N)$ is $\mathbf{u}=0$; for $\alpha=0$ any weak solution$\mathbf{u}\in L^2(\Bbb R^N)$ is $\mathbf{u}=0$.

Blow-up, zero $\alpha$ limit and the Liouville type theorem for the Euler-Poincar\'{e} equations