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

In this paper we study the Euler-Poincar'{e} equations in ${\mathbb{R}}^N$. We prove local existence of weak solutions in $W^{2,p}({\mathbb{R}}^N),$ $p>N$, and local existence of unique classical solutions in $H^k({\mathbb{R}}^N)$, $k>N\mathrm{/}2+3$, 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 $\alpha \to 0$, provided that the limiting solution belongs to $C([0,T);H^k({\mathbb{R}}^N))$ with $k>N\mathrm{/}2+3$. 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$\alpha=0$anyweaksolution$$\mathbf{u}$\in L^2(\Bbb R^N)$is$$\mathbf{u}$=0$.