Blow-up, zero $α$ limit and the Liouville type theorem for the Euler-Poincaré equations

In this paper we study the Euler-Poincaré 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($α=0$) 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 $α\to0$, 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-Poincaré equations we prove a Liouville type theorem. Namely, for $α>0$ any weak solution $\mathbf{u}\in H^1(\Bbb R^N)$ is $\mathbf{u}=0$; for $α=0$ any weak solution $\mathbf{u}\in L^2(\Bbb R^N)$ is $\mathbf{u}=0$.

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