Blow-up, Zero α Limit and the Liouville Type Theorem for the Euler-Poincaré Equations

Abstract

In this paper we study the Euler-Poincaré 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/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 → 0, provided that the limiting solution belongs to ${C([0, T);H^k(\mathbb {R}^N))}$ with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution ${\mathbf{u} \in H^1(\mathbb{R}^N)}$ is u=0; for α= 0 any weak solution ${\mathbf{u} \in L^2(\mathbb{R}^N)}$ is u=0.