Blow-up, Zero alpha Limit and the Liouville Type Theorem for the Euler-Poincar, Equations

Abstract

In this paper we study the Euler-Poincar, equations in . We prove local existence of weak solutions in , and local existence of unique classical solutions in , 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 -> 0, provided that the limiting solution belongs to with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincar, equations we prove a Liouville type theorem. Namely, for alpha > 0 any weak solution is u=0; for alpha= 0 any weak solution is u=0.