Chae, D; Liu, J-G, *Blow-up, Zero α Limit and the Liouville Type Theorem for the Euler-Poincaré Equations*,
Communications in Mathematical Physics, vol. 314 no. 3
(2012),
pp. 671-687 [doi] .
**Abstract:**

*In this paper we study the Euler-Poincaré equations in ℝ N. We prove local existence of weak solutions in W 2,p(ℝ N),p>N, and local existence of unique classical solutions in H k(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(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 u ∈ H 1(ℝ N) is u=0; for α= 0 any weak solution u ∈ L 2(ℝ N) is u=0. © 2012 Springer-Verlag.*