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

We prove local existence of weak solutions in W 2,p(RN), p> N, and local existence of unique clas-sical solutions in Hk(RN), 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); Hk(RN)) with k> N/2 + 3. For the station-ary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α> 0 any weak solution u ∈ H1(RN) is u = 0; for α = 0 any weak solution u ∈ L2(RN) is u = 0