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

19 pages

References in corpus (3)

• Soliton Dynamics in Computational Anatomy
• The Three Dimensional Viscous Camassa-Holm Equations, and Their Relation to the Navier-Stokes Equations and Turbulence Theory
• Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

Cited by in corpus (5)

• On the Euler-Poincaré equation with non-zero dispersion
• Non-uniform continuous dependence on initial data of solutions to the Euler-Poincaré system
• Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces
• On the continuity of the solution map of the Euler-Poincaré equations in Besov spaces
• Non-uniform dependence for higher dimensional Camassa-Holm equations in Besov spaces