Blow-up, zero $\alpha$ limit and the Liouville type theorem for the Euler-Poincar\'{e} equations

TL;DR

This paper investigates the Euler-Poincaré equations in bb^N, establishing local existence, blow-up phenomena, convergence as dispersion vanishes, and Liouville theorems for stationary solutions, advancing understanding of their mathematical properties.

Contribution

It provides new results on local existence, blow-up criteria, zero dispersion limits, and Liouville theorems for stationary solutions of the Euler-Poincare9 equations.

Findings

Finite time blow-up for zero dispersion solutions.

Weak solutions converge as b1 a0 0 with sharp rate.

Stationary weak solutions are trivial (b7=0).

Abstract

In this paper we study the Euler-Poincar\'{e} equations in $\Bbb R^N$. We prove local existence of weak solutions in $W^{2,p}(\Bbb R^N),$ $p>N$, and local existence of unique classical solutions in $H^k (\Bbb R^N)$, $k>N/2+3$, 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 $\alpha\to0$, provided that the limiting solution belongs to $C([0, T);H^k(\Bbb R^N))$ with $k>N/2 +3$. For the {\em stationary weak solutions} of the Euler-Poincar\'{e} equations we prove a Liouville type theorem. Namely, for $\alpha>0$ any weak solution $\mathbf{u}\in H^1(\Bbb R^N)$ is $\mathbf{u}=0$; for $\alpha=0$ any weak solution $\mathbf{u}\in L^2(\Bbb R^N)$ is $\mathbf{u}=0$.