The uniform existence time and Zero-Alpha limit problem of the Euler-Poincaré equations

Abstract

We consider the Cauchy problem of the Euler-Poincaré equations in ${\mathbb{R}}^d$ with a varying dispersion parameter $\hat{I}\pm$. Based on the convex entropy structure and the modified commutator estimates, we have proved that the Euler-Poincaré equations have a uniform existence time with respect to $\hat{I}\pm$ in Sobolev spaces $H^s.$ Combined with the Bona-Simth method, we obtain convergence of the solutions to the Euler-Poincaré equations as $\hat{I}\pm \to 0$ in the same space where the initial data are located.