The uniform existence time and Zero-Alpha limit problem of the Euler-Poincar\'e equations

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