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

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