Global solvability and convergence of the Euler-Poincaré regularization of the two-dimensional Euler equations

Abstract

We study the Euler-Poincaré equations that are the regularized Euler equations derived from the Euler-Poincaré framework. It is noteworthy to remark that the Euler-Poincaré equations are a generalization of two well-known regularizations, the vortex blob method and the Euler-$\alpha$ equations. We show the global existence of a unique weak solution for the two-dimensional (2D) Euler-Poincaré equations with the initial vorticity in the space of Radon measure. This is a remarkable feature of these equations since the existence of weak solutions with the Radon measure initial vorticity has not been established in general for the 2D Euler equations. We also show that weak solutions of the 2D Euler-Poincaré equations converge to those of the 2D Euler equations in the limit of the regularization parameter when the initial vorticity belongs to the space of integrable and bounded functions.

23 pages, no figures

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