Global solvability and convergence of the Euler-Poincar\'e regularization of the two-dimensional Euler equations

TL;DR

This paper proves the global existence and convergence of solutions for the Euler-Poincaré regularization of 2D Euler equations, extending known regularizations and establishing solutions with measure-valued initial vorticity.

Contribution

It demonstrates the global solvability of the 2D Euler-Poincaré equations with measure initial vorticity and their convergence to classical Euler solutions as regularization vanishes.

Findings

Global existence of weak solutions with Radon measure initial vorticity.

Convergence of solutions to classical Euler equations as regularization parameter tends to zero.

Euler-Poincaré equations generalize vortex blob and Euler-alpha regularizations.

Abstract

We study the Euler-Poincar\'e equations that are the regularized Euler equations derived from the Euler-Poincar\'e framework. It is noteworthy to remark that the Euler-Poincar\'e 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-Poincar\'e 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-Poincar\'e 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.