We study the filtered Euler equations that are the regularized Euler equations derived by filtering the velocity field. The filtered Euler equations are a generalization of two well-known regularizations of incompressible inviscid flows, the Euler-$\alpha$ equations and the vortex blob method. We show the global existence of a unique weak solution for the two-dimensional (2D) filtered Euler equations with initial vorticity in the space of Radon measure that includes point vortices and vortex sheets. Moreover, a sufficient condition for the global well-posedness is described in terms of the filter and thus our result is applicable to various filtered models. We also show that weak solutions of the 2D filtered Euler equations converge to those of the 2D Euler equations in the limit of the regularization parameter provided that initial vorticity belongs to the space of bounded functions.

中文翻译：

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具有测值初始涡度的二维滤波欧拉方程的整体可解性

我们研究滤波后的Euler方程，这是通过对速度场进行滤波而得出的正则化Euler方程。滤波后的欧拉方程是不可压缩无粘性流的两个众所周知的正则化，欧拉-α方程和涡旋斑点法的推广。我们显示了在包含点涡旋和涡旋片在内的Radon测度空间中具有初始涡度的二维（2D）滤波的Euler方程的唯一弱解的整体存在。此外，根据滤波器描述了全局适定性的充分条件，因此我们的结果适用于各种滤波模型。