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Convergence of filtered weak solutions to the 2D Euler equations with measure-valued vorticity

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Abstract

We study weak solutions of the two-dimensional (2D) filtered Euler equations for measure-valued initial vorticity. The filtered Euler equations are a regularized model based on a spatial filtering to the Euler equations and have a unique global weak solution for initial vorticity in the space of a finite Radon measure. In this paper, we treat the vortex sheet problem and the point-vortex problem on the 2D filtered Euler equations. We prove that vortex sheet solutions of the 2D filtered Euler equations converge to those of the 2D Euler equations in the limit of the filtering parameter when initial vortex sheet has a distinguished sign. In addition, we make it clear what kind of condition should be imposed on the spatial filter to show the convergence and, according to the condition, our result is applicable to well-known regularized models such as the Euler-\(\alpha \) model and the vortex blob model. We also show that filtered point vortices converge to a solution of the point-vortex system provided that collision of point vortices does not occur.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP19J00064.

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  1. Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Japan

    Takeshi Gotoda

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Gotoda, T. Convergence of filtered weak solutions to the 2D Euler equations with measure-valued vorticity. J. Evol. Equ. 20, 1485–1509 (2020). https://doi.org/10.1007/s00028-020-00563-4

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