Convergence of filtered weak solutions to the 2D Euler equations with vortex sheet initial data

TL;DR

This paper proves that solutions of the 2D filtered Euler equations with vortex sheet initial data converge to solutions of the classical 2D Euler equations as the filtering parameter approaches zero, under certain sign conditions.

Contribution

It establishes the convergence of vortex sheet solutions of the filtered Euler equations to those of the Euler equations, clarifying conditions on the spatial filter for this convergence.

Findings

Vortex sheet solutions of filtered Euler equations converge to Euler solutions as filter vanishes.

Convergence holds for initial vortex sheets with a distinguished sign.

Results apply to models like Euler-α and vortex blob methods.

Abstract

We study weak solutions of the two-dimensional (2D) filtered Euler equations whose vorticity is a finite Radon measure and velocity has locally finite kinetic energy, which is called the vortex sheet solution. The filtered Euler equations are a regularized model based on a spatial filtering to the Euler equations. The 2D filtered Euler equations have a unique global weak solution for measure valued initial vorticity, while the 2D Euler equations require initial vorticity to be in the vortex sheet class with a distinguished sign for the existence of global solutions. In this paper, 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 provided that initial vortex sheet has a distinguished sign. We also show that a simple application of our proof yields the convergence of the vortex blob method for vortex sheet solutions. Moreover, we make it clear what kind of condition should be imposed on the spatial filter to show the convergence results and, according to the condition, these results are applicable to well-known regularized models like the Euler-$\alpha$ model and the vortex blob model.