  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.

Analysis of PDEs