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We study weak solutions of the two-dimensional (2D) filtered Euler equationswhose vorticity is a finite Radon measure and velocity has locally finitekinetic energy, which is called the vortex sheet solution. The filtered Eulerequations are a regularized model based on a spatial filtering to the Eulerequations. The 2D filtered Euler equations have a unique global weak solutionfor measure valued initial vorticity, while the 2D Euler equations requireinitial vorticity to be in the vortex sheet class with a distinguished sign forthe existence of global solutions. In this paper, we prove that vortex sheetsolutions of the 2D filtered Euler equations converge to those of the 2D Eulerequations in the limit of the filtering parameter provided that initial vortexsheet has a distinguished sign. We also show that a simple application of ourproof yields the convergence of the vortex blob method for vortex sheetsolutions. Moreover, we make it clear what kind of condition should be imposedon the spatial filter to show the convergence results and, according to thecondition, these results are applicable to well-known regularized models likethe Euler-$\alpha$ model and the vortex blob model.

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