On concentration in vortex sheets

The question of energy concentration in approximate solution sequences $$u^\epsilon $$ u ϵ , as $$\epsilon \rightarrow 0$$ ϵ → 0 , of the two-dimensional incompressible Euler equations with vortex-sheet initial data is revisited. Building on a novel identity for the structure function in terms of vorticity, the vorticity maximal function is proposed as a quantitative tool to detect concentration effects in approximate solution sequences. This tool is applied to numerical experiments based on the vortex-blob method, where vortex sheet initial data without distinguished sign are considered, as introduced in Krasny (J Fluid Mech 167:65–93, 1986). Numerical evidence suggests that no energy concentration appears in the limit of zero blob-regularization $$\epsilon \rightarrow 0$$ ϵ → 0 , for the considered initial data.