On concentration in vortex sheets

04/03/2020

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by   Samuel Lanthaler, et al.

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The question of energy concentration in approximate solution sequences u^ϵ, as 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 [R. Krasny, J. Fluid Mech. 167:65-93 (1986)]. Numerical evidence suggests that no energy concentration appears in the limit of zero blob-regularization ϵ→ 0, for the considered initial data.