On concentration in vortex sheets
share
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.
READ FULL TEXT