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On concentration in vortex sheets

by S. Lanthaler

(Report number 2020-23)

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

Keywords: vortex sheets, vortex blob, concentration, energy conservation, vorticity, structure function

@Techreport{L20_896,
  author = {S. Lanthaler},
  title = {On concentration in vortex sheets},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-23.pdf },
  year = {2020}
}

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