On concentration in vortex sheets

Samuel Lanthaler

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arXiv · 2004.01537

On concentration in vortex sheets

Samuel Lanthaler

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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.

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Samuel Lanthaler. 2020-04-03. On concentration in vortex sheets. https://arxiv.org/abs/2004.01537

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