Numerical evidence of nonuniqueness in the evolution of vortex sheets
ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 2, pp. 225-237.

We consider a special configuration of vorticity that consists of a pair of externally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary weak solution. Preliminary simulations presented here suggest this is indeed the case. We establish a convergence theorem for the vortex blob method that applies to this problem. This theorem and the preliminary calculations we carried out support the existence of two distinct weak solutions with the same initial data.

DOI: 10.1051/m2an:2006012
Classification: 35Q35, 65M12, (Secondary) 76B03, (Primary) 76M23
Keywords: nonuniqueness, vortex sheets, vortex methods, Euler equations
Lopes Filho, Milton C. ; Lowengrub, John ; Nussenzveig Lopes, Helena J. ; Zheng, Yuxi 1

1 Departament of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. Research supported in part by the NSF-DMS grants 9703711, 0305497, 0305114 and by the Sloan Foundation.
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Lopes Filho, Milton C.; Lowengrub, John; Nussenzveig Lopes, Helena J.; Zheng, Yuxi. Numerical evidence of nonuniqueness in the evolution of vortex sheets. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 2, pp. 225-237. doi : 10.1051/m2an:2006012. http://archive.numdam.org/articles/10.1051/m2an:2006012/

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