Numerical evidence of nonuniqueness in the evolution of vortex sheets
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, p. 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 : https://doi.org/10.1051/m2an:2006012
Classification:  35Q35,  65M12,  (Secondary) 76B03,  (Primary) 76M23
Keywords: nonuniqueness, vortex sheets, vortex methods, Euler equations
@article{M2AN_2006__40_2_225_0,
     author = {Lopes Filho, Milton C. and Lowengrub, John and Nussenzveig Lopes, Helena J. and Zheng, Yuxi},
     title = {Numerical evidence of nonuniqueness in the evolution of vortex sheets},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {2},
     year = {2006},
     pages = {225-237},
     doi = {10.1051/m2an:2006012},
     zbl = {1124.76010},
     mrnumber = {2241821},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_2_225_0}
}
Lopes Filho, Milton C.; Lowengrub, John; Nussenzveig Lopes, Helena J.; Zheng, Yuxi. Numerical evidence of nonuniqueness in the evolution of vortex sheets. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, pp. 225-237. doi : 10.1051/m2an:2006012. http://www.numdam.org/item/M2AN_2006__40_2_225_0/

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