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.
Mots clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {225--237},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
doi = {10.1051/m2an:2006012},
mrnumber = {2241821},
zbl = {1124.76010},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2006012/}
}
TY - JOUR AU - Lopes Filho, Milton C. AU - Lowengrub, John AU - Nussenzveig Lopes, Helena J. AU - Zheng, Yuxi TI - Numerical evidence of nonuniqueness in the evolution of vortex sheets JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 225 EP - 237 VL - 40 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006012/ DO - 10.1051/m2an:2006012 LA - en ID - M2AN_2006__40_2_225_0 ER -
%0 Journal Article %A Lopes Filho, Milton C. %A Lowengrub, John %A Nussenzveig Lopes, Helena J. %A Zheng, Yuxi %T Numerical evidence of nonuniqueness in the evolution of vortex sheets %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 225-237 %V 40 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006012/ %R 10.1051/m2an:2006012 %G en %F 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: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 2, pp. 225-237. doi : 10.1051/m2an:2006012. http://www.numdam.org/articles/10.1051/m2an:2006012/
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