TY - JOUR

T1 - Numerical evidence of nonuniqueness in the evolution of vortex sheets

AU - Lopes Filho, Milton C.

AU - Lowengrub, John

AU - Nussenzveig Lopes, Helena J.

AU - Zheng, Yuxi

N1 - Funding Information:
1 Departamento de Matematica, IMECC-UNICAMP, Caixa Postal 6065, Campinas, SP 13081-970, Brasil. mlopes@ime.unicamp.br; hlopes@ime.unicamp.br Research supported in part by CNPq grant # 300.962/91-6 and FAPESP grants # 96/07635-4 and # 97/13855-0.
Funding Information:
2 Department of Mathematics, Univ. of California at Irvine, Irvine, CA 92697, USA. lowengrb@math.uci.edu Partially supported by the National Science Foundation, Division of Mathematical Sciences, and the Minnesota Supercomputer Institute.
Funding Information:
3 Departament of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. yzheng@math.psu.edu Research supported in part by the NSF-DMS grants 9703711, 0305497, 0305114 and by the Sloan Foundation. 21 06 2006 03 2006 21 6 2006 21 6 2006 40 2 m2an/2006/02 225 237 21 1 2005 © EDP Sciences, SMAI, 2006 2006 EDP Sciences, SMAI

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

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U2 - 10.1051/m2an:2006012

DO - 10.1051/m2an:2006012

M3 - Article

AN - SCOPUS:33745318679

VL - 40

SP - 225

EP - 237

JO - Mathematical Modelling and Numerical Analysis

JF - Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 2

ER -