Numerical evidence of nonuniqueness in the evolution of vortex sheets

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.
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.
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.

Received: 21 January 2005

Abstract


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.