Geodesic
Numerical evidence of nonuniqueness in the evolution of vortex sheets
Lopes Filho, Milton C. ; Lowengrub, John  ; Nussenzveig Lopes, Helena J. ; Zheng, Yuxi1
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.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 40 (2006) no. 2, pp. 225-237

Voir la notice de l'article provenant de la source Numdam

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. 
@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 },
     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://geodesic.mathdoc.fr/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: Mathematical Modelling and Numerical Analysis 
PY  - 2006
SP  - 225
EP  - 237
VL  - 40
IS  - 2
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/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: Mathematical Modelling and Numerical Analysis 
%D 2006
%P 225-237
%V 40
%N 2
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/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: Mathematical Modelling and Numerical Analysis , Tome 40 (2006) no. 2, pp. 225-237. doi: 10.1051/m2an:2006012

Cité par Sources :