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/

[1] G.R. Baker and J.T. Beale, Vortex blob methods applied to interfacial motion. J. Comput. Phys. 196 (2004) 233-258. | Zbl 1115.76380

[2] R. Caflisch and O. Orellana, Long time existence for a slightly perturbed vortex sheet. Comm. Pure Appl. Math. 39 (1986) 807-838. | Zbl 0603.76039

[3] R. Caflisch and O. Orellana, Singularity solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20 (1989) 293-307. | Zbl 0697.76029

[4] J.-Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Partial Differential Equations 21 (1996) 1771-1779. | Zbl 0876.35087

[5] A. Chorin and P. Bernard, Discretization of a vortex sheet with an example of roll-up. J. Comput. Phys. 13 (1973) 423-429. | Zbl 0273.76022

[6] J.-M. Delort, Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4 (1991) 553-586. | Zbl 0780.35073

[7] R. Diperna and A. Majda, Concentrations and regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. XL (1987) 301-345. | Zbl 0850.76730

[8] R. Diperna and A. Majda, Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow. J. Am. Math. Soc. 1 (1988) 59-95. | Zbl 0707.76026

[9] J. Duchon and R. Robert, Global vortex sheet solutions of Euler equations in the plane. Comm. Partial Differential Equations 73 (1988) 215-224. | Zbl 0667.35046

[10] D. Ebin, Ill-posedness of the Rayleigh-Taylor and Helmholtz problem for incompressible fluids. Comm. Partial Differential Equations 73 (1988) 1265-1295. | Zbl 0683.76041

[11] L.C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics 74 A.M.S., Providence, RI (1990). | MR 1034481 | Zbl 0698.35004

[12] C. Greengard and E. Thomann, On DiPerna-Majda concentration sets for two-dimensional incompressible flow. Comm. Pure Appl. Math. 41 (1988) 295-303. | Zbl 0652.76014

[13] R. Krasny, Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65 (1986) 292-313. | Zbl 0591.76059

[14] R. Krasny, Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech. 184 (1987) 123-155. | Zbl 1142.76348

[15] R. Krasny and M. Nitsche, The onset of chaos in vortex sheet flow. J. Fluid Mech. 454 (2002) 47-69. | Zbl 1014.76009

[16] G. Lebeau, Régularité du problème de Kelvin-Helmholtz pour l'équation d'Euler 2D. ESAIM: COCV 8 (2002) 801-825. | Numdam | Zbl 1070.35504

[17] J.G. Liu and Z.P. Xin, Convergence of vortex methods for weak solutions to the 2D Euler equations with vortex sheet data. Comm. Pure Appl. Math. XLVIII (1995) 611-628. | Zbl 0829.35098

[18] M.C. Lopes Filho, H.J. Nussenzveig Lopes and Y.X. Zheng, Convergence of the vanishing viscosity approximation for superpositions of confined eddies. Commun. Math. Phys. 201 (1999) 291-304. | Zbl 0942.76012

[19] M.C. Lopes Filho, H.J. Nussenzveig Lopes and E. Tadmor, Approximate solutions of the incompressible Euler equations with no concentrations. Ann. I. H. Poincaré-An. 17 (2000) 371-412. | Numdam | Zbl 0965.35110

[20] M.C. Lopes Filho, H.J. Nussenzveig Lopes and Z.P. Xin, Existence of vortex sheets with reflection symmetry in two space dimensions. Arch. Rational Mech. Anal. 158 (2001) 235-257. | Zbl 1058.35176

[21] M.C. Lopes Filho, H.J. Nussenzveig Lopes and M.O. Souza, On the equation satisfied by a steady Prandtl-Munk vortex sheet. Comm. Math. Sci. 1 (2003) 68-73. | Zbl 1080.76018

[22] A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana U. Math J. 42 (1993) 921-939. | Zbl 0791.76015

[23] A. Majda, G. Majda and Y.X. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case. Physica D 74 (1994) 268-300. | Zbl 0813.35091

[24] M. Nitsche, M.A. Taylor and R. Krasny, Comparison of regularizations of vortex sheet motion, Proc. 2nd MIT Conf. Comput. Fluid and Solid Mech., K.J. Bathe Ed., Elsevier, Cambridge, MA (2003).

[25] W.R.C. Phillips and D.I. Pullin, On a generalization of Kaden's problem. J. Fluid Mech. 104 (1981) 45-53. | Zbl 0494.76025

[26] D.I. Pullin, On similarity flows containing two branched vortex sheets, in Mathematical Aspects of Vortex Dynamics, R. Caflisch Ed., SIAM (1989) 97-106. | Zbl 0671.76028

[27] V. Scheffer, An inviscid flow with compact support in space-time. J. Geom. Anal. 3 (1993) 343-401. | Zbl 0836.76017

[28] S. Schochet, The weak vorticity formulation of the 2D Euler equations and concentration-cancellation. Comm. P.D.E. 20 (1995) 1077-1104. | Zbl 0822.35111

[29] S. Schochet, Point-vortex method for periodic weak solutions of the 2-D Euler equations. Comm. Pure Appl. Math. XLIX (1996) 911-965. | Zbl 0862.35092

[30] A. Shnirelman, On the non-uniqueness of weak solutions of the Euler equations. Comm. Pure Appl. Math. L (1997) 1261-1286. | Zbl 0909.35109

[31] P.L. Sulem, C. Sulem, C. Bardos and U. Frisch, Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability. Comm. Math. Phys. 80 (1981) 485-516. | Zbl 0476.76032

[32] G. Tryggvason, W. Dahn and K. Sbeih, Fine structure of rollup by viscous and inviscid simulation. J. Fluids Eng.-T ASME 113 (1991) 31-36.

[33] I. Vecchi and S.J. Wu, On L${}^{1}$-vorticity for 2-D incompressible flow. Manuscripta Math. 78 (1993) 403-412. | Zbl 0807.35115

[34] M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. École Norm. S. Ser. 4 32 (1999) 769-812. | Numdam | Zbl 0938.35128

[35] S.J. Wu, Recent progress in mathematical analysis of vortex sheets, in Proceedings of the ICM, Beijing (2002) Vol. III, 233-242. | Zbl 1007.76005

[36] V. Yudovich, Non-stationary flow of an ideal incompressible liquid (in Russian), Zh. Vych. Mat. 3 (1963) 1032-1066.

[37] V. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal, incompressible fluid. Math. Res. Lett. 2 (1995) 27-38. | Zbl 0841.35092