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Desingularization of vortex rings in 3 dimensional Euler flows. (English) Zbl 1451.35120

Summary: In this paper, we are concerned with nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler fluids. We focus on the case when the vorticity function has a simple discontinuity, which corresponding to a jump in vorticity at the boundary of the cross-section of the vortex ring. Using the vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in several kinds of domains. The precise localization of the asymptotic singular vortex filament is proved to depend on the circulation and the velocity at far fields of the vortex ring, and the geometry of the domains. Some qualitative and asymptotic properties are also established.

MSC:

35Q31 Euler equations
76B47 Vortex flows for incompressible inviscid fluids
35B40 Asymptotic behavior of solutions to PDEs
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[1] Ambrosetti, A.; Mancini, G., On Some Free Boundary Problems. Recent Contributions to Nonlinear Partial Differential Equations, Research Notes in Mathematis, vol. 50, 24-36 (1981), Pitman: Pitman Boston · Zbl 0451.00012
[2] Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[3] Ambrosetti, A.; Struwe, M., Existence of steady vortex rings in an ideal fluid, Arch. Ration. Mech. Anal., 108, 97-109 (1989) · Zbl 0694.76012
[4] Ambrosetti, A.; Yang, J., Asymptotic behaviour in planar vortex theory, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 1, 4, 285-291 (1990) · Zbl 0709.76027
[5] Amick, C. J.; Fraenkel, L. E., The uniqueness of Hill’s spherical vortex, Arch. Ration. Mech. Anal., 92, 2, 91-119 (1986) · Zbl 0609.76018
[6] Amick, C. J.; Fraenkel, L. E., The uniqueness of a family of steady vortex rings, Arch. Ration. Mech. Anal., 100, 3, 207-241 (1988) · Zbl 0694.76011
[7] Ao, W.; Dávila, J.; Del Pino, M.; Musso, M.; Wei, J., Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation, preprint · Zbl 1477.35153
[8] Badiani, T. V.; Burton, G. R., Vortex rings in \(\mathbb{R}^3\) and rearrangements, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457, 1115-1135 (2001) · Zbl 0991.76012
[9] Bartsch, T.; Pistoia, A., Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations, SIAM J. Appl. Math., 75, 726-744 (2015) · Zbl 1317.35073
[10] Bartsch, T.; Pistoia, A.; Weth, T., N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations, Commun. Math. Phys., 297, 653-686 (2010) · Zbl 1195.35250
[11] Batchelor, G. K., On steady laminar flow with closed streamlines at large Reynolds number, J. Fluid Mech., 1, 177-190 (1956) · Zbl 0070.42004
[12] Benedetto, D.; Caglioti, E.; Marchioro, C., On the motion of a vortex ring with sharply concentrated vorticity, Math. Methods Appl. Sci., 23, 2, 147-168 (2000) · Zbl 0956.35109
[13] Benjamin, T. B., The alliance of practical and analytic insights into the nonlinear problems of fluid mechanics, (Applications of Methods of Functional Analysis to Problems of Mechanics. Applications of Methods of Functional Analysis to Problems of Mechanics, Lecture Notes in Math., vol. 503 (1976), Springer-Verlag: Springer-Verlag Berlin), 8-29 · Zbl 0369.76048
[14] Berestycki, H.; Brezis, H., On a free boundary problem arising in plasma physics, Nonlinear Anal., 4, 3, 415-436 (1980) · Zbl 0437.35032
[15] Berger, M. S.; Fraenkel, L. E., Nonlinear desingularization in certain free-boundary problems, Commun. Math. Phys., 77, 149-172 (1980) · Zbl 0454.35087
[16] Burton, G. R., Vortex rings in a cylinder and rearrangements, J. Differ. Equ., 70, 333-348 (1987) · Zbl 0648.35029
[17] Burton, G. R., Rearrangements of functions, maximization of convex functionals, and vortex rings, Math. Ann., 276, 2, 225-253 (1987) · Zbl 0592.35049
[18] Burton, G. R., Global nonlinear stability for steady ideal fluid flow in bounded planar domains, Arch. Ration. Mech. Anal., 176, 149-163 (2005) · Zbl 1064.76053
[19] Buttà, P.; Marchioro, C., Time evolution of concentrated vortex rings, J. Math. Fluid Mech., 22, 2, Article 19 pp. (2020) · Zbl 1433.76029
[20] Caffarelli, L. A.; Friedman, A., The shape of axisymmetric rotating fluid, J. Funct. Anal., 35, 109-142 (1980) · Zbl 0439.35068
[21] Caffarelli, L. A.; Friedman, A., Asymptotic estimates for the plasma problem, Duke Math. J., 47, 705-742 (1980) · Zbl 0466.35033
[22] Cao, D.; Liu, Z.; Wei, J., Regularization of point vortices for the Euler equation in dimension two, Arch. Ration. Mech. Anal., 212, 179-217 (2014) · Zbl 1293.35223
[23] Cao, D.; Peng, S.; Yan, S., Planar vortex patch problem in incompressible steady flow, Adv. Math., 270, 263-301 (2015) · Zbl 1480.35318
[24] Cao, D.; Guo, Y.; Peng, S.; Yan, S., Local uniqueness for vortex patch problem in incompressible planar steady flow, J. Math. Pures Appl., 131, 9, 251-289 (2019) · Zbl 1436.35153
[25] Cao, D.; Zhan, W.; Zou, C., On desingularization of steady vortex for the lake equations, preprint · Zbl 1507.35150
[26] Dávila, J.; Del Pino, M.; Musso, M.; Wei, J., Gluing methods for vortex dynamics in Euler flows, Arch. Ration. Mech. Anal., 235, 3, 1467-1530 (2020) · Zbl 1439.35382
[27] Dávila, J.; Del Pino, M.; Musso, M.; Wei, J., Travelling helices and the vortex filament conjecture in the incompressible Euler equations, preprint · Zbl 1490.35262
[28] Dekeyser, J., Desingularization of a steady vortex pair in the lake equation, preprint · Zbl 1445.76030
[29] Dekeyser, J., Asymptotic of steady vortex pair in the lake equation, SIAM J. Math. Anal., 51, 2, 1209-1237 (2019) · Zbl 1445.76030
[30] de Valeriola, S.; Van Schaftingen, J., Desingularization of vortex rings and shallow water vortices by semilinear elliptic problem, Arch. Ration. Mech. Anal., 210, 2, 409-450 (2013) · Zbl 1294.35083
[31] Douglas, R. J., Rearrangements of functions on unbounded domains, Proc. R. Soc. Edinb. A, 124, 621-644 (1994) · Zbl 0818.49010
[32] Fraenkel, L. E., On the method of matched asymptotic expansions Part III: two boundary-value problems, Math. Proc. Camb. Philos. Soc., 65, 1, 263-284 (1969) · Zbl 0187.24104
[33] Fraenkel, L. E., On steady vortex rings of small cross-section in an ideal fluid, Proc. R. Soc. Lond. Ser. A, 316, 29-62 (1970) · Zbl 0195.55101
[34] Fraenkel, L. E., Examples of steady vortex rings of small cross-section in an ideal fluid, J. Fluid Mech., 51, 119-135 (1972) · Zbl 0231.76013
[35] Fraenkel, L. E.; Berger, M. S., A global theory of steady vortex rings in an ideal fluid, Acta Math., 132, 13-51 (1974) · Zbl 0282.76014
[36] Friedman, A., Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics (1982), Wiley: Wiley New York · Zbl 0564.49002
[37] Friedman, A.; Turkington, B., Vortex rings: existence and asymptotic estimates, Trans. Am. Math. Soc., 268, 1, 1-37 (1981) · Zbl 0497.76031
[38] Helmholtz, H., On integrals of the hydrodynamics equations which express vortex motion, J. Reine Angew. Math., 55, 25-55 (1858) · ERAM 055.1448cj
[39] Hill, M. J.M., On a spherical vortex, Philos. Trans. R. Soc. Lond. A, 185, 213-245 (1894) · JFM 25.1471.01
[40] Jerrard, R. L.; Seis, C., On the vortex filament conjecture for Euler flows, Arch. Ration. Mech. Anal., 224, 1, 135-172 (2017) · Zbl 1371.35205
[41] Lamb, H., Hydrodynamics Cambridge Mathematical Library (1932), Cambridge University Press: Cambridge University Press Cambridge · JFM 58.1298.04
[42] Li, G.; Yan, S.; Yang, J., An elliptic problem related to planar vortex pairs, SIAM J. Math. Anal., 36, 1444-1460 (2005) · Zbl 1076.35039
[43] Meleshko, V. V.; Gourjii, A. A.; Krasnopolskaya, T. S., Vortex ring: history and state of the art, J. Math. Sci., 187, 772-806 (2012)
[44] Ni, W. M., On the existence of global vortex rings, J. Anal. Math., 37, 208-247 (1980) · Zbl 0457.76020
[45] Smets, D.; Van Schaftingen, J., Desingularization of vortices for the Euler equation, Arch. Ration. Mech. Anal., 198, 3, 869-925 (2010) · Zbl 1228.35171
[46] Tadie, On the bifurcation of steady vortex rings from a Green function, Math. Proc. Camb. Philos. Soc., 116, 3, 555-568 (1994) · Zbl 0853.35135
[47] Teman, R., Remarks on a free boundary value problem arising in plasma physics, Commun. Pure Appl. Math., 563-585 (1997) · Zbl 0355.35023
[48] Thomson, W., Mathematical and Physical Papers, IV (1910), Cambridge University Press: Cambridge University Press Cambridge · JFM 41.0019.02
[49] Turkington, B., On steady vortex flow in two dimensions. I, II, Commun. Partial Differ. Equ., 8, 999-1030 (1983), 1031-1071 · Zbl 0523.76014
[50] Turkington, B., Vortex rings with swirl: axisymmetric solutions of the Euler equations with nonzero helicity, SIAM J. Math. Anal., 20, 1, 57-73 (1989) · Zbl 0667.76038
[51] Yang, J., Global vortex rings and asymptotic behaviour, Nonlinear Anal., 25, 5, 531-546 (1995) · Zbl 0845.76017
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