On singular vortex patches, II: Long-time dynamics

Elgindi, Tarek; Jeong, In-Jee

doi:10.1090/tran/8134

On singular vortex patches, II: Long-time dynamics
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Abstract:

In a companion paper [arXiv:1903.00833], we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the above-mentioned paper. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time.

References

A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys. 152 (1993), no. 1, 19–28. MR 1207667, DOI 10.1007/BF02097055
Jacob Burbea, Motions of vortex patches, Lett. Math. Phys. 6 (1982), no. 1, 1–16. MR 646163, DOI 10.1007/BF02281165
T. F. Buttke, The observation of singularities in the boundary of patches of constant vorticity, Phys. Fluids A 1 (1989), 1283–1285.
E. Caglioti and C. Maffei, Asymptotic behaviour of vortex patches: a case of confinement, Boll. Un. Mat. Ital. B (7) 10 (1996), no. 2, 261–276 (English, with Italian summary). MR 1397348
E. Caglioti and C. Maffei, Scattering theory: a possible approach to the homogenization problem for the Euler equations, Rend. Mat. Appl. (7) 17 (1997), no. 3, 445–475 (English, with English and Italian summaries). MR 1608704
J. A. Carrillo and J. Soler, On the evolution of an angle in a vortex patch, J. Nonlinear Sci. 10 (2000), no. 1, 23–47. MR 1730570, DOI 10.1007/s003329910002
Angel Castro, Diego Córdoba, and Javier Gómez-Serrano, Uniformly rotating smooth solutions for the incompressible 2D Euler equations, Arch. Ration. Mech. Anal. 231 (2019), no. 2, 719–785. MR 3900813, DOI 10.1007/s00205-018-1288-3
J.-Y. Chemin, Persistance des structures géométriques liées aux poches de tourbillon, Séminaire sur les Équations aux Dérivées Partielles, 1990–1991, École Polytech., Palaiseau, 1991, pp. Exp. No. XIII, 11 (French). MR 1131586
Kyudong Choi, On the estimate of distance traveled by a particle in a disk-like vortex patch, Appl. Math. Lett. 97 (2019), 67–72. MR 3955684, DOI 10.1016/j.aml.2019.05.020
Albert Cohen and Raphael Danchin, Multiscale approximation of vortex patches, SIAM J. Appl. Math. 60 (2000), no. 2, 477–502. MR 1740256, DOI 10.1137/S0036139997319785
P. Constantin and E. S. Titi, On the evolution of nearly circular vortex patches, Comm. Math. Phys. 119 (1988), no. 2, 177–198. MR 968694, DOI 10.1007/BF01217737
Raphaël Danchin, Évolution temporelle d’une poche de tourbillon singulière, Comm. Partial Differential Equations 22 (1997), no. 5-6, 685–721 (French, with English summary). MR 1452164, DOI 10.1080/03605309708821280
Raphaël Danchin, Évolution d’une singularité de type cusp dans une poche de tourbillon, Rev. Mat. Iberoamericana 16 (2000), no. 2, 281–329 (French, with English and French summaries). MR 1809342, DOI 10.4171/RMI/276
D. G. Dritschel and M. E. McIntyre, Does contour dynamics go singular?, Phys. Fluids A 2 (1990), no. 5, 748–753. MR 1050012, DOI 10.1063/1.857728
David G. Dritschel, Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics, J. Comput. Phys. 77 (1988), no. 1, 240–266. MR 954310, DOI 10.1016/0021-9991(88)90165-9
David G. Dritschel, The repeated filamentation of two-dimensional vorticity interfaces, J. Fluid Mech. 194 (1988), 511–547. MR 988302, DOI 10.1017/S0022112088003088
D.G. Dritschel, Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasi-geostrophic flows, J. Fluid Mech. 191 (1988) 575–581.
Tarek M. Elgindi and In-Jee Jeong, Finite-time singularity formation for strong solutions to the Boussinesq system, Ann. PDE 6 (2020), no. 1, Paper No. 5, 50. MR 4098032, DOI 10.1007/s40818-020-00080-0
Tarek M. Elgindi and In-Jee Jeong, The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain, arXiv:2001.07840.
Tarek M. Elgindi and In-Jee Jeong, On singular vortex patches, I: Well-posedness issues, Memoirs of the AMS, to appear, arXiv:1903.00833.
Tarek M. Elgindi and In-Jee Jeong, Symmetries and critical phenomena in fluids, Comm. Pure Appl. Math. doi:10.1002/cpa.21829.
Tarek M. Elgindi and In-Jee Jeong, Finite-time singularity formation for strong solutions to the axi-symmetric 3D Euler equations, Ann. PDE 5 (2019), no. 2, Paper No. 16, 51. MR 4029562, DOI 10.1007/s40818-019-0071-6
Javier Gomez-Serrano, Jaemin Park, Jia Shi, and Yao Yao, Symmetry in stationary and uniformly-rotating solutions of active scalar equations, arXiv:1908.01722.
Taoufik Hmidi and Joan Mateu, Bifurcation of rotating patches from Kirchhoff vortices, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5401–5422. MR 3543554, DOI 10.3934/dcds.2016038
Taoufik Hmidi and Joan Mateu, Degenerate bifurcation of the rotating patches, Adv. Math. 302 (2016), 799–850. MR 3545942, DOI 10.1016/j.aim.2016.07.022
Taoufik Hmidi and Joan Mateu, Existence of corotating and counter-rotating vortex pairs for active scalar equations, Comm. Math. Phys. 350 (2017), no. 2, 699–747. MR 3607460, DOI 10.1007/s00220-016-2784-7
Taoufik Hmidi, Joan Mateu, and Joan Verdera, Boundary regularity of rotating vortex patches, Arch. Ration. Mech. Anal. 209 (2013), no. 1, 171–208. MR 3054601, DOI 10.1007/s00205-013-0618-8
Alexander Kiselev and Chao Li, Global regularity and fast small-scale formation for Euler patch equation in a smooth domain, Comm. Partial Differential Equations 44 (2019), no. 4, 279–308. MR 3941226, DOI 10.1080/03605302.2018.1546318
Alexander Kiselev, Lenya Ryzhik, Yao Yao, and Andrej Zlatoš, Finite time singularity for the modified SQG patch equation, Ann. of Math. (2) 184 (2016), no. 3, 909–948. MR 3549626, DOI 10.4007/annals.2016.184.3.7
Alexander Kiselev and Vladimir Šverák, Small scale creation for solutions of the incompressible two-dimensional Euler equation, Ann. of Math. (2) 180 (2014), no. 3, 1205–1220. MR 3245016, DOI 10.4007/annals.2014.180.3.9
Paolo Luzzatto-Fegiz and Charles H. K. Williamson, Investigating stability and finding new solutions in conservative fluid flows through bifurcation approaches, Nonlinear physical systems, Mech. Eng. Solid Mech. Ser., Wiley, Hoboken, NJ, 2014, pp. 203–221. MR 3330803
Andrew Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S187–S220. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861488, DOI 10.1002/cpa.3160390711
Philip S. Marcus, Jupiter’s great red spot and other vortices, Annual Review of Astronomy and Astrophysics 31 (1993), no. 1, 523–569.
N. S. Nadirashvili, Wandering solutions of the two-dimensional Euler equation, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 70–71 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 220–221 (1992). MR 1139875, DOI 10.1007/BF01085491
Lorenzo M. Polvani and David G. Dritschel, Wave and vortex dynamics on the surface of a sphere, J. Fluid Mech. 255 (1993), 35–64. MR 1244223, DOI 10.1017/S0022112093002381
R. Prieto, B. D. McNoldy, S. R. Fulton, and W. H. Schubert, A Classification of Binary Tropical Cyclone Like Vortex Interactions*, Monthly Weather Review, 131 (2003), 2656.
D. I. Pullin and D. W. Moore, Remark on a result of D. G. Dritschel: “The repeated filamentation of two-dimensional vorticity interfaces” [J. Fluid. Mech. 194 (1988), 511–547; MR0988302 (89m:76042)], Phys. Fluids A 2 (1990), no. 6, 1039–1041. MR 1053650, DOI 10.1063/1.857641
P. G. Saffman, Vortex dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. MR 1217252
Philippe Serfati, Une preuve directe d’existence globale des vortex patches $2$D, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 6, 515–518 (French, with English and French summaries). MR 1270072
Thomas C. Sideris and Luis Vega, Stability in $L^1$ of circular vortex patches, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4199–4202. MR 2538580, DOI 10.1090/S0002-9939-09-10048-5
Y. H. Wan and M. Pulvirenti, Nonlinear stability of circular vortex patches, Comm. Math. Phys. 99 (1985), no. 3, 435–450. MR 795112, DOI 10.1007/BF01240356

Additional Information

Tarek M. Elgindi
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093
MR Author ID: 990694
Email: telgindi@ucsd.edu.
In-Jee Jeong
Affiliation: School of Mathematics, Korea Institute for Advanced Study, 85 Hoegi-ro, Cheongnyangri-dong, Dongdaemun-gu, Seoul, South Korea
MR Author ID: 1055009
Email: ijeong@kias.re.kr
Received by editor(s): October 17, 2019
Received by editor(s) in revised form: February 12, 2020
Published electronically: July 8, 2020
Additional Notes: The first author was partially supported by grant number NSF DMS-1817134.
The second author was supported by a KIAS Individual Grant MG066202 at Korea Institute for Advanced Study, the Science Fellowship of POSCO TJ Park Foundation, and the National Research Foundation of Korea grant (No. 2019R1F1A1058486).
Journal: Trans. Amer. Math. Soc. 373 (2020), 6757-6775
DOI: https://doi.org/10.1090/tran/8134
MathSciNet review: 4155190