Long time confinement of vorticity around a stable stationary point vortex in a bounded planar domain
Donati, Martin1 ; Iftimie, Dragoș1
1 Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43 bd. du 11 novembre, Villeurbanne Cedex F-69622, France
Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1461-1485.
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In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simply-connected bounded domain. The domain and the stationary point vortex must satisfy a condition expressed in terms of the conformal mapping from the domain to the unit disk. Explicit examples are discussed at the end.

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DOI : 10.1016/j.anihpc.2020.11.009
Mots-clés : Perfect incompressible flows, Point-vortex system, Confinement of vorticity
Donati, Martin 1 ; Iftimie, Dragoș 1

1 Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43 bd. du 11 novembre, Villeurbanne Cedex F-69622, France 
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Donati, Martin; Iftimie, Dragoș. Long time confinement of vorticity around a stable stationary point vortex in a bounded planar domain. Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1461-1485. doi : 10.1016/j.anihpc.2020.11.009. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.11.009/

[1] Ahlfors, L. Complex Analysis, McGraw Hill Publishing Co., New York, 1966 | MR | Zbl

[2] Buttà, P.; Marchioro, C. Long time evolution of concentrated Euler flows with planar symmetry, SIAM J. Math. Anal., Volume 50 (2018) no. 1, pp. 735-760 | DOI | MR | Zbl

[3] Cao, D.; Wang, G. Euler evolution of a concentrated vortex in planar bounded domains, 2018 | arXiv

[4] Goodman, J.; Hou, T.; Lowengrub, J. Convergence of the point vortex method for 2-D Euler equations, Commun. Pure Appl. Math., Volume 43 (1990) no. 3, pp. 415-430 | DOI | MR | Zbl

[5] Gustafsson, B. On the Motion of a Vortex in Two-Dimensional Flow of an Ideal Fluid in Simply and Multiply Connected Domains, Royal Institute of Technology, 1979 (Trita-MAT-1979-7)

[6] Han, Z.; Zlatos, A. Euler equations on general planar domains, 2020 | arXiv | MR | Zbl

[7] Hartman, P. Ordinary Differential Equations, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104, 1910 | MR | Zbl

[8] Helmholtz, H. Über integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math., Volume 55 (1858), pp. 25-55 | MR | Zbl

[9] Iftimie, D. Large time behavior in perfect incompressible flows, Partial Differential Equations and Applications, Sémin. Congr., vol. 15, Soc. Math. France, Paris, 2007, pp. 119-179 | MR | Zbl

[10] Iftimie, D.; Marchioro, C. Self-similar point vortices and confinement of vorticity, Commun. Partial Differ. Equ., Volume 43 (2018) no. 3, pp. 347-363 | DOI | MR | Zbl

[11] Iftimie, D.; Sideris, T.C.; Gamblin, P. On the evolution of compactly supported planar vorticity, Commun. Partial Differ. Equ., Volume 24 (1999) no. 9–10, pp. 159-182 | MR | Zbl

[12] Kanas, S.; Sugawa, T. On conformal representations of the interior of an ellipse, Ann. Acad. Sci. Fenn., Math., Volume 31 (2006) no. 2, pp. 329-348 | MR | Zbl

[13] Lacave, C.; Zlatoš, A. The Euler equations in planar domains with corners, Arch. Ration. Mech. Anal., Volume 234 (2019) no. 1, pp. 57-79 | DOI | MR | Zbl

[14] Marchioro, C.; Pulvirenti, M. Vortex Methods in Two-Dimensional Fluid Dynamics, Lecture Notes in Physics, Springer-Verlag, 1984 | MR | Zbl

[15] Marchioro, C.; Pulvirenti, M. Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, Springer, New York, 1993 | MR | Zbl

[16] Marchioro, C.; Pulvirenti, M. Vortices and localization in Euler flows, Commun. Math. Phys., Volume 154 (1993) no. 1, pp. 49-61 | DOI | MR | Zbl

[17] Wolibner, W. Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z., Volume 37 (1933) no. 1, pp. 698-726 | DOI | JFM | MR | Zbl

[18] Yudovich, V. Non-stationary flow of an ideal incompressible liquid, USSR Comput. Math. Math. Phys., Volume 3 (1963) no. 6, pp. 1407-1456 | DOI | MR | Zbl

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