Concentrated solutions with helical symmetry for the 3D Euler equation and rearrangments. (English) Zbl 1542.76006
The authors consider the 3D incompressible Euler equation and study the existence and stability of concentrated traveling-rotating helical vortex. They construct a family of 3D Euler flows with helical symmetry, where the vorticity field is concentrated on a given traveling-rotating helical vortex filament. The solutions are obtained by maximization of the energy over the set of rearrangments of a fixed bounded function with compact support, and the solutions are shown to tend asymptotically to singular helical vortex filament evolving by the binormal curvature flow. The authors also show nonlinear stability of the maximizers under \(Lp\) perturbation.
Reviewer: Fatma Gamze Düzgün (Ankara)
MSC:
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 35Q31 | Euler equations |
Keywords:
energy maximization; orbital stability; singular helical vortex filament; binormal curvature flow; nonlinear maximizer stabilityReferences:
| [1] | Ao, W.; Liu, Y.; Wei, J., Clustered travelling vortex rings to the axisymmetric three-dimensional incompressible Euler flows, Physica D, 434, Article 133258 pp., 2022 · Zbl 1495.76025 |
| [2] | Arnol’d, V. I., Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Sov. Math. Dokl., 162, 773-777, 1965 · Zbl 0141.43901 |
| [3] | Arnol’d, V. I., Variational principles for three-dimensional steady-state flows of an ideal fluid, J. Appl. Math. Mech., 29, 1002-1008, 1965 · Zbl 0163.19807 |
| [4] | Arnol’d, V. I., On an a priori estimate in the theory of hydrodynamic stability, Am. Math. Soc. Transl., 79, 267-269, 1969 · Zbl 0191.56303 |
| [5] | Benvenutti, M., Nonlinear stability for stationary helical vortices, Nonlinear Differ. Equ. Appl., 27, 2, Article 15 pp., 2020 · Zbl 1437.35554 |
| [6] | 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 |
| [7] | Burton, G. R., Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 6, 4, 295-319, 1989 · Zbl 0677.49005 |
| [8] | 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 |
| [9] | Caffarelli, L. A.; Friedman, A., Asymptotic estimates for the plasma problem, Duke Math. J., 47, 705-742, 1980 · Zbl 0466.35033 |
| [10] | Cao, D.; Lai, S., Helical symmetry vortices for 3D incompressible Euler equations, J. Differ. Equ., 360, 67-89, 2023 · Zbl 1538.76038 |
| [11] | 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 |
| [12] | Cao, D.; Wan, J., Structure of Green’s function of elliptic equations and helical vortex patches for 3D incompressible Euler equations, Math. Ann., 388, 2627-2669, 2024 · Zbl 1533.35247 |
| [13] | Cao, D.; Wan, J., Helical vortices with small cross-section for 3D incompressible Euler equation, J. Funct. Anal., 284, Article 109836 pp., 2023 · Zbl 1508.76027 |
| [14] | 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 |
| [15] | Dávila, J.; del Pino, M.; Musso, M.; Wei, J., Travelling helices and the vortex filament conjecture in the incompressible Euler equations, Calc. Var. Partial Differ. Equ., 61, 119, 2022 · Zbl 1490.35262 |
| [16] | 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 |
| [17] | Da Rios, L. S., Sul moto dun liquido indefinito con un filetto vorticoso di forma qualunque, Rend. Circ. Mat. Palermo (1884-1940), 22, 1, 117-135, 1906 · JFM 37.0764.01 |
| [18] | Ettinger, B.; Titi, E. S., Global existence and uniqueness of weak solutions of three-dimensional Euler equations with helical symmetry in the absence of vorticity stretching, SIAM J. Math. Anal., 41, 1, 269-296, 2009 · Zbl 1303.76006 |
| [19] | Fraenkel, L. E., On steady vortex rings of small cross-section in an ideal fluid, Proc. R. Soc. Lond. A, 316, 29-62, 1970 · Zbl 0195.55101 |
| [20] | 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 |
| [21] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 2001, Springer: Springer Berlin · Zbl 1042.35002 |
| [22] | Helmholtz, H., On integrals of the hydrodynamics equations which express vortex motion, J. Reine Angew. Math., 55, 25-55, 1858 · ERAM 055.1448cj |
| [23] | 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 |
| [24] | Lamb, H., Hydrodynamics Cambridge Mathematical Library, 1932, Cambridge University Press: Cambridge University Press Cambridge · JFM 58.1298.04 |
| [25] | Levi-Civita, T., Sull’attrazione esercitata da una linea materiale in punti prossimi alla linea stessa, Rend. R. Accad. Lincei, 17, 3-15, 1908 · JFM 39.0830.01 |
| [26] | Lin, C. C., On the motion of vortices in two dimension - I. Existence of the Kirchhoff-Routh function, Proc. Natl. Acad. Sci. USA, 27, 570-575, 1941 · Zbl 0063.03560 |
| [27] | Majda, A.; Bertozzi, A., Vorticity and Incompressible Flow, 2002, Cambridge University Press: Cambridge University Press Cambridge · Zbl 0983.76001 |
| [28] | Marchioro, C.; Pulvirenti, M., Mathematical Theory of Incompressible Nonviscous Fluids, 1994, Springer: Springer New York · Zbl 0789.76002 |
| [29] | Smets, D.; Van Schaftingen, J., Desingularization of vortices for the Euler equation, Arch. Ration. Mech. Anal., 198, 3, 869-925, 2010 · Zbl 1228.35171 |
| [30] | Turkington, B., On steady vortex flow in two dimensions, I, II, Commun. Partial Differ. Equ., 8, 999-1071, 1983 · Zbl 0523.76015 |
| [31] | Thomson, W., Mathematical and Physical Papers, IV, 1910, Cambridge University Press: Cambridge University Press Cambridge · JFM 41.0019.02 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
