A numerical study of some questions in vortex rings theory
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 18 (1984) no. 1, p. 7-85
@article{M2AN_1984__18_1_7_0,
     author = {Berestycki, Henri and Fernandez Cara, Enrique and Glowinski, Roland},
     title = {A numerical study of some questions in vortex rings theory},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {18},
     number = {1},
     year = {1984},
     pages = {7-85},
     zbl = {0594.76021},
     mrnumber = {727602},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1984__18_1_7_0}
}
Berestycki, Henri; Fernandez Cara, Enrique; Glowinski, Roland. A numerical study of some questions in vortex rings theory. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 18 (1984) no. 1, pp. 7-85. http://www.numdam.org/item/M2AN_1984__18_1_7_0/

[1] A. Ambrosetti and G. Mancini, to appear.

[2] A. Ambrosetti and G. Mancini, Remarks on some free boundary problems, In Recent Contributions to Nonlinear Partial Differential Equations. H. Berestycki and H. Brézis éd. Pitman, London, 1981. | MR 639743 | Zbl 0477.35084

[3] J. F. G. Auchmuty and R. Beals, Variational solutions of some nonlinear free boundary problems, Arch. Rat. Mech. Anal. 43 (1971), pp. 255-271. | MR 337260 | Zbl 0225.49013

[4] J. F. G. Auchmuty and T. B. Benjamin, Rearrangement existence proofs for vortex rings, In préparation.

[5] P. Benilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, à paraître | Zbl 1150.35406

[6] T. B. Benjamin, The alliance of practical and analytic insights into the nonlinear problems of fluid mechanics, in Applications of Methods of Functional Analysis to Problems of Mechanics, pp. 8-29. Lecture Notes in Math. N° 503. Springer-Verlag, New York, 1976. | MR 671099 | Zbl 0369.76048

[7] H. Berestycki, Thèse de Doctorat d'État ès-Sciences, Univ. P. et M. Curie (Paris VI), 1980.

[8] H. Berestycki, Some free boundary problems in plasma physics and fluid mechanics, In Applications of Nonlinear Analysis to the Physical Sciences. H. Amann, N. Bazley and K. Kirchgassner ed. Pitman, London 1981. | MR 659699 | Zbl 0503.76127

[9] H. Berestycki, Quelques questions à la théorie des tourbillons stationnaires dans un fluide idéal, J. Math. Pures Appl., to appear.

[10] H. Berestycki and H. Brezis, Sur certains problèmes de frontière libre, Compte Rendus Acad. Se. Paris, série A, 283 (1976), pp. 1091-1094. | MR 427812 | Zbl 0342.35014

[11] H. Berestycki and H. Brezis, On a free boundary problem arising in plasma physics, Nonlinear Analysis, 4 (1980), pp. 415-436. | MR 574364 | Zbl 0437.35032

[12] H. Berestycki and P. L. Lions, A direct variational approach to the global theory of vortex rings in an ideal fluid, To appear.

[13] H. Berestycki and C. Stuart, Sur des méthodes itératives pour la résolution de certains problèmes de valeurs propres non linéaires, Note C.R.A.S., to appear.

[14] H. Berestycki and C. Stuart, Some itérative schemes for nonlinear eigenvalue problems, To appear.

[15] M. S. Berger and L. E. Fraenkel, Nonlinear desingularization in certain free-boundary problems, Comra. Math. Phys. 77 (1980), pp. 149-172. | MR 589430 | Zbl 0454.35087

[16] N. Bourbaki, Elément de Mathématiques : Livre VI, Intégration. Actualités Scient. Ind. Hermann, Paris 1963-67.

[17] H. Brezis, Some variational problems of the Thomas-Fermi type, In Variational Inequalities. Cottle, Gianessi and Lions éd., J. Wiley and Sons, New York 1980. | MR 578739 | Zbl 0643.35108

[18] H. Brezis, R. Benguria and F. H. Lieb, The Thomas-Fermi Von Weizacker theory ofatoms and molécules, To appear in Comm. Math. Phys. | MR 612246 | Zbl 0478.49035

[19] J. P. Christiansen and N. J. Zabusky, Instability, coalescence and fission offinit e area vortex structures, J. Fluid Mech., 61 (1973), pp.219-243. | Zbl 0266.76039

[20] P. G. Ciarlet, The Finite Element Method for Elliptic Problem. North-Holland, Amsterdam, 1978. | MR 520174 | Zbl 0383.65058

[21] P G Ciarlet and P A Raviat, Maximum pnnciple and uniform convergence for the finite element method, Computer Methods in Applied Mechanics and Engineering, 2 (1973), pp 17-31 | MR 375802 | Zbl 0251.65069

[22] L Collatz, Functional Analysis and Numencal Mathematics, Academic Press, New York, 1966 | MR 205126 | Zbl 0148.39002

[23] G S Deem and N J Zabusky, Vortex waves statwnary V-vortex waves , Phys Rev Letters, 40 (1978), pp 859-862 e

[24] M J Esteban, Thèse de Doctorat de 3e Cycle, Univ P et M Curie (Paris VI), 1981

[25] M J Esteban and P L Lions, To appear

[26] E Fernandez Cara, Méthodes numériques pour des problèmes nonlinéaires apparaissant dans la théorie des tourbillons stationnaires d'un fluide idéal, Rapport de Recherche INRIA n° 39, October 1980

[27] E, Fernandez Cara, To appear

[28] L E Fraenkel and M S Berger, On the global theory of vortex rings inanidéal fluid, Acta Math 132 (1974), pp 13-51 | MR 422916 | Zbl 0282.76014

[29] A Friedman and B Turkington, Asymptotic estimates for an axisymmetric rotating fluid, J Functional Anal, to appear | MR 578929 | Zbl 0435.35014

[30] B Gidas, Ni, Wei-Ming and L Niremberg, Symmetry and related properties via the maximum principle, Comm Math Phys 68 (1979), pp 209-243 | MR 544879 | Zbl 0425.35020

[31] R Glowinski, Numencal Methods for Nonlinear Variational Problems, 2nd édition To appear | Zbl 0536.65054

[32] J W Kitchen, Concerning the convergence of iterates tofixed points, Stud Math 27 (1966), pp 247-249 | MR 200759 | Zbl 0143.16601

[33] H Lamb, Hydrodynamics (6th ed) Cambridge, 1932 | JFM 58.1298.04 | Zbl 0828.01012

[34] L Lichtenstein, Uber einige existenz probleme der Hydrodynamik, Math Z , 23 (1925), pp 89-154 | JFM 51.0658.01 | MR 1544733

[35] L Lichtenstein, Grundlagen der Hydrodynamik, Sprmger-Veriag, Berlin 1929

[36] E H Lieb and B Simon, The Thomas-Fermi theory of atoms, molecules and solids Advances in Math , 23 (1977), pp 22-116 1 3 | MR 428944 | Zbl 0938.81568

[37] P L Lions, Mimmization problems in L 1 (R 3 ) and applications to free boundary problems, To appear | Zbl 0467.49028

[38] P Lions, Minimization problems in L 1 (R 3 ), To appear | Zbl 0464.49019

[39] Ni, Wei-Ming, On the existence of global vortex rings, To appear | MR 583638 | Zbl 0457.76020

[40] J Norbury, Steady planar vortex pairs inan idéal fluid, Comm Pure Appl Math , 28 (1975), pp 679-700 | MR 399645 | Zbl 0338.76015

[41] J Norbury, A steady vortex ring close to Hill's spherical vortex, Proc Camb Phil Soc , 72 (1972), pp 253-284 | MR 302044 | Zbl 0256.76016

[42] J Norbury, A family of steady vortex rings, J Fluid Mech ,57 (1973), pp 417-431 | Zbl 0254.76018

[43] R T Pierrehumbert, A family of steady translating vortex pairs with distnbuted vorticity, J Fluid Mech, 99 (1980), pp 129-144 | Zbl 0473.76034

[44] J P Puel, Sur un problème de valeur propre non linéaire et de frontière libre, Compte Rendus Ac Sc Paris, série A, 284 (1977), pp 861-863 | MR 436755 | Zbl 0362.35024

[45] P G Saffman, The velocity of vortex rings, Studies in Appl Math , 49 (1970), pp 371-379 | Zbl 0224.76032

[46] P G Saffman, Dynamics of vorticity, J Fluid Mech , 106 (1981), pp 49-58 | Zbl 0465.76020

[47] P G Saffman and J C Schatzman, Properties of a vortex street of finite vortices, To appear | MR 632900 | Zbl 0484.76032

[48] M. Sermange, Etude numérique des bifurcations et de la stabilité des solutions des équations de Grad-Shafranov. In IVe Colloque International sur les Méthodes de Calcul Scientifique et Technique, Versailles, 10-14 déc. 1974. | Zbl 0441.76092

[49] R. Temam, A nonlinear eigenvalue problem : the shape at equilibrium of a confined plasma, Arch. Rat. Mech. Anal., 60 (1975), pp. 51-73. | MR 412637 | Zbl 0328.35069

[50] R. Temam, Remarks on a free boundary problem arising in plasma physics, Comm. P.D.E., 2 (1977), pp. 563-585. | MR 602544 | Zbl 0355.35023

[51] B. Turkington, Inviscid flows with vorticity, In Proceedings of the Montecatini Conference on Free Boundary Problems, June 1981. To appear.