User menu

Desingularization of vortex rings and shallow water vortices by a semilinear elliptic problem

Bibliographic reference de Valeriola, Sébastien ; Van Schaftingen, Jean. Desingularization of vortex rings and shallow water vortices by a semilinear elliptic problem. In: Archive for Rational Mechanics and Analysis, Vol. 210, no.2, p. 409-450 (2013)
Permanent URL
  1. Ambrosetti, A., Mancini, G.: On some free boundary problems. Recent contributions to nonlinear partial differential equations. Research Notes in Mathematics, Vol. 50, pp. 24–36. Pitman, Boston, 1981
  2. Ambrosetti Antonio, Rabinowitz Paul H, Dual variational methods in critical point theory and applications, 10.1016/0022-1236(73)90051-7
  3. Ambrosetti A., Struwe M., Existence of steady vortex rings in an ideal fluid, 10.1007/bf01053458
  4. Ambrosetti A., Yang J.F.: 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)
  5. Amick C.J., Fraenkel L.E., The uniqueness of Hill's spherical vortex, 10.1007/bf00251252
  6. Badiani T.V., Burton G.R.: Vortex rings in $${\mathbb{R}^3}$$ and rearrangements. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2001), 1115–1135 (2009)
  7. Benedetto D., Caglioti E., Marchioro C., On the motion of a vortex ring with a sharply concentrated vorticity, 10.1002/(sici)1099-1476(20000125)23:2<147::aid-mma108>;2-j
  8. Berestycki Henri, Brezis Haîm, On a free boundary problem arising in plasma physics, 10.1016/0362-546x(80)90083-8
  9. Fraenkel L. E., Berger M. S., A global theory of steady vortex rings in an ideal fluid, 10.1007/bf02392107
  10. Berger M. S., Fraenkel L. E., Nonlinear desingularization in certain free-boundary problems, 10.1007/bf01982715
  11. Buffoni B., Nested axi-symmetric vortex rings, 10.1016/s0294-1449(97)80133-3
  12. Burton G.R, Vortex rings in a cylinder and rearrangements, 10.1016/0022-0396(87)90155-0
  13. Burton G. R., Rearrangements of functions, maximization of convex functionals, and vortex rings, 10.1007/bf01450739
  14. BURTON G. R., Vortex-rings of prescribed impulse, 10.1017/s0305004102006631
  15. Burton Geoffrey R., Preciso Luca, Existence and isoperimetric characterization of steady spherical vortex rings in a uniform flow in R, 10.1017/s0308210500003292
  16. Camassa Roberto, Holm Darryl D., Levermore C.David, Long-time effects of bottom topography in shallow water, 10.1016/0167-2789(96)00117-0
  17. CAMASSA ROBERTO, HOLM DARRYL D., LEVERMORE C. DAVID, Long-time shallow-water equations with a varying bottom, 10.1017/s0022112097006721
  18. Cao, D., Liu, Z., Wei, J.: Regularization of point vortices for the Euler equation in dimension two (available at: arXiv:1208.3002)
  19. Cao, D., Liu, Z., Wei, J.: Regularization of point vortices for the Euler equation in dimension two, part II (available at: arXiv:1208.5540)
  20. De Giorgi E.: Sulla differenziabilità à e l’analiticità à delle estremali degli integrali multipli regolari. Mem. Acad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3), 25–43 (1957)
  21. Fraenkel L. E., On Steady Vortex Rings of Small Cross-Section in an Ideal Fluid, 10.1098/rspa.1970.0065
  22. Fraenkel L. E., Examples of steady vortex rings of small cross-section in an ideal fluid, 10.1017/s0022112072001107
  23. Fraenkel L. E., A lower bound for electrostatic capacity in the plane, 10.1017/s0308210500020114
  24. Friedman A.: Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics. Wiley, New York (1982)
  25. Friedman Avner, Turkington Bruce, Vortex rings: existence and asymptotic estimates, 10.1090/s0002-9947-1981-0628444-6
  26. Gehring F.W., Osgood B.G.: Uniform domains and the quasihyperbolic metric. J. Anal. Math. 36(1979), 50–74 (1980)
  27. Gehring F. W., Palka B. P., Quasiconformally homogeneous domains, 10.1007/bf02786713
  28. Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001)
  29. Helmholtz H., Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen., 10.1515/crll.1858.55.25
  30. Hill M. J. M., On a Spherical Vortex, 10.1098/rsta.1894.0006
  31. Lamb H.: Hydrodynamics Cambridge Mathematical Library. 6th edn Cambridge University Press, Cambridge (1932)
  32. Li Gongbao, Yan Shusen, Yang Jianfu, An Elliptic Problem Related to Planar Vortex Pairs, 10.1137/s003614100343055x
  33. Lindén, H.: Hyperbolic-Type Metrics. Quasiconformal Mappings and Their Applications. Narosa, New Delhi, pp. 151–164 (2007)
  34. Lions P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case, part I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1((2), 109–145 (1984)
  35. Lions P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case, part II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(4), 223–283 (1984)
  36. Maz’ya V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. 2rd edn Grundlehren der Mathematischen Wissenschaften, Vol. 342. Springer (2011)
  37. Ni Wei-Ming, On the existence of global vortex rings, 10.1007/bf02797686
  38. Norbury J., A steady vortex ring close to Hill's spherical vortex, 10.1017/s0305004100047083
  39. Norbury J., A family of steady vortex rings, 10.1017/s0022112073001266
  40. Norbury J., Steady planar vortex pairs in an ideal fluid, 10.1002/cpa.3160280602
  41. Rabinowitz Paul, Minimax Methods in Critical Point Theory with Applications to Differential Equations, ISBN:9780821807156, 10.1090/cbms/065
  42. Rabinowitz Paul H., On a class of nonlinear Schr�dinger equations, 10.1007/bf00946631
  43. RICHARDSON G., Vortex motion in shallow water with varying bottom topography and zero Froude number, 10.1017/s0022112099008393
  44. Smets Didier, Van Schaftingen Jean, Desingularization of Vortices for the Euler Equation, 10.1007/s00205-010-0293-y
  45. Tadie, On the bifurcation of steady vortex rings from a Green function, 10.1017/s0305004100072819
  46. Thomson, W.:Mathematical and Physical Papers, IV. Cambridge, 1910
  47. Weinstein Alexander, Generalized axially symmetric potential theory, 10.1090/s0002-9904-1953-09651-3
  48. Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, Vol. 24. Birkhäuser Boston Inc., Boston, 1996
  50. Jianfu Yang, Global vortex rings and asymptotic behaviour, 10.1016/0362-546x(93)e0018-x