Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Rearrangements in steady vortex flows with circulation

Authors: Alan R. Elcrat and Kenneth G. Miller
Journal: Proc. Amer. Math. Soc. 111 (1991), 1051-1055
MSC: Primary 35Q35; Secondary 58D25, 76C05
MathSciNet review: 1043409
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that a steady two-dimensional flow, in which a finite vortex is in equilibrium with the irrotational flow past an obstacle, can be obtained as the solution of a variational problem in the class of rearrangements of a fixed function in $ {L^p}$. The main step is to establish a bound on the support of the vorticity. The advantage of this approach, as in the recent works of Burton, Benjamin, and Auchmuty, is that the profile function of the vorticity is determined by the rearrangement class in which solutions are sought.

References [Enhancements On Off] (What's this?)

  • [1] V. I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, Berlin, 1978. MR 0690288 (57:14033b)
  • [2] J. F. Auchmuty and T. B. Benjamin, unpublished manuscript.
  • [3] T. B. Benjamin, The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics, Applications of methods of functional analysis to problems of mechanics, Lecture Notes in Mathematics, vol. 503, Springer-Verlag, Berlin, 1976, pp. 8-29. MR 0671099 (58:32375)
  • [4] H. J. Brascamp, E. H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974), 227-237. MR 0346109 (49:10835)
  • [5] G. R. Burton, Steady symmetric vortex pairs and rearrangements, Proc. Royal Soc. Edinburgh 108A (1988), 269-290. MR 943803 (89f:35178)
  • [6] -, Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. Henri Poincaré. Analyse Nonlineare 6 (1989), 295-319. MR 998605 (90h:58017)
  • [7] A. R. Elcrat and K. G. Miller, Steady vortex flows with circulation past asymmetric obstacles, Comm. Partial Differential Equations 12 (1987), 1095-1115. MR 886341 (88f:76013)
  • [8] B. Turkington, On steady vortex flows in two dimensions, I, Comm. Partial Differential Equations 8 (1983), 999-1030. MR 702729 (85g:35110)
  • [9] -, On steady vortex flows in two dimensions, II, Comm. Partial Differential Equations 8 (1983), 1031-1071.
  • [10] -, On the evolution of a concentrated vortex in an ideal fluid, Arch. Rational Mech. Anal. 97 (1987), 75-87. MR 856310 (87h:76036)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35Q35, 58D25, 76C05

Retrieve articles in all journals with MSC: 35Q35, 58D25, 76C05

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society