×

zbMATH — the first resource for mathematics

Eigenvalues associated with the vortex patch in 2-D Euler equations. (English) Zbl 1058.76012
Summary: We consider Dirichlet eigenvalue problem associated with a vortex patch for two-dimensional Euler equations. We show that the eigenvalues grow at most doubly exponentially in time. As an application, we derive bounds on the growth of some geometric quantities like the diameter and the inscription radius of the patch. We also discuss the growth of the perimeter of the patch. In particular, we give a double exponential bound on the growth of certain portion of the boundary of the patch.

MSC:
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ancona, A.: On strong barriers and an inequality of Hardy for domains in Rn. J. London Math. Soc. (2) 34, 274-290 (1986) · Zbl 0629.31002
[2] Majda, A.: Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math. 39, S187?S220 (1986) · Zbl 0595.76021
[3] Bertozzi, A.L.: Existence, uniqueness and a characterisation of solutions to the contour dynamics equation. PhD Thesis, Princeton University, 1991
[4] Bertozzi, A.L., Constantin, P.: Global regularity for vortex patches. Comm. Math. Phys. 152, 19-28 (1993) · Zbl 0771.76014 · doi:10.1007/BF02097055
[5] Chemin, J.Y.: Perfect incompressible fluids. The Clarendon Press, Oxford University Press, New York, 1998 · Zbl 0927.76002
[6] Chemin, J.Y.: Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. Ecole Norm. Sup. (4) 26, 517-542 (1993) · Zbl 0779.76011
[7] Chemin, J.Y.: Sur le mouvement des particules d?un fluide parfait, incompressible, bidimensionnel. Invent. Math. 103, 599-629 (1991) · Zbl 0739.76010 · doi:10.1007/BF01239528
[8] Danchin, R.: Évolution d?une singularité de type cusp dans une poche de tourbillon. Rev. Mat. Iberoamericana 16, 281-329 (2000) · Zbl 1158.35404
[9] Fuglede, B.: Continuous domain dependence of the eigenvalues of the Dirichlet Laplacian and related operators in Hilbert space. J. Funct. Anal., 167, 183-200 (1999) · Zbl 0948.47047 · doi:10.1006/jfan.1999.3442
[10] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Classics in Mathematics. Springer-Verlag, Berlin, 2001 · Zbl 1042.35002
[11] Iftimie, D., Sideris, T.C., Gamblin, P.: On the evolution of compactly supported planar vorticity. Comm. Partial Diff. Eqs. 24, 1709-1730 (1999) · Zbl 0937.35137
[12] Lions, P.L.: Mathematical topics in fluid mechanics, vol. 1, incompressible models. Oxford University Press, New York, 1996 · Zbl 0866.76002
[13] Marchioro, C., Pulvirenti, M.: Mathematical theory of incompressible nonviscous fluids. Springer-Verlag, New York, 1994 · Zbl 0789.76002
[14] Yudovich, V.: Non stationary flows of an ideal incompressible fluid. Zhurnal Vych Matematika 3, 1032-1066 (1963)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.