Abstract
We study the time evolution of a non-viscous incompressible two-dimensional fluid when the initial vorticity is concentrated inN small disjoint regions of diameter ε. We prove that the time evolved vorticity is also concentrated inN regions of diameterd, vanishing as ε→0. As a consequence we give a rigorous proof of the validity of the point vortex system. The same problem is examined in the context of the vortex-wave system.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Helmholtz, H.: On the integrals of the hydrodynamical equations which express vortex motion. Phil. Mag.33, 485 (1867); Kirchhoff, G.: Vorlesungen über Math. Phys., Leipzig: Teuber 1876 Poincaré, H.: Théories des Tourbillons. Paris: George Carré 1893 Kelvin, J.: Mathematical and physical papers. Cambridge: Cambridge University Press 1910
Marchioro, C., Pulverenti, M.: Vortex methods in two-dimensional flud mechancis. Lecture, Notes in Physics, Vol. 203, Berlin, Heidelberg, New York: Springer 1984
Marchioro, C., Pulvirenti, M.: Mathematical Theory of incompressible non-viscous fluids. Berlin, Heidelberg, New York: Springer (to appear)
Marchioro, C., Pulvirenti, M.: Euler evolution for singular data and vortex theory. Commun. Math. Phys.91, 563 (1983)
Marchioro, C.: Euler evolution for singular initial data and vortex theory: a global solution. Commun. Math. Phys.116, 45 (1988)
Turkington, B.: On the evolution of a concentrated vortex in an ideal, fluid. Arch. Rat. Mech. An.97, 75 (1987)
Marchioro, C., Pagani, E.: Evolution of two concentrated vortices in a two-dimensional bounded domain. Math. Meth. Appl. Sci.8, 328 (1986)
Marchioro, C.: On the vanishing viscosity limit for two-dimensional Navier-Stokes equations with singular initial data. Math. Meth. Appl. Sci.12, 463 (1990)
Marchioro, C., Pulvirenti, M.: On vortex-wave system. Mechanics, analysis, and geometry: 200 years after Lagrange. M. Francaviglia (ed.), Amsterdam: Elsevier Science 1991
Author information
Authors and Affiliations
Additional information
Communicated by J.L. Lebowitz
Research partially supported by MURST, (Ministero dell'Università e della Ricerca Scientifica e Tecnologica), CNR (Consiglio Nazionale delle Ricerche-Gruppo Nazionale per la Fisica Matematica) and CNR contract n.92.00544.01
Rights and permissions
About this article
Cite this article
Marchioro, C., Pulvirenti, M. Vortices and localization in Euler flows. Commun.Math. Phys. 154, 49–61 (1993). https://doi.org/10.1007/BF02096831
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02096831