Bounds on the growth of the support of a vortex path. (English) Zbl 0839.76010

Summary: We study the time evolution of the support of a vortex patch evolving in \(\mathbb{R}^2\) according to the Euler equation for an incompressible fluid and we bound its growth. Furthermore, we discuss the same problem in the framework of a simplified model. Finally, we consider a similar problem for the Navier-Stokes flow.


76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
35Q30 Navier-Stokes equations
Full Text: DOI


[1] [Dri88] Dritschel, D.G.: Nonlinear Stability Bounds for Inviscid, Two-Dimensional, Parallel or Circular Flows with Monotonic vorticity, and Analogous Three-dimensional Quasi-Geostrophis Flows. J. Fluid Mech.191, 575–581 (1988) · Zbl 0643.76059
[2] [Dri89] Dritschel, D.G.: Contour Dynamics and Contour Surgery: Numerical Algorithms for Extended, High-Resolution Modelling of Vortex Dynamics in Two-Dimensional, Inviscid, Incompressible Flows. Computer Phys. Reports10, 77–146 (1989)
[3] [Lam32] Lamb, H.: Hydrodynamics, 6. ed. Cambridge: Cambridge University Press, 1932
[4] [Mar88] Marchioro, C.: Euler Evolution for Singular Initial Data and Vortex Theory: A Global Solution. Commun. Math. Phys.116, 45–55 (1988) · Zbl 0654.76017
[5] [Ma85] Marchioro, C., Pulvirenti, M.: Some Considerations on the Nonlinear Stability of Stationary Planar Euler Flows. Commun. Math. Phys.100, 343–354 (1985) · Zbl 0625.76060
[6] [Ma92] Marchioro, C., Pulvirenti, M.: Vortices and localization in Euler flows. Commun. Math. Phys.154, 49–61 (1993) · Zbl 0774.35058
[7] [MaP94] Marchioro, C., Pulvirenti, M.: Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96, Berlin, Heidelberg, New York: Springer 1994 · Zbl 0789.76002
[8] [WaP85] Wan, Y.H., Pulvirenti, M.: Nonlinear Stabilityof Circular Vortex Patch. Commun. Math. Phys.99, 435–450 (1985) · Zbl 0584.76062
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.