# Time evolution of vortex rings with large radius and very concentrated vorticity

@inproceedings{Cavallaro2021TimeEO, title={Time evolution of vortex rings with large radius and very concentrated vorticity}, author={Guido Cavallaro and Carlo Marchioro}, year={2021} }

We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on N annuli of radii ≈ r0 and thickness ε. We prove that when r0 = | log ε|α, α > 2, the vorticity field of the fluid converges as ε → 0 to the point vortex model, at least for a small but positive time. This result generalizes a previous paper that assumed a power law for the relation between r0 and ε.

## 2 Citations

Stability of the two-dimensional point vortices in Euler flows

- Mathematics
- 2022

∂tω + u · ∇ω = 0 ω(0, x) = ω0(x). We are interested in the cases when the initial vorticity has the form ω0 = ω0,ǫ + ω0p,ǫ, where ω0,ǫ is concentrated near M disjoint points p0m and ω0p,ǫ is a small…

Global time evolution of concentrated vortex rings

- Mathematics, PhysicsZeitschrift für angewandte Mathematik und Physik
- 2022

We study the time evolution of an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside N small disjoint rings of…

## References

SHOWING 1-10 OF 32 REFERENCES

On the motion of a vortex ring with a sharply concentrated vorticity

- Mathematics, Environmental Science
- 2000

We study an incompressible non-viscous fluid with axial symmetry without swirl, in the case when the vorticity is supported in an annulus. It is well known that there exist particular initial data…

Time Evolution of Concentrated Vortex Rings

- Physics
- 2019

We study the time evolution of an incompressible fluid with axisymmetry without swirl when the vorticity is sharply concentrated. In particular, we consider N disjoint vortex rings of size…

Vortices and localization in Euler flows

- Mathematics
- 1993

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…

Large smoke rings with concentrated vorticity

- Mathematics, Physics
- 1999

In this paper we study an incompressible inviscid fluid when the initial vorticity is sharply concentrated in N disjoint regions. This problem has been well studied when a planar symmetry is present,…

On steady vortex rings of small cross-section in an ideal fluid

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1970

This paper is concerned with vortex rings, in an unbounded inviscid fluid of uniform density, that move without change of form and with constant velocity when the fluid at infinity is at rest. The…

Vanishing viscosity limit for an incompressible fluid with concentrated vorticity

- Mathematics
- 2007

We study an incompressible fluid with a sharply concentrated vorticity moving in the whole space. First, we review some known results for the inviscid case, which prove, in particular situations, the…

On the Inviscid Limit for a Fluid with a Concentrated Vorticity

- Mathematics
- 1998

Abstract:We study the time evolution of a viscous incompressible fluid in ℝ2 when the initial vorticity is sharply concentrated in N regions of diameter ε. We prove that in the zero viscosity limit…

Long Time Evolution of Concentrated Euler Flows with Planar Symmetry

- PhysicsSIAM J. Math. Anal.
- 2018

A toy model is analyzed that shows a similar behavior to an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks, showing that in some cases this happens for quite long times.

Concentrated Euler flows and point vortex model

- Mathematics
- 2015

This paper is an improvement of previous results on concentrated Euler flows and their connection with the point vortex model. Precisely, we study the time evolution of an incompressible two…

Interaction of Vortices in Weakly Viscous Planar Flows

- Mathematics
- 2011

We consider the inviscid limit for the two-dimensional incompressible Navier–Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that…