Long time confinement of vorticity around a stable stationary point vortex in a bounded planar domain

Donati, Martin; Iftimie, Dragoş

doi:10.1016/j.anihpc.2020.11.009

Cited by 6 publications

(6 citation statements)

References 10 publications

(26 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In that same article, the authors proved that when the initial vorticity is concentrated near the center of a disk, namely that Ω = D(0, 1), N = 1, and z 1 = 0, then we obtain the same power-law lower-bound τ ε,β ≥ ε −ξ0 than with expanding self-similar configurations. This result has been generalized to other bounded domains in [7]. This is due to a strong stability property induced by the shape of the boundary.…”

Section: 4mentioning

confidence: 73%

“…The long time confinement problem consists of obtaining a lower-bound on τ ε,β in order to describe how long the approximation of a concentrated solution of equations (1) by the point-vortex model (α-PVS) remains valid. Results have been obtained in [2,7,5]. In the following, we recall some of them and state our main results, starting with the case α = 1.…”

Section: 1mentioning

confidence: 99%

See 1 more Smart Citation

Construction of unstable concentrated solutions of the Euler and gSQG equations

Donati

2024

DCDS

Self Cite

View full text Add to dashboard Cite

In this paper we construct solutions to the Euler and gSQG equations that are concentrated near unstable stationary configurations of pointvortices. Those solutions are themselves unstable, in the sense that their localization radius grows from order ε to order ε β (with β < 1) in a time of order | ln ε|. In particular, this proves that the logarithmic lower-bound obtained in previous papers (in particular [P. Buttà and C. Marchioro, Long time evolution of concentrated Euler flows with planar symmetry, SIAM J. Math. Anal., 50(1):735-760, 2018]) about vorticity localization in Euler and gSQG equations is optimal. In addition, we construct unstable solutions of the Euler equations in bounded domains concentrated around a single unstable stationary point. To achieve this we construct a domain whose Robin's function has a saddle point.

show abstract

Section: 4mentioning

confidence: 73%

Section: 1mentioning

confidence: 99%

Construction of unstable concentrated solutions of the Euler and gSQG equations

Donati

2024

DCDS

Self Cite

View full text Add to dashboard Cite

show abstract

“…This point-vortex model is used to describe vortex phenomena arising in different problems of fluid mechanics (see for instance: [13,20,28,30,33] and references therein). The question of a rigorous derivation of the point-vortex system from the Euler equations (also called desingularisation problem, or localisation problem) is a standard problem that is linked to the problem of confinement and localisation of vorticity [8,34,49]. Another important link between the point-vortex system and the underlying PDE is the series of result concerning mean-field limits.…”

Section: Introductionmentioning

confidence: 99%

Hölder regularity for collapses of point-vortices

Donati,

Godard-Cadillac

2023

Nonlinearity

Self Cite

View full text Add to dashboard Cite

The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. In these models the kernel of the Biot–Savart law is a power function of exponent − α . It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to the time of collapse. The Hölder exponent obtained is 1 / ( α + 1 ) and this exponent is proved to be optimal for all α by exhibiting an example of a three-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given point-vortex has an accumulation point in the interior of the domain as t → T, then it converges towards this point and displays the same Hölder continuity property. A partial result for point-vortices that collapse with the boundary is also established: we prove that their distance to the boundary is Hölder regular.

show abstract

“…This point-vortex model is used to describe vortex phenomena arising in different problems of fluid mechanics (for instance: [10,15,19,20,21] and references therein). The question of a rigorous derivation of the point-vortex system from the Euler equations (also called desingularization problem, or localization problem) is a standard problem that is linked to the problem of confinement and localization of vorticity [8,22,29].…”

Section: Introductionmentioning

confidence: 99%

Hölder regularity for collapses of point vortices

Donati¹,

Godard-Cadillac²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. This consists in a Biot-Savart law with a kernel being a power function of exponent −α. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the vorticies have a regularity Hölder at T the time of collapse. The Hölder exponent obtained is 1/(α + 1) and this exponent is proved to be optimal for all α by exhibiting an example of a 3-vortex collapse.The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given vortex has an adherence point in the interior of the domain as t → T , then it converges towards this point and show the same Hölder continuity property.

show abstract

Long time confinement of vorticity around a stable stationary point vortex in a bounded planar domain

Cited by 6 publications

References 10 publications

Construction of unstable concentrated solutions of the Euler and gSQG equations

Construction of unstable concentrated solutions of the Euler and gSQG equations

Hölder regularity for collapses of point-vortices

Hölder regularity for collapses of point vortices

Contact Info

Product

Resources

About