Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data

Donati, Martin

doi:10.1137/21m1413213

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…We have that the function g i (t) is bounded by a constant that depends only on Ω, see the proof of corollary 3.6 in[35]. Concerning h i (t), the definitions of c m (t) and β m respectively at(1.21) and(1.20) give that this term is bounded by standard elliptic estimates.…”

mentioning

confidence: 86%

“…As a consequence, it is possible to proceed to an analogous construction and give a meaning to the point-vortex system in bounded smooth domains. This consists in considering the Euler 2D equation in bounded domain with impermeability condition at the boundary and to write this equation in term of vorticity (see [10,31,32,35]). In the case of smooth simply connected bounded domains, we obtain the following ODE:…”

Section: Hölder Regularity For Point-vortices In Smooth Bounded Domainsmentioning

confidence: 99%

“…where C Ω is a constant that depends only on Ω and where dist(x, A) := inf y∈A |x − y|. Indeed, we know (see for instance [35]) that γ Ω satisfies…”

Section: Hölder Regularity For Point-vortices In Smooth Bounded Domainsmentioning

confidence: 99%

“…In [33], the authors prove that when the domain is a disc, the collapses are improbable in the sense that the set of initial datum leading to a collapse in finite time has a vanishing Lebesgue measure. This result has been extended to any bounded smooth domains (simply connected and multi-connected domains) by [35]. In [34, chapter 4], the authors obtained the improbability of collapses when the domain is the whole plane, under the non-neutral clusters hypothesis (1.11) presented hereafter.…”

Section: Introductionmentioning

confidence: 96%

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Hölder regularity for collapses of point-vortices

Donati,

Godard-Cadillac

2023

Nonlinearity

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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.

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mentioning

confidence: 86%

Section: Hölder Regularity For Point-vortices In Smooth Bounded Domainsmentioning

confidence: 99%

“…where C Ω is a constant that depends only on Ω and where dist(x, A) := inf y∈A |x − y|. Indeed, we know (see for instance [35]) that γ Ω satisfies…”

Section: Hölder Regularity For Point-vortices In Smooth Bounded Domainsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

Hölder regularity for collapses of point-vortices

Donati,

Godard-Cadillac

2023

Nonlinearity

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“…This is not a very restrictive hypothesis since it is known that the point-vortex system has a global solution for almost any initial data, in the sense of the Lebesgue measure. This was proved for Euler point-vortex dynamics (namely equations (α-PVS) for α = 1) in the torus [9], in bounded domains [22], and in the plane 2 [19]. For the general α-model (α-PVS), it was proved in the plane 3 [4,13].…”

Section: 1mentioning

confidence: 99%

Construction of unstable concentrated solutions of the Euler and gSQG equations

Donati

2024

DCDS

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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.

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A Control Problem with Passive Particles Driven by Point Vortices on the Sphere

Balsa

Gama

2022

Communications in Computer and Information Science

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Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data

Cited by 7 publications

References 11 publications

Hölder regularity for collapses of point-vortices

Hölder regularity for collapses of point-vortices

Construction of unstable concentrated solutions of the Euler and gSQG equations

A Control Problem with Passive Particles Driven by Point Vortices on the Sphere

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