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In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simply-connected bou...
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In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simply-connected bou...
Citations
... In the literature, (P1) and (P2) are often referred to as desingularization of vortices. The study of (P1) was initiated by Marchioro and Pulvirenti [26], then followed by many authors [8,14,15,17,[25][26][27]33]. Roughly speaking, the answer to (P1) is positive, although some open problems still remain, such as the optimal concentration rate. ...
In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the polar axis. More importantly, these solutions ``converges" to a pair of point vortices with equal strength and opposite signs. The construction is achieved by maximizing the energy-impulse functional relative to a family of suitable rearrangement classes and analyzing the asymptotic behavior of the maximizers. Based on their variational characterization, we also prove the stability of these rotating solutions with respect to odd-symmetric perturbations.
... In the present paper, we try to push further the analysis and prove that any solution to the Euler equation concentrated near this type of configuration remains concentrated for a very long time following the vortex crystal prediction. More precisely, if the support of the initial vorticity lies within disks of size ε, we prove that it remains within disks of size ε β , with β < 1/2, for a time at least of order ε −α for some α > 0. This extends previous results, see [3,7], where such long time confinement was obtained for configurations satisfying either N = 1 (only one vortex), or that point-vortices had to move away from each other very fast. Here, we prove that we need neither of those assumptions, but instead that the vortices are disposed according to the previously described polygonal configuration. ...
... There are other configurations found in [3] that satisfy this improved bound on the time of confinement when one only considers a single blob of vorticity (N = 1), in the plane, or at the center of a disk. In [7], it is proved that, for N = 1 in some special bounded and simply connected domains with a suitable choice of z 1 , the solution satisfies that τ ε,β ≥ ε −α for any α < min(β, 2 − 4β). ...
... Let us formulate a few remarks about Theorem 3.4 and its hypotheses. As mentioned in the introduction, the big difference with the similar long-time confinement results obtained in [3] and [7] is that we neither assume that N = 1, nor that all the distances between the point-vortex grow quickly. ...
In this paper, we control the growth of the support of particular solutions to the Euler two-dimensional equations, whose vorticity is concentrated near special vortex crystals. These vortex crystals belong to the classical family of regular polygons with a central vortex, where we choose a particular intensity for the central vortex to have strong stability properties. A special case is the regular pentagon with no central vortex which also satisfies the stability properties required for the long-time confinement to work.
... In (2.23) the condition a > |b| is required. This corresponds to the ellipticity of the critical point ξ 0 and it is necessary for the dynamics to be confined locally near ξ 0 , as explained in [31]. ...
We examine the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic motion. We show that for the single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of these non-degeneracy conditions, which are satisfied by a broad class of domains, including convex ones. The proof uses Nash-Moser scheme and KAM techniques, in the spirit of the recent work of Hassainia-Hmidi-Masmoudi on the leapfrogging motion, combined with complex geometry tools. Additionally, we employ a vortex duplication mechanism to generate synchronized time-periodic motion of multiple vortices. This approach can be, for instance, applied to desingularize the motion of two symmetric dipoles (with four vortices) in a disc or a rectangle. To our knowledge, this is the first result showing the existence of non-rigid time-periodic motion for Euler equations in generic simply-connected bounded domain. This answers an open problem that has been pointed in the literature, for example by Bartsch-Sacchet.
... 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. ...
... 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. ...
... Known results on critical points of the Robin's function. In this section we refer to [7] and recall the following results. For more details on the Robin's function we refer the reader to [15,11]. ...
... The point vortex model is only an approximate model, and rigorous analysis on its connection with the vorticity equation with concentrated vorticity is an interesting research topic. We refer the interested readers to [18,[30][31][32][33][34][35]39] for some deep results in this respect. ...
In this paper, we study the stability of two-dimensional steady Euler flows with sharply concentrated vorticity in a bounded domain. These flows are obtained as maximizers of the kinetic energy on some isovortical surface, under the constraint that the vorticity is compactly supported in a finite number of disjoint regions of small diameter. We prove the nonlinear stability of these flows when the vorticity is sufficiently concentrated in one small region, or in two small regions with opposite signs. The proof is achieved by showing that these flows constitute a compact and isolated set of local maximizers of the kinetic energy on the isovortical surface. The separation property of the stream function plays a crucial role in validating isolatedness.
... The long time confinement problem consists in 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 [2,8,5]. In the following, we recall some of them and state our main results, starting with the case α = 1. ...
... 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 powerlaw lower-bound τ ε,β ≥ ε −ξ 0 than with expanding self-similar configurations. This result has been generalized to other bounded domains in [8]. This is due to a strong stability property induced by the shape of the boundary. ...
... In this section we refer to [8] and recall the following results. For more details on the Robin's function we refer the reader to [16,12]. ...
In this paper we construct solutions to the Euler and gSQG equations that are concentrated near unstable stationary configurations of point-vortices. Those solutions are themselves unstable, in the sense that their localization radius grows from order \eps to order \eps^\beta (with ) in a time of order |\ln\eps|. This proves
in particular that the logarithmic lower-bound obtained in previous papers (in particular [P. Buttà and C. Marchioro, \emph{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.
... For such initial vorticities, Buttà and Marchioro proved in [10] that the solution ω ǫ (x, t) remains concentrated near p(t) during a time scale t ≈ | log ǫ|. They also show that when M = 1, p 0 1 = 0 (a single vortex concentrated near origin) and if the space domain R 2 is replaced by B R (0), then the solution remains concentrated near origin during a time scale t ≈ ǫ −a for some a > 0. In [19], Donati and Iftimie generalized the result to simply connected bounded domains. ...
We consider the two-dimensional incompressible Euler equation We are interested in the cases when the initial vorticity has the form , where is concentrated near M disjoint points and is a small perturbation term. First, we prove that for such initial vorticities, the solution admits a decomposition , where remains concentrated near M points and remains small for . Second, we give a quantitative description when the initial vorticity has the form , where we do not assume to have compact support. Finally, we prove that if remains separated for all , then remains concentrated near M points at least for , where is small and converges to 0 as .
... This point-vortex model is used to describe vortex phenomena arising in different problems of fluid mechanics (see for instance: [10,15,21,23,24] 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 [7,25,32]. ...
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 models. In these models the kernel of the Biot-Savart law is a power function of exponent . It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to, and including, the time of collapse. The Hölder exponent obtained is and this exponent is proved to be optimal for all 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 point-vortex has an accumulation point in the interior of the domain as , 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.
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.
In this paper, we prove that in bounded planar domains with C^{2,\alpha} boundary, for almost
every initial condition in the sense of the Lebesgue measure, the point-vortex system has
a global solution, meaning that there is no collision between two point-vortices or with
the boundary. This extends the work previously done in [C. Marchioro and M. Pulvirenti,
Vortex Methods in Two-Dimensional Fluid Dynamics, Springer-Verlag, 1984] for the disk.
The proof requires the construction of a regularized dynamics that approximates the real
dynamics and some strong inequalities for the Green's function of the domain. The estab-
lishment of some useful estimates is discussed and the details of the proof are given in the
original article [M. Donati, SIAM J. Math. Anal., 54 (2022), pp. 79--113].