Existence of non-trivial non-concentrated compactly supported stationary solutions of the 2D Euler equation with finite energy

Gómez-Serrano, Javier; Park, Jaemin; Shi, Jia

doi:10.48550/arxiv.2112.03821

Cited by 4 publications

(5 citation statements)

References 72 publications

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“…In this case, ψ ε has a line of critical points. The recent work [100] gives examples of this phenomenon for non-radial vortex patches with compactly supported velocity field on the plane. If ψ has no critical points (on a multiply connected domain) or isolated critical points, there always will exist a single global map F .…”

Section: D Fluids: a Tale Of Isolation Wandering And Long-time Strifementioning

confidence: 95%

Singularity formation in the incompressible Euler equation in finite and infinite time

Drivas¹,

Elgindi²

2022

Preprint

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Section: D Fluids: a Tale Of Isolation Wandering And Long-time Strifementioning

confidence: 95%

Singularity formation in the incompressible Euler equation in finite and infinite time

Drivas¹,

Elgindi²

2022

Preprint

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“…To the best of our knowledge, the asymmetric tripole patch solutions are new both for the Euler and gSQG equations. Furthermore, this appears to be the first existence proof for stationary solutions to the gSQG equations involving multiple patches; see [27] for stationary solutions to the Euler equations with multiple multi-layered patches and [26] for stationary doubly connected solutions to the gSQG equations. (i) For any ε > 0 sufficiently small, there are two strictly convex domains O ε 1 , O ε 2 , 1-fold symmetric, C 1+β perturbations of the unit disc, and real numbers…”

Section: 3mentioning

confidence: 99%

“…See [6] for related results where point vortices are instead desingularized into doubly-connected patches. We mention that, using more sophisticated Nash-Moser techniques, Gómez-Serrano, Park and Shi [27] have very recently constructed stationary configurations of multi-layered patches with finite kinetic energy. Also, García and Haziot [23] have combined ideas from [36] and [32] to prove a global bifurcation result for co-rotating and counter-rotating pairs.…”

mentioning

confidence: 99%

Multipole Vortex Patch Equilibria for Active Scalar Equations

Hassainia¹,

Wheeler²

2022

SIAM J. Math. Anal.

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We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The configurations can be uniformly rotating, uniformly translating, or completely stationary. Using a technique first introduced by Hmidi and Mateu [36] for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular in the point-vortex limit. Provided the point vortex equilibrium is non-degenerate in a natural sense, solutions can then be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we construct new families of asymmetric translating and rotating pairs, as well as stationary tripoles. We also show how the techniques can be adapted for highly symmetric configurations such as regular polygons, body-centered polygons and nested regular polygons by integrating the appropriate symmetries into the formulation of the problem.

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“…On the other hand, there are very recent constructions of compactly supported solutions with noncircular streamlines. In [12] nontrivial patch solutions with three layers are built; here ω has the form 3 i=1 c i 1 D i for some c i ∈ R and some domains D i which are perturbations of concentric disks. In the forthcoming paper [24] a different example is given via a nonradial solution of a semilinear problem in a perturbed annulus.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Results for Compactly Supported Steady Solutions of the 2D Euler Equations

Ruiz

2023

Arch Rational Mech Anal

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In this paper we present symmetry results regarding compactly supported solutions of the 2D steady Euler equations. Assuming that $$\Omega = \{x \in {\mathbb {R}}^2:\ u(x) \ne 0\}$$ Ω = { x ∈ R 2 : u ( x ) ≠ 0 } is an annular domain, we prove that the streamlines of the flow are circular. We are also able to remove the topological condition on $$\Omega $$ Ω if we impose regularity and nondegeneracy assumptions on u at $$\partial \Omega $$ ∂ Ω . The proof uses the corresponding stream function solves an elliptic semilinear problem $$-\Delta \phi = f(\phi )$$ - Δ ϕ = f ( ϕ ) with $$\nabla \phi =0$$ ∇ ϕ = 0 at the boundary. One of the main difficulties in our study is that f is not Lipschitz continuous near the boundary values. However, $$f(\phi )$$ f ( ϕ ) vanishes at the boundary values and then we can apply a local symmetry result of F. Brock to conclude. In the case $$\partial _{\nu } u \ne 0$$ ∂ ν u ≠ 0 at $$\partial \Omega $$ ∂ Ω this argument is not possible. In this case we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of f. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, The Hopf lemma and The Serrin corner lemma for elliptic linear operators with singular coefficients.

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Existence of non-trivial non-concentrated compactly supported stationary solutions of the 2D Euler equation with finite energy

Cited by 4 publications

References 72 publications

Singularity formation in the incompressible Euler equation in finite and infinite time

Singularity formation in the incompressible Euler equation in finite and infinite time

Multipole Vortex Patch Equilibria for Active Scalar Equations

Symmetry Results for Compactly Supported Steady Solutions of the 2D Euler Equations

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