Stability of radially symmetric, monotone vorticities of 2D Euler equations

Choi, Kyudong; Lim, Deokwoo

doi:10.1007/s00526-022-02231-6

Cited by 9 publications

(5 citation statements)

References 32 publications

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“…The first result of this paper is about the stability of such a x 2 −independent solution when it is non-negative and monotone with bounded support. Theorem 1.1 is analogous to that of radial and monotone solution in R 2 from the recent work [12].…”

Section: Resultsmentioning

confidence: 75%

“…Sideris-Vega [24] presented a L 1 −stability of a disc patch 1 Br , r > 0 among patches with a compact support in R 2 . The most recent work [12] extended the work in [20] to show a weighted L 1 −stability of a non-negative, radially symmetric, and non-increasing solution among non-negative solutions in L ∞ (R 2 ). In addition, [1] proved an orbital stability, admitting translation, of the Lamb dipole by using a variational method.…”

Section: Resultsmentioning

confidence: 89%

“…In order to prove Theorem 1.1, we needed to reproduce the main ideas that were used in the proof of [12,Lemma 3.3], such as the conservation of the measure of level set, the total mass, the angular impulse, and properties of symmetric decreasing rearrangement, including the rearrangement estimate established in [20]. We use the conservation of the level set measure, the total mass, and the horizontal impulse.…”

Section: Key Ideasmentioning

confidence: 99%

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Stability of monotone vorticities in the half cylinder and infinite perimeter growth for patches

Choi¹,

Jeong²,

Lim³

2022

Preprint

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We consider the incompressible Euler equations in the half cylinder R>0 × T. In this domain, any vorticity which is independent of x2 defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted L 1 norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in x1. This includes stability of cylindrical patches {x1 < α}, α > 0. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.

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Section: Resultsmentioning

confidence: 75%

Section: Resultsmentioning

confidence: 89%

Section: Key Ideasmentioning

confidence: 99%

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Stability of monotone vorticities in the half cylinder and infinite perimeter growth for patches

Choi¹,

Jeong²,

Lim³

2022

Preprint

Self Cite

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“…We refer the reader to [2,41] for some historical reviews of the achievements in experimental, analytical, and numerical studies of vortex rings. Though numerous contributions to stability of steady solutions to 2D Euler equations have been achieved, for which we refer the interested reader to [1,13,14,22,25,26,50] and references therein, the works on the stability of vortex rings seem to be relatively rare. Recently, Choi [24] established the orbital stability of Hill's vortex by a combination of the variational framework due to [32], the uniqueness result [4] and the concentrated compactness lemma of Lions [39].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Uniqueness and stability of steady vortex rings

Cao¹,

Lai²,

Qin³

et al. 2022

Preprint

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In this paper, we are concerned with the uniqueness and nonlinear stability of vortex rings for the 3D Euler equation. We prove the uniqueness of a special family of vortex rings with a small cross-section, which are global classical solutions to the 3D Euler equation. We also establish the nonlinear stability of these vortex rings based on a combination of the uniqueness and a general stability criteria.

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“…These three ingredients are also essential in the nonlinear stability analysis of many other stationary Euler flows. See [1,5,9,10,[13][14][15]26] for example. The variational characterizations, which states that the sinusoidal flows in Theorem 1.4 are exactly the set of maximizers of the conserved functional E relative to all isovortical flows to them, are the most important step in the whole proof.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nonlinear stability of sinusoidal Euler flows on a flat torus

Wang¹,

Zuo²

2022

Preprint

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Sinusoidal flows are an important class of explicit stationary solutions of the two-dimensional incompressible Euler equations on a flat torus. For such flows, the steam functions are eigenfunctions of the negative Laplacian. In this paper, we prove that any sinusoidal flow related to some least eigenfunction is, up to phase translations, nonlinearly stable under L p norm of the vorticity for any 1 < p < +∞, which improves a classical stability result by Arnold based on the energy-Casimir method. The key point of the proof is to distinguish least eigenstates with fixed amplitude from others by using isovortical property of the Euler equations.

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Stability of radially symmetric, monotone vorticities of 2D Euler equations

Cited by 9 publications

References 32 publications

Stability of monotone vorticities in the half cylinder and infinite perimeter growth for patches

Stability of monotone vorticities in the half cylinder and infinite perimeter growth for patches

Uniqueness and stability of steady vortex rings

Nonlinear stability of sinusoidal Euler flows on a flat torus

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