Knots and links in steady solutions of the Euler equation

Enciso, Alberto; Peralta‐Salas, Daniel

doi:10.4007/annals.2012.175.1.9

Cited by 110 publications

(147 citation statements)

References 18 publications

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“…An immediate consequence of this is that we recover our previous result on the existence of knotted vortex lines in steady solutions to the Euler equation [5]. It is worth emphasizing, however, that the proofs of these results are totally different, as we will discuss in Section 5, which is reflected in the fact that the behavior of the vortex lines in a neighborhood of the knots in the aforementioned reference and in a neighborhood of the central curve Φ(γ i ) is very different too.…”

Section: Realization Theorem For Thin Vortex Tubessupporting

confidence: 83%

“…To conclude, we shall recall our previous realization theorem for (possibly knotted and linked) vortex lines [5] and discuss the differences and similarities of the proofs. The statement is as follows: The proof of both this result and of the realization theorem for vortex tubes (Theorem 3.1) is based on the construction of a local Beltrami field with a collection of robust invariant sets (either invariant tori, in the case of vortex tubes, or periodic trajectories, for vortex lines), which is then approximated by a global Beltrami field.…”

Section: Realization Of Vortex Linesmentioning

confidence: 96%

“…It should be remarked that the decay of the field u stated in Theorem 5.1 cannot be found in the original paper [5]. The reason for this is that in [5] we used an approximation theorem for Beltrami fields weaker than the one we have developed for the Step 3 of the proof of Theorem 3.1, so that we could not provide any control whatsoever on the growth at infinity of the global Beltrami field.…”

Section: Realization Of Vortex Linesmentioning

confidence: 99%

“…In Section 4 we will give some ideas on the proof of this result. Finally, Section 5 will be devoted to the comparison between the proof of the theorem for vortex tubes and that of our previous realization theorem for vortex lines [5].…”

Section: Theorem 11 ([6] Informal Statement) Take a Collection Of mentioning

confidence: 99%

“…The reason for this is that in [5] we used an approximation theorem for Beltrami fields weaker than the one we have developed for the Step 3 of the proof of Theorem 3.1, so that we could not provide any control whatsoever on the growth at infinity of the global Beltrami field. With the new approximation theorem, one readily obtains the statement of Theorem 5.1.…”

Section: Realization Of Vortex Linesmentioning

confidence: 99%

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Thin vortex tubes in the stationary Euler equation

Enciso¹,

Peralta‐Salas²

2014

Journées Équations Aux Dérivées Partielles

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Tubes de vorticité étroits dans l'équation d'Euler stationnaire RésuméOn expose quelques nouveaux résultats sur l'existence de solutions stationnaires à l'équation d'Euler sur R 3 avec un ensemble de tubes de vorticité étroits (qui peuvent être noués et entrelacés) qu'on peut prescrire a priori. AbstractIn this paper we outline some recent results concerning the existence of steady solutions to the Euler equation in R 3 with a prescribed set of (possibly knotted and linked) thin vortex tubes.

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Section: Realization Theorem For Thin Vortex Tubessupporting

confidence: 83%

Section: Realization Of Vortex Linesmentioning

confidence: 96%

Section: Realization Of Vortex Linesmentioning

confidence: 99%

Section: Theorem 11 ([6] Informal Statement) Take a Collection Of mentioning

confidence: 99%

Section: Realization Of Vortex Linesmentioning

confidence: 99%

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Thin vortex tubes in the stationary Euler equation

Enciso¹,

Peralta‐Salas²

2014

Journées Équations Aux Dérivées Partielles

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Applications of a formula on Beltrami flow

Zeng

Zhang

2018

Math Methods in App Sciences

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In this note, we obtain uniqueness results for Beltrami flow in both bounded and unbounded domain with nonempty boundary by establishing an elementary but useful formula involving operators div and curl. We also use this formula to deal with Maxwell and Stokes eigenvalue problems. KEYWORDSBeltrami flow, Liouville-type theorem, star-shaped domain, the first Maxwell eigenvalue, the first Stokes eigenvalue INTRODUCTIONIn this note, we study the Beltrami flow, that is, a vector field u, which satisfies the systemBy establishing an elementary but useful identity, we obtain uniqueness results for Beltrami flow in both bounded and unbounded domain with nonempty boundary. As another interesting application, we use this identity to deal with Maxwell and Stokes eigenvalue problems.Since curlu × u = (u · ∇)u − ∇|u| 2 ∕2, each Beltrami flow will then give a special solution to the stationary Euler system. We refer the reader to Arnold and Khesin 1 for the basic properties of Beltrami flows. For some recent results, see previous studies 2-6 and references therein. We mention here that the Beltrami flows are also called force-free magnetic fields in magnetohydrodynamics, since the term curlu × u models the Lorentz force when u represents the magnetic field, see previous works 7-9 and references therein.Enciso and Peralta-Salas 3 constructed Beltrami fields, which satisfy curlu = u in R 3 for nonzero constant and fall off as |u(x)| < C|x| −1 at infinity. In particular, they are in L (R 3 , R 3 ) for all p > 3. A similar result can be also found in Lei et al. 10 Recently, Nadirashvili 11 proved a Liouville-type theorem for the globally defined Beltrami flow. He proved that when Ω = R 3 , a C 1 Beltrami flow satisfying either u ∈ L (R 3 , R 3 ), ∈ [2,3] or |u(x)| = o(|x| −1 ) as |x| → +∞ is in fact trivial, ie, u ≡ 0 in R 3 . Chae and Constantin 12 gave a new and elementary proof to a similar result, which partially covers the result of Nadirashvili. Chae and Wolf 13 succeeded in covering the result of Nadirashvili and got some improvements. Concerned with the exterior problem, Enciso, Poyato, and Soler 5 considered a related system 3632

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On the Topology of the Magnetic Lines of Solutions of the MHD Equations

Ciampa

2024

SEMA SIMAI Springer Series

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Knots and links in steady solutions of the Euler equation

Cited by 110 publications

References 18 publications

Thin vortex tubes in the stationary Euler equation

Thin vortex tubes in the stationary Euler equation

Applications of a formula on Beltrami flow

On the Topology of the Magnetic Lines of Solutions of the MHD Equations

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