Beltrami Fields with a Nonconstant Proportionality Factor are Rare

Enciso, Alberto; Peralta‐Salas, Daniel

doi:10.1007/s00205-015-0931-5

Cited by 48 publications

(64 citation statements)

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“…Our objective in this section is to show that, in fact, any generalized Beltrami field possesses a local partial stability property which can be essentially regarded as a local version of Theorem 3.7. We recall that, in view of the results in [20], one cannot prove a full stability result even in arbitrarily small open sets, so we regard this partial stability (where partial is understood in a very precise sense) as a satisfactory counterpart to the results in this paper.…”

Section: Local Stability Of Generalized Beltrami Fieldsmentioning

confidence: 85%

“…where a ∞ is called the far field pattern of a, and reads as It is apparent that a ∞ is uniquely determined from formula (20). Hence, we can define the following well-defined linear and one to one map…”

Section: Definition 21mentioning

confidence: 99%

“…Hence, the first step is to show the existence of Beltrami field that possess the desired collection of vortex tubes and which is defined in a neigborhood of the boundary tori. Since the tubes can be chosen analytic without loss of generality (by slightly deformating the initial vortex tube configuration), one could try to do that using a kind of Cauchy-Kowalewski theorem that was proved in [20]. However, this would lead to a local Beltrami field defined in a region whose complement is not connected, and this should prevent us from applying any kind of approximation theorem (this restriction is already present in the classical theorem of Runge, and appears in all the approximations theorems known to date).…”

Section: 1mentioning

confidence: 99%

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Stability Results, Almost Global Generalized Beltrami Fields and Applications to Vortex Structures in the Euler Equations

Enciso

Poyato

Soler

2017

Commun. Math. Phys.

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Strong Beltrami fields, that is, vector fields in three dimensions whose curl is the product of the field itself by a constant factor, have long played a key role in fluid mechanics and magnetohydrodynamics. In particular, they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarily complicated topology. On the contrary, there are very few results about the existence of generalized Beltrami fields, that is, divergence-free fields whose curl is the field times a non-constant function. In fact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) have been recently shown to be rare, in the sense that for "most" proportionality factors there are no nontrivial Beltrami fields of high enough regularity (e.g., of class C 6,α ), not even locally.Our objective in this work is to show that, nevertheless, there are "many" Beltrami fields with non-constant factor, even realizing arbitrarily complicated vortex structures. This fact is relevant in the study of turbulent configurations. The core results are an "almost global" stability theorem for strong Beltrami fields, which ensures that a global strong Beltrami field with suitable decay at infinity can be perturbed to get "many" Beltrami fields with non-constant factor of arbitrarily high regularity and defined in the exterior of an arbitrarily small ball, and a "local" stability theorem for generalized Beltrami fields, which is an analogous perturbative result which is valid for any kind of Beltrami field (not just with a constant factor) but only applies to small enough domains.The proof relies on an iterative scheme of Grad-Rubin type. For this purpose, we study the Neumann problem for the inhomogeneous Beltrami equation in exterior domains via a boundary integral equation method and we obtain Hölder estimates, a sharp decay at infinity and some compactness properties for these sequences of approximate solutions. Some of the parts of the proof are of independent interest.

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Section: Local Stability Of Generalized Beltrami Fieldsmentioning

confidence: 85%

Section: Definition 21mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Stability Results, Almost Global Generalized Beltrami Fields and Applications to Vortex Structures in the Euler Equations

Enciso

Poyato

Soler

2017

Commun. Math. Phys.

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“…such that ∇ × w = α w with α = 0 a real constant. This problem is both intrinsic and technical: on one hand the solution of (1) for general inhomogeneous proportionality factors is locally obstructed by the inhomogeneity, leading to Beltrami fields with lowregularity [17,18,19]. On the other hand any Beltrami field with nonzero proportionality factor is inevitably nonintegrable (it cannot satisfy h = 0).…”

Section: Introductionmentioning

confidence: 99%

Local representation and construction of Beltrami fields

Sato

Yamada

2019

Physica D: Nonlinear Phenomena

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A Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie-Darboux theorem of differential geoemtry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold-Beltrami-Childress flow with two of the parameters set to zero. Furthermore, a Beltrami flow admits two local invariants, a coordinate representing the physical plane of the flow, and an angular momentum-like quantity in the direction across the plane. As a consequence of the theorem, we derive a method to construct Beltrami fields with given proportionality factor. This method, based on the solution of the eikonal equation, guarantees the existence of Beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal. We construct several solenoidal and nonsolenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors. arXiv:1809.03136v1 [math-ph]

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“…When f is constant, the divergence-free condition is redundant and u is called a strong Beltrami field. Strong Beltrami fields are well-studied; see, e.g., [2], [3].…”

Section: Introductionmentioning

confidence: 99%

Beltrami Fields with Nonconstant Proportionality Factor

Clelland

Klotz

2019

Arch Rational Mech Anal

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We consider the question raised by Enciso and Peralta-Salas in [4]: What nonconstant functions f can occur as the proportionality factor for a Beltrami field u on an open subset U ⊂ R 3 ? We also consider the related question: For any such f , how large is the space of associated Beltrami fields? By applying Cartan's method of moving frames and the theory of exterior differential systems, we are able to improve upon the results given in [4]. In particular, the answer to the second question depends crucially upon the geometry of the level surfaces of f . We conclude by giving a complete classification of Beltrami fields that possess either a translation symmetry or a rotation symmetry. This is clearly an important result; unfortunately, the operator P is extremely cumbersome to compute. Moreover, this necessary condition for f is almost certainly not sufficient; the proof of the theorem shows that there is, in fact, a hierarchy of differential constraints that

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Beltrami Fields with a Nonconstant Proportionality Factor are Rare

Cited by 48 publications

References 20 publications

Stability Results, Almost Global Generalized Beltrami Fields and Applications to Vortex Structures in the Euler Equations

Stability Results, Almost Global Generalized Beltrami Fields and Applications to Vortex Structures in the Euler Equations

Local representation and construction of Beltrami fields

Beltrami Fields with Nonconstant Proportionality Factor

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