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

Abstract

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.

Applications of a formula on Beltrami flow

“…Since the half space is both a star‐shaped domain and the complement of the closure of the star‐shaped domain ; therefore, either u · ν =0 on { x 3 =0} or u × ν = 0 on { x 3 =0} will give u ≡ 0 in . (ii) Note that the star‐shaped domain D in our theorem can be both bounded and unbounded; thus, it contains some domains that cannot be covered by the results of Enciso et al…”

“…The authors are grateful to their supervisor, Prof Xingbin Pan, for guidance and constant encouragement. They would like to thank Prof Hairong Yuan for bringing them the reference . This work was partially supported by the National Natural Science Foundation of China grant no.…”

“…Chae and Wolf succeeded in covering the result of Nadirashvili and got some improvements. Concerned with the exterior problem, Enciso, Poyato, and Soler considered a related system for some f which, is a compactly supported perturbation of a constant proportionality factor , ie, f = λ + ϕ for some , where Ω is a C k +1, α bounded domain homeomorphic to an Euclidean ball. They showed that if is transverse to ∂ Ω at some point outside the support of ϕ , then there exists no ϵ >0 such that | u ( x )|= O (| x | −1− ϵ ) as | x |→+ ∞ , otherwise u ≡ 0 .…”

Local representation and construction of Beltrami fields

“…In Refs. [10,11,12] it is shown that vortex tubes of arbitrarily complex topology can be realized by means of Beltrami fields. Despite the centrality of Beltrami fields in the study of fluid flows and plasma turbulence [13,14], there are some aspects pertaining to their geometrical properties that have not yet been clarified.…”

Beltrami Fields with Nonconstant Proportionality Factor

“…In [4] and [5], the authors undertake a study of Beltrami fields on open subsets of R 3 , with a primary focus on Beltrami fields with nonconstant proportionality factor. Both local and global issues are considered, and the most significant result is that Beltrami fields of this type are rare, in the sense that most nonconstant functions f cannot occur as the proportionality factor for any nonvanishing Beltrami field, even locally.…”