Applications of a formula on Beltrami flow

Abstract

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

An improved Liouville type theorem for Beltrami flows

“…They play a key role in the proof of the main theorem. The first lemma comes from [17,Lemma 3.2]. In the original version, α ≥ 0 and u ∈ H 1 (D, R 3 ).…”

General three-dimensional equilibria for gravitating ideal magnetohydrodynamics of field-aligned steady incompressible flows with an application to solar prominences

“…Beltrami flows are also called force-free magnetic fields in MHD, since the term (∇ ∧ v) ∧ v models the Lorentz force when v represents the magnetic field, see Moawad (2014) and references therein. By establishing an elementary identity, uniqueness results for Beltrami flow in both bounded and unbounded domains with a non-empty boundary were obtained by Zeng & Zhang (2018). They used that identity to deal with Maxwell and Stokes eigenvalue problems.…”

Comparison results for eigenvalues of curl curl operator and Stokes operator

“…Moreover, if the domain is convex, Pauly [12] proved that α 1 = β 1 ≥ µ 2 . For three-dimensional bounded and star-shaped C 1,1 domains, Zeng and the author [16] obtained that α 1 = β 1 < γ 1 . For two-dimensional bounded domains, Kelliher [9] showed that γ k > λ k holds for any positive integer k. For two-dimensional simply connected bounded domains, since the Stokes eigenvalue problem can be rewritten as the clamped buckling plate problem, one can obtain that γ 1 > λ 2 , see [14] or [5].…”