Solvability of Equations for Motion of a Vortex Filament with or without Axial Flow

Abstract

It is known that the motion of a vortex filament with axial flow in a perfect fluid is approximately described by a generalization of the localized induction equation. The unique solvability of the initial value problem for it is first established by parabolic regularization. (1.2)x t = x s x x ss + a{x sss 4-(3/2)* ss x (x s x x ss )}, when the vortex filament has an axial flow within its thin vortex core. Here a is a real constant representing the magnitude of the axial flow effect.

A Third-Order Dispersive Flow for Closed Curves into Kähler Manifolds

“…We shall prove the following. We remark that Theorem 1.2 generalizes the results of Nishiyama and Tani in [15] and [19]. In other words, the proof of Theorem 1.2 will explain the reason why the global existence theorem of (1.1)-(1.2) holds in case that N = S 2 .…”

“…In [15] and [19], Nishiyama and Tani treated (1.1)-(1.2) in case N = S 2 with K = 1, and proved a time-global existence theorem by using the following conserved quantity:…”

“…In case N = S 2 , a = 0 and b = a/2, Nishiyama and Tani made use of some conservation laws to prove the global existence theorem in [15] and [19]. These conservation laws were discovered by Zakharov and Shabat in the study of the Hirota equation.…”

Generalized Hasimoto Transform of One-Dimensional Dispersive Flows into Compact Riemann Surfaces

“…In other words, if one see the metric tensor g as a real symmetric matrix or a hermitian matrix, a commutator [g, I∂ xxx + J∂ xx ] eliminates the bad part of the second and the first order terms, where I is the identity matrix. It is worth to mention that Nishiyama and Tani studied existence theorems of the initial value problem for the third-order two-sphere valued model (15) in [12] and [17]. The necessary and sufficient condition of the L 2 -well-posedness of the initial value problem for (43) was given in [11] under some technical condition.…”

On asymptotic behavior of solutions to Korteweg–de Vries type equations related to vortex filament with axial flow

“…Laurey [14] (see also Staffilani [19]) proved the global well-posedness for the initial value problem of (1.1) in Sobolev space H 1 (R). Tani and Nishiyama [22] proved an existence of the solution to (1.6) with some initial-boundary condition.…”