2002/02/28 by Almut Burchard, Lawrence E. Thomas
Mathematics · Physics and Astronomy · #math.AP #math-ph #math.MP #msc:35Q72
published as Communications in PDE 28 (2003), 271-300 · 31 pages. Revised introduction
arxiv created 2002/06/11 · arxiv updated 2009/11/30
The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order elliptic equation for its tension. A Hasimoto transformation is used to rewrite the equations in convenient coordinates for the space and time derivatives of the tangent vector. A feature of this elastica is that it exhibits time-dependent monodromy. A frame parallel-transported along the elastica is in general only quasi-periodic, resulting in time-dependent boundary conditions for the coordinates. This complication is addressed by a gauge transformation, after which a contraction mapping argument can be applied. Local existence and uniqueness of elastica solutions are established for initial data in suitable Sobolev spaces.