On the Cauchy Problem for a Dynamical Euler's Elastica

Almut Burchard, Lawrence E. Thomas

Type: Article

Publication Date: 2003-01-06

Citations: 20

DOI: https://doi.org/10.1081/pde-120019382

Abstract

Abstract 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.

Locations

Communications in Partial Differential Equations - View
arXiv (Cornell University) - View - PDF

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+	On Schrödinger maps	2002	Andrea R. Nahmod Atanas Stefanov Karen Uhlenbeck
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+	None	2001	Sergiù Klainerman Igor Rodnianski
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+	On the Laplace Operator Penalized by Mean Curvature	1998	Evans M. Harrell Michael Loss
+	On the cauchy problem for harmonic maps defined on two‐dimensional Minkowski space	1980	Chaohao Gu
+ PDF Chat	Singularities of harmonic maps	1997	Robert Hardt