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The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

Part of: General higher-order equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  14 August 2017

Eiji Onodera
Eiji Onodera*
Affiliation:
Department of Mathematics, Kochi University, 2-5-1 Akebono-cho, Kochi 780-8520, Japan (onodera@kochi-u.ac.jp)
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Extract

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

Keywords

fourth-order dispersive partial differential equationvortex filamentHeisenberg spin chain systemslocal existence and uniquenessloss of derivativesgauge transformation

MSC classification

Secondary: 35Q35: PDEs in connection with fluid mechanics 35G61: Initial-boundary value problems for nonlinear higher-order systems 35Q40: PDEs in connection with quantum mechanics 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 6 , December 2017 , pp. 1243 - 1277
Copyright
Copyright © Royal Society of Edinburgh 2017 

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