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The motion of whips and chains

By Stephen C. Preston

Abstract

We study the motion of an inextensible string (a whip) fixed at one point in the absence of gravity, satisfying the equations $$ \eta_{tt} = \partial_s(\sigma \eta_s), \qquad \sigma_{ss}-\lvert \eta_{ss}\rvert^2 = -\lvert \eta_{st}\rvert^2, \qquad \lvert \eta_s\rvert^2 \equiv 1 $$ with boundary conditions $\eta(t,1)=0$ and $\sigma(t,0)=0$. We prove local existence and uniqueness in the space defined by the weighted Sobolev energy $$ \sum_{\ell=0}^m \int_0^1 s^{\ell} \lvert \partial_s^{\ell}\eta_t\rvert^2 \, ds + \int_0^1 s^{\ell+1} \lvert \partial_s^{\ell+1}\eta\rvert^2 \, ds, $$ when $m\ge 3$. In addition we show persistence of smooth solutions as long as the energy for $m=3$ remains bounded. We do this via the method of lines, approximating with a discrete system of coupled pendula (a chain) for which the same estimates hold.Comment: 47 pages, 8 figure

Topics: Mathematics - Analysis of PDEs
Publisher: 'Elsevier BV'
Year: 2011
DOI identifier: 10.1016/j.jde.2011.05.005
OAI identifier: oai:arXiv.org:1105.1944

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