The motion of whips and chains

Details The motion of whips and chains The motion of whips and chains

2011-05-10

arxiv.org/abs/1105.1944v1

Mathematics

Analysis of PDEs

47 pages, 8 figures

Scientific paper

10.1016/j.jde.2011.05.005

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.

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