The motion of whips and chains

We study the motion of an inextensible string (a whip) fixed at one point in the absence of gravity, satisfying the equations $η_{tt} = \partial_s(ση_s), \qquad σ_{ss}-\lvert η_{ss}\rvert^2 = -\lvert η_{st}\rvert^2, \qquad \lvert η_s\rvert^2 \equiv 1$ with boundary conditions $η(t,1)=0$ and $σ(t,0)=0$. We prove local existence and uniqueness in the space defined by the weighted Sobolev energy $\sum_{\ell=0}^m \int_0^1 s^{\ell} \lvert \partial_s^{\ell}η_t\rvert^2 \, ds + \int_0^1 s^{\ell+1} \lvert \partial_s^{\ell+1}η\rvert^2 \, ds,$ 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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