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 equationsηtt=∂s(σηs),σss−|ηss|2σ=−|ηst|2,|ηs|2≡1 with boundary conditions η(t,1)=0η(t,1)=0 and σ(t,0)=0σ(t,0)=0. We prove local existence and uniqueness in the space defined by the weighted Sobolev energy∑ℓ=0m∫01sℓ|∂sℓηt|2ds+∫01sℓ+1|∂sℓ+1η|2ds, when m⩾3m⩾3. In addition we show persistence of smooth solutions as long as the energy for m=3m=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.