A priori estimates for solutions to equations of motion of an inextensible hanging string

TL;DR

This paper derives a priori estimates and proves uniqueness for solutions to the equations of motion of an inextensible hanging string under gravity, considering degeneracy at the free end and using weighted Sobolev spaces.

Contribution

It introduces new a priori estimates for inextensible string equations, accounting for tension degeneracy and establishing solution uniqueness under natural stability conditions.

Findings

A priori estimates in weighted Sobolev spaces.

Proof of solution uniqueness.

Analysis of tension degeneracy at the free end.

Abstract

We consider the initial boundary value problem to equations of motion of an inextensible hanging string of finite length under the action of the gravity. We also consider the problem in the case without any external forces. In this problem, the tension of the string is also an unknown quantity. It is determined as a unique solution to a two-point boundary value problem, which is derived from the inextensibility of the string together with the equation of motion, and degenerates linearly at the free end. We derive a priori estimates for solutions to the initial boundary value problem in weighted Sobolev spaces under a natural stability condition. The necessity for the weights results from the degeneracy of the tension. Uniqueness of solutions is also proved.