Well-posedness of the initial boundary value problem for the motion of an inextensible hanging string

TL;DR

The paper addresses the local well-posedness of the initial boundary value problem for the motion of an inextensible hanging string under gravity, governed by nonlinear, nonlocal hyperbolic equations that degenerate at the free end. The authors reformulate the problem by removing the unit-speed constraint and employ a quasilinearization strategy introducing $\bm{y}=\ddot{\bm{x}}$ and $\nu=\ddot{\tau}$ to overcome derivative loss, proving local existence in weighted Sobolev spaces for $m\ge4$ under a stability condition $\frac{\tau}{s}\ge c_0$. For the critical cases $m=4,5$, they construct smooth initial-data approximations satisfying higher-order compatibility conditions and pass to the limit using a priori estimates, obtaining a unique, strongly time-continuous solution. The results extend previous work on nonlocal, degenerate hyperbolic systems and provide a rigorous foundation for the dynamics of an inextensible hanging string, including the role of compatibility and stability in well-posedness.

Abstract

We consider the motion of an inextensible hanging string of finite length under the action of the gravity. The motion is governed by nonlinear and nonlocal hyperbolic equations, which is degenerate at the free end of the string. We show that the initial boundary value problem to the equations of motion is well-posed locally in time in weighted Sobolev spaces at the quasilinear regularity threshold under a stability condition. This paper is a continuation of our preceding articles.

Figures (1)

• Figure 1.1: Hanging String

Theorems & Definitions (18)

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• proof : Proof of Lemma \ref{['lem:reg']}
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