Integro-differential equations and functional analysis

Existence of strong solutions for compressible elastic curves in the energy conservation system

Chiharu Kosugi

Yamaguchi University, Yamaguchi, Japan

Abstract: In this paper, we consider initial and boundary value problems for the beam equation system accompanying by a function having a singularity point for the nonlinear strain, called a compressible stress function. This problem is constructed as the mathematical model describing motions of closed elastic curves on $\mathbb{R}^{2}$ in our previous work. It is known that the energy derived from the system is conserved. For this problem we have already proved existence and uniqueness of weak solutions. Also, we have obtained results for existence and uniqueness of the strong solutions to the problem with the viscosity term. Our aim of this paper is not only to establish existence and uniqueness of a strong solution to the present problem, but also convergence of solutions to the problem with the viscosity term as the viscosity coefficient tends to $0$. The key to this proof is the uniform estimate for the fourth derivative with respect to the space of solutions.

Keywords: beam equation, nonlinear strain, compressible elastic curve, energy method.

UDC: 517.9

MSC: 35Q74, 35G31, 74B20

Received: 27.09.2024
Revised: 04.02.2025
Accepted: 05.02.2025

Language: English

DOI: 10.26516/1997-7670.2025.53.85