Local well posedness for a system of quasilinear pdes modelling suspension bridges

TL;DR

This paper establishes local well-posedness for a quasilinear PDE system modeling suspension bridges, capturing the nonlinear interactions of longitudinal and torsional motions with advanced analytical techniques.

Contribution

It introduces a novel analytical framework using paradifferential calculus and Kato's iterative scheme to prove well-posedness for a complex bridge model.

Findings

Proved local existence and uniqueness of solutions.

Developed a modified energy method for a quasilinear system.

Applied paradifferential calculus to handle nonlinearities.

Abstract

In this paper we provide a local well posedness result for a quasilinear beam-wave system of equations on a one-dimensional spatial domain under periodic and Dirichlet boundary conditions. This kind of systems provides a refined model for the time-evolution of suspension bridges, where the beam and wave equations describe respectively the longitudinal and torsional motion of the deck. The quasilinearity arises when one takes into account the nonlinear restoring action of deformable cables and hangers. To obtain the a priori estimates for the solutions of the linearized equation we build a modified energy by means of paradifferential changes of variables. Then we construct the solutions of the nonlinear problem by using a quasilinear iterative scheme \`a la Kato.