Large Time Existence for Thin Vibrating Plates

TL;DR

This paper proves the long-time existence of strong solutions for a nonlinear wave model of thin vibrating plates, connecting 3D elastodynamics to 2D plate equations as thickness approaches zero.

Contribution

It establishes the existence of solutions for large times in thin plate models and quantifies convergence rates to classical plate equations as thickness diminishes.

Findings

Existence of strong solutions for small thickness and large times.

Convergence of 3D solutions to 2D plate equations as thickness tends to zero.

Explicit convergence rate for the linear Germain-Lagrange case.

Abstract

We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large times under appropriate scaling of the initial values such that the limit system as $h\to 0$ is either the nonlinear von K\'arm\'an plate equation or the linear fourth order Germain-Lagrange equation. In the case of the linear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.