# Existence, Uniqueness and Regularity of Piezoelectric Partial Differential Equations

## TL;DR

This paper investigates the mathematical properties of the PDEs governing piezoelectric materials, establishing existence, uniqueness, regularity, and long-term behavior of solutions to improve understanding and simulation accuracy.

## Contribution

It provides rigorous proofs of well-posedness and regularity for the coupled PDE system modeling piezoelectric behavior, including effects of Rayleigh damping.

## Key findings

- Proved existence and uniqueness of solutions
- Established regularity and long-term behavior of solutions
- Applied Galerkin approximation and energy estimates

## Abstract

Piezoelectric appliances have become hugely important in the past century and computer simulations play an essential part in the modern design process thereof. While much work has been invested into the practical simulation of piezoelectric ceramics there still remain open questions regarding the partial differential equations governing the piezoceramics. The piezoelectric behavior of many piezoceramics can be described by a second order coupled partial differential equation system. This consists of an equation of motion for the mechanical displacement in three dimensions and a coupled electrostatic equation for the electric potential. Furthermore, an additional Rayleigh damping approach makes sure that a more realistic model is considered. In this work we analyze existence, uniqueness and regularity of solutions to theses equations and give a result concerning the long-term behavior. The well-posedness of the initial boundary value problem in a bounded domain with sufficiently smooth boundary is proved by Galerkin approximation in the discretized weak version, followed by an energy estimation using Gronwall inequality and using the weak limit to show the results in the infinite dimensional space. Initial conditions are given for the mechanical displacement and the velocity.