Existence, uniqueness and regularity of piezoelectric partial differential equations

Abstract

Piezoelectric devices belong to the most prominent examples of smart materials. They find their applications in sensors and actuators, e.g. in the context of ultrasonic applications, tomography, cavitation-based cleaning. Nowadays, the design of new piezoelectric devices is generally accompanied by a computer-aided design, i.e. some models are used to predict the mechanical–electrical coupling of the new products. The coupling is described by a set of second-order coupled partial differential equations. For the mechanical part, this system comprises the equation of motion for the mechanical displacement in three dimensions and for the electric part, an electrostatic potential equation is employed. Coupling terms and an additional Rayleigh damping approach ensure the validness of the model. In this work, we analyze the existence, uniqueness and regularity of the solutions to these equations and give a result concerning the long-term behavior. The assumptions mainly on the material parameters involved are quite natural and allow meaningful physical interpretation.

Keywords:

• Partial differential equation
• Piezoelectricity
• Well-posedness

2010 Mathematics Subject Classifications:

• 35G15
• 35A01
• 35B65

Acknowledgments

The second author greatly appreciates funding by the German Research Foundation (DFG) under the support code WA 1607/14-1. The authors would like to thank Johannes Lankheit for the valuable discussions on some technical details which arose during the preparation of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author greatly appreciates funding by the German Research Foundation (DFG) under the support code WA 1607/14-1.