Existence of large-data solutions to a thermo-piezoelectric system and forward operator analysis for associated inverse problems

TL;DR

This paper proves the existence of solutions for a thermo-piezoelectric system and analyzes the forward operator for inverse problems, providing a foundation for parameter identification in complex materials.

Contribution

It establishes global-in-time existence of weak solutions under minimal assumptions and analyzes the structural properties of the forward operator in inverse problems.

Findings

Existence of global weak solutions for the thermo-piezoelectric system.

Well-definedness and boundedness of the forward operator.

Fréchet differentiability of the forward operator.

Abstract

We consider an inverse problem governed by the initial-boundary value problem for the thermoviscoelastic Kelvin-Voigt system \begin{align*}\left\{ \begin{array}{l} \rho(z,t) u_{tt}- \left(\Gamma(\Theta) u_{zt} +p(z,t) u_z -\beta\Theta\right)_z=0\\ b(z,t) \Theta_t-\left(k(z,t)\Theta_z\right)_z - \Gamma(\Theta) u_{zt}^2+\beta \Theta u_{zt}=0, \end{array} \right. \end{align*} in an open bounded interval $\Omega\subset\mathbb{R}$, for the evolution of the displacement variable $u$, and the temperature $\Theta\geq 0$. Assuming the material coefficients $\rho$, $\Gamma$, $p$, $b$, $\beta$ and $k$ are strictly positive and bounded, a global-in-time existence result is established for weak solutions. The present manuscript demonstrates that this can be achieved under energy- and entropy-minimal assumptions, in the sense that global weak solutions are shown to exist for any initial data $$u_0\in W^{1,2}(\Omega),\quad u_{0t}\in L^2(\Omega)\quad\text{and}\quad 0\le\Theta_0\in L^2(\Omega).$$ The qualitative analysis of the evolution problem then allows to model and analyze the structural properties of the corresponding forward operator that naturally arises in inverse parameter identification settings. Therein, two modeling approaches of the observation operator as approximations of the electrical surface charge are presented and results on their well-definedness and boundedness are established. With the results on well-definedness and boundedness of the model operator, established in this paper as well, results on well-definedness, boundedness and continuous Fr\'{e}chet differentiability of the forward operator are presented.