Weak solutions for a thermoelectric problem with power-type boundary effects

TL;DR

This paper investigates the existence of weak solutions for a coupled thermoelectric problem involving Peltier and Seebeck effects with power-type boundary conditions, using analytical methods and time discretization techniques.

Contribution

It introduces a novel analytical framework for coupled elliptic-parabolic thermoelectric systems with nonsmooth data and power-type boundary effects, providing existence results via Rothe's method.

Findings

Existence of weak solutions established under smallness conditions.

Development of quantitative estimates for the coupled system.

Application of Rothe method with fixed point and compactness arguments.

Abstract

This paper deals with thermoelectric problems including the Peltier and Seebeck effects. The coupled elliptic and doubly quasilinear parabolic equations for the electric and heat currents are stated, respectively, accomplished with power-type boundary conditions that describe the thermal radiative effects. To verify the existence of weak solutions to this coupled problem (Theorem 1), analytical investigations for abstract multi-quasilinear elliptic-parabolic systems with nonsmooth data are presented (Theorem 2 and 3). They are essentially approximated solutions based on the Rothe method. It consists on introducing time discretized problems, establishing their existence, and then passing to the limit as the time step goes to zero. The proof of the existence of time discretized solutions relies on fixed point and compactness arguments. In this study, we establish quantitative estimates to clarify the smallness conditions.