A limit model for thermoelectric equations

TL;DR

This paper investigates the behavior of thermoelectric equations under high heat conductivity, establishing existence of solutions and deriving a limit model for the asymptotic regime in complex boundary conditions.

Contribution

It proves the existence of weak solutions for a multidimensional thermistor problem with minimal assumptions and derives a limit model for high thermal conductivity scenarios.

Findings

Existence of weak solutions under minimal assumptions.

Two different existence results based on conductivity assumptions.

Derivation of a limit model for high heat conductivity asymptotics.

Abstract

We analyze the asymptotic behavior corresponding to the arbitrary high conductivity of the heat in the thermoelectric devices. This work deals with a steady-state multidimensional thermistor problem, considering the Joule effect and both spatial and temperature dependent transport coefficients under some real boundary conditions in accordance with the Seebeck-Peltier-Thomson cross-effects. Our first purpose is that the existence of a weak solution holds true under minimal assumptions on the data, as in particular nonsmooth domains. Two existence results are studied under different assumptions on the electrical conductivity. Their proofs are based on a fixed point argument, compactness methods, and existence and regularity theory for elliptic scalar equations. The second purpose is to show the existence of a limit model illustrating the asymptotic situation.