Sufficient conditions to the existence for solutions of a thermoelectrochemical problem

TL;DR

This paper establishes conditions under which solutions exist for a complex thermoelectrochemical model incorporating nonlinear radiation, with parameters depending on space and temperature, using fixed point theory and elliptic/parabolic estimates.

Contribution

It introduces a novel existence analysis for a nonlinear, parameter-dependent thermoelectrochemical problem with radiation effects, extending previous models.

Findings

Derived smallness conditions for data ensuring solution existence

Applied the theoretical results to electrolysis of molten sodium chloride

Utilized Gehring-Giaquinta-Modica theory for solution estimates

Abstract

A mathematical model of nonlinear radiation is introduced into a thermoelectrochemical problem, and its qualitative analysis is focused on existence of solutions. The main objective is the nonconstant character of each parameter, that is, the coefficients are assumed to be depend on the spatial variable and the temperature. Making recourse of known estimates of solutions for some auxiliary elliptic and parabolic problems, which are explicitly determined by the Gehring-Giaquinta-Modica theory, we find sufficient smallness conditions on the data to the existence of the original solutions via the Schauder fixed point argument. These conditions may provide useful informations for numerical as well as real applications. We conclude with an example of application, namely the electrolysis of molten sodium chloride.