Weak solutions for multiquasilinear elliptic-parabolic systems. Application to thermoelectrochemical problems

TL;DR

This paper proves the existence of weak solutions for a complex coupled elliptic-parabolic system with discontinuous coefficients, incorporating nonlinear thermal effects, and provides explicit estimates useful for real-world thermoelectrochemical applications.

Contribution

It introduces explicit constants in quantitative estimates for weak solutions of biquasilinear systems, considering general boundary conditions and nonlinear thermal effects.

Findings

Existence of weak solutions established for the coupled system.

Explicit expressions for constants in estimates provided.

Application demonstrated in thermoelectrochemical cell modeling.

Abstract

This paper investigates the existence of weak solutions of biquasilinear boundary value problem for a coupled elliptic-parabolic system of divergence form with discontinuous leading coefficients. The mathematical framework addressed in the article considers the presence of an additional nonlinearity in the model which reflects the radiative thermal boundary effects in some applications of interest. The results are obtained via the Rothe-Galerkin method. Only weak assumptions are made on the data and the boundary conditions are allowed to be on a general form. The major contribution of the current paper is the explicit expressions for the constants appeared in the quantitative estimates that are derived. These detailed and explicit estimates may be useful for the study on nonlinear problems that appear in the real world applications. In particular, they clarify the smallness conditions. In conclusion, we illustrate how the above results may be applied to the thermoelectrochemical phenomena in an electrolysis cell. This problem has several applications as for instance to optimize the cell design and operating conditions.