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

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 powertype 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. 2010 Mathematics Subject Classification. 35K51, 35R05, 35J62, 35Q79.

Weak solutions for multiquasilinear elliptic–parabolic systems: application to thermoelectrochemical problems

“…for every i, j = 1, · · · , I + 1, and for all v ∈ H 1 (Ω). Thanks to Propositions 5.1, 5.2 and 5.4, we may pass to the limit in (47) and (49), as M tends to infinity, concluding that u i and φ verify, respectively, (7), for i = 1, · · · , I, and (9).…”

“…The paper [9] deals with modeling of quasilinear thermoelectric phenomena, including the Peltier and Seebeck effects. In [5], the spatial distribution of the variables such as the electrolyte temperature, which is subject to local cell conditions, is studied.…”