A limit model for thermoelectric equations

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.Comment: 20 page

Radiative effects for some bidimensional thermoelectric problems

“…Let us consider the following problem that extends the thermoelectric problems, which were introduced in [10,11], in the sense of that the thermoelectric coefficient is assumed to be a given but arbitrary nonlinear function. The electrical current density j and the energy flux density J = q + φj, with q being the heat flux vector, satisfy    ∇ · j = 0 in Ω −j · n = g on Γ N j · n = 0 on Γ    ∇ · J = 0 in Ω J · n = 0 on Γ N −J · n = f λ (θ)|θ| ℓ−2 θ − γ(θ)θ ℓ−1 e on Γ, (1) for ℓ ≥ 2.…”

Explicit estimates for solutions of nonlinear radiation-type problems

“…Case n > 2: Setting M 1 = S 2,2 , and M 2 = K 2,2 , by using (6), and w β ∈ W 1,2 (Ω) ֒→ L q 1 χ (Ω), (7), and w β ∈ W 1,2 (Ω) ֒→ L qχ (∂Ω) with qχ = 2(n − 1)/(n − 2) i.e. χ = (s − 1)(n − 1)/[s(n − 2)] > 1 if s > n − 1.…”

“…Thermal effects on steady-state physical and technological models, whatever they are from mechanical engineering, electrochemistry, biomedical engineering, to mention a few, appear as an additional elliptic equation with a nonlinear radiation-type boundary condition into the coupled PDE system under study [5][6][7][8]. These form a boundary value problem constituted by an elliptic quasilinear second order equation in divergence form with the leading coefficient depending on the spatial variable and on the solution itself.…”

Explicit Estimates for Solutions of Mixed Elliptic Problems

“…Then, we argue as in the above case, concluding (13) with κ = 4. For both cases, we can extract a subsequence of u m , still denoted by u m , such that it weakly converges to u in W 1,q (Ω), where u ∈ V q solves the limit problem (9)…”

“…E ∈ V p ′ is either the Green or the Neumann functions, E = G and E = N, in accordance with Propositions 5.1 and 5.2, respectively. To this end, we take v = E ∈ V p ′ and v = u ∈ V p as test functions in (9) and (34), respectively, obtaining the Green representation formula In the presence of the Hardy-Littlewood-Sobolev inequality, we prove the following W 1,p -estimate. Proposition 6.1.…”