  This paper investigates the existence of weak solutions to two problems set
of elliptic equations in adjoining domains, with Beavers--Joseph--Saffman and
regularized Butler--Volmer boundary conditions being prescribed on the common
interfaces, porous-fluid and membrane, respectively. Mathematically, the
modeling tool is the coupled Stokes/Darcy problem, which consists of the Stokes
equation on one part of the domain coupled to the Darcy equation, where the
flow velocities are small and mainly driven by the pressure gradient in porous
medium, completed by the thermoelectrochemical (TEC) system, which consists of
the energy equation and the mass transport associated with electrochemical
reactions, where the fluxes are given by generalized Fourier, Fick and Ohm
laws, by including the Dufour--Soret and Peltier--Seebeck cross effects, in the
multidimensional domain. The present model includes macrohomogeneous models for
both hydrogen and methanol crossover. The novelty in the presented model lies
in the presence of the Joule effect into the Stokes/Darcy-TEC system altogether
to the quasilinear character given by temperature dependence of the physical
parameters such as the viscosities and the diffusion coefficients, by the
concentration-temperature dependence of cross-effects coefficients, and by the
pressure dependence of the permeability. The purpose of the present work is to
derive quantitative estimates for solutions to explicit smallness conditions on
the data. We use fixed point and compactness arguments based on the
quantitative estimates of approximated solutions.
