A Steady-state System in Nonequilibrium Thermodynamics including Thermal and Electrical Effects

The steady-state equations for a charged gas or uid consisting of several components, exposed to an electric eld, are considered. These equations form a system of strongly coupled, quasilinear elliptic equations which in some situations can be derived from the Boltzmann equation. The model uses the duality between the thermodynamic uxes and the thermo-dynamic forces. Physically motivated mixed Dirichlet-Neumann boundary conditions are prescribed. The existence of generalized solutions is proven. The key of the proof is a transformation of the problem by using the entropic variables, or electro-chemical potentials, which symmetrize the equations. The uniqueness of weak solutions is shown under the assumption that the boundary data are not far from the thermal equilibrium. A general uniqueness result cannot be expected for physical reasons.