A SYSTEM OF PARABOLIC EQUATIONS IN NONEQUILIBRIUM THERMODYNAMICS INCLUDING THERMAL AND ELECTRICAL EFFECTS

The time-dependent equations for a charged gas or fluid consisting of several components, exposed to an electric field, are considered. Thes e equations form a system of strongly coupled, quasilinear parabolic e quations which in some situations can be derived from the Boltzmann eq uation. The model uses the duality between the thermodynamic fluxes an d the thermodynamic forces. Physically motivated mixed Dirichlet-Neuma nn boundary conditions and initial conditions are prescribed. The exis tence of weak solutions is proven. The key of the proof is (i) a trans formation of the problem by using the entropic variables, or electro-c hemical potentials, which symmetrizes the equations, and (ii) a priori estimates obtained by using the entropy function. Finally, the entrop y inequality is employed to show the convergence of the solutions to t he thermal equilibrium state as the time tends to infinity.