The time-dependent equations for a charged gas or fluid consisting of several components, exposed to an electric field, are considered. These equations form a system of strongly coupled, quasilinear parabolic equations which in some situations can be derived from the Boltzmann equation. The model uses the duality between the thermodynamic fluxes and the thermodynamic forces. Physically motivated mixed Dirichlet-Neumann boundary conditions and initial conditions are prescribed. The existence of weak solutions is proven. The key of the proof is (i) a transformation of the problem by using the entropic variables, or electro-chemical potentials, which symmetrizes the equations, and (ii) a priori estimates obtained by using the entropy function. Finally, the entropy inequality is employed to show the convergence of the solutions to the thermal equilibrium state as the time tends to infinity.
Degond, P., Génieys, S., & Jüngel, A. (1997). A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects. Journal Des Mathematiques Pures et Appliquees, 76(10), 991–1015. https://doi.org/10.1016/S0021-7824(97)89980-1