A Generalized Poisson--Nernst--Planck--Navier--Stokes Model on the Fluid with the Crowded Charged Particles: Derivation and Its Well-Posedness

We derive a hydrodynamic model of the compressible conductive fluid by using an energetic variational approach, which could be called a generalized Poisson--Nernst--Planck--Navier--Stokes system. This system characterizes the micro-macro interactions of the charged fluid and the mutual friction between the crowded charged particles. In particular, it reveals the cross-diffusion phenomenon which does not happen in the fluid with the dilute charged particles. The cross-diffusion is tricky; however, we develop a general method to show that the system is globally asymptotically stable under small perturbations around a constant equilibrium state. Under some conditions, we also obtain the optimal decay rates of the solution and its derivatives of any order. Our method will apply equally well to a class of cross-diffusion systems if their linearized diffusion matrices are diagonally dominant.