Well-posedness on a new hydrodynamic model of the fluid with the dilute charged particles

We use an energetic variational approach to derive a new hydrodynamic model, which could be called a generalized Poisson–Nernst–Planck–Navier–Stokes system. Such the system could describe the dynamics of the compressible conductive fluid with the dilute charged particles and be used to analyze the interactions between the macroscopic fluid motion and the microscopic charge transportation. Then, we develop a general method to obtain the unique local classical solution, the unique global solution under small perturbations and the optimal decay rates of the solution and its derivatives of any order.