Global weak solutions for the compressible Poisson-Nernst-Planck-Navier-Stokes System

Abstract: We consider the compressible Poisson-Nernst-Planck-Navier-Stokes (PNPNS) system of equations, governing the transport of charged particles under the influence of the self-consistent electrostatic potential, in a three-dimensional bounded domain. We prove the existence of global weak solutions for the initial-boundary value problem with no-slip boundary condition for the fluid's velocity, blocking boundary condition for the ionic concentrations and inhomogeneous Robin boundary condition for the electrostatic potential, without restrictions on the size of the initial data. We derive the crucial energy dissipation of the system and prove the weak sequential stability of solutions of the Poisson-Nernst-Planck subsystem with respect to the velocity field of the fluid, which enables the proof of the existence of global weak solutions for the PNPNS system. We also study the large-time behavior of the solutions and justify the incompressible limit of the compressible PNPNS system as applications of the weak sequential stability of the solutions. New techniques and estimates are developed to overcome the difficulties from the strong interaction of the fluid with the ion particles and the physical boundary conditions.