Global weak solutions for the compressible Poisson–Nernst–Planck–Navier–Stokes system
Daniel Marroquin, Dehua Wang
Source abstract
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 to the initial‐boundary value problem with a no‐slip boundary condition for the fluid velocity, a blocking boundary condition for the ionic concentrations, and an 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 to 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 to the PNPNS system. We also study the large‐time behavior of the solutions and justify the incompressible limit of the compressible PNPNS system as an application of the weak sequential stability of the solutions. New techniques and estimates are developed to overcome the difficulties arising from the strong interaction between the fluid and the ion particles, as well as from the physical boundary conditions.
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