Global Weak Solutions for an Incompressible Charged Fluid with Multi-Scale Couplings: Initial-Boundary Value Problem

The Cauchy problem for the Poisson-Nernst-Planck/Navier-Stokes model was inves-tigated by the first author in [Transport Theory Statist. Phys. 31 (2002), 333–366], where a local existence-uniqueness theory was demonstrated, based upon Kato’s framework for examining evolution equations. In this article, the existence of a global weak solution is proved to hold for the model, in the case of the initial-boundary value problem. Connection of the above analysis to significant applications is ad-dressed, including bio-hybrid devices in neuronal cell monitoring, bio-reactor devices in tissue engineering and microfluidic devices in Lab-On-Chip technology. Key words: Navier-Stokes, Poisson-Nernst-Planck, initial boundary problem for hybrid systems, Rothe’s method, slip boundary condition