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 investigated by the first author in [J.W. Jerome, An analytical approach to charge transport in a moving medium, 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 initialboundary-value problem. Connection of the above analysis to significant applications is addressed, including bio-hybrid devices in neuronal cell monitoring, bio-reactor devices in tissue engineering and microfluidic devices in Lab-On-Chip technology.