Global weak solutions for an incompressible charged fluid with multi-scale couplings: Initial–boundary-value problem Original Research Article

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 initial–boundary-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.