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

By: Joseph W. Jerome and Riccardo Sacco

The Cauchy problem for the Poisson-Nernst-Planck/Navier-Stokes model was investigated 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 addressed, including bio-hybrid devices in neuronal cell monitoring, bio-reactor devices in tissue engineering and microfluidic devices in Lab-On-Chip technology. This paper has been presented at WCNA-5, held in Orlando in July, 2008, and has appeared electronically in Nonlinear Analysis: vol. 71 (2009), pp. e2487--e2497. doi:10.1016/