Friday, June 14, 2013

1306.3053 (Hao Wu et al.)

On transport of ionic solutions: from kinetic laws to continuum

Hao Wu, Tai-Chia Lin, Chun Liu
The Poisson-Nernst-Planck (PNP) system is a conventional macroscopic continuum model to describe the transport and distribution of ionic species in different media and solvents. In order to justify such a model for dilute solutions of multi-species charged particles, rather than employing the spatial coarse graining (averaging) we study a diffusion limit of Vlasov-Poisson-Fokker-Planck (VPFP) systems on a bounded domain with reflection boundary conditions of charge distributions. Here the VPFP system has a small parameter coming from the hypotheses of the scaled thermal velocity and mean free path of charged particles. Under the global neutrality assumption, we prove that as the small parameter tends to zero, solutions of VPFP systems converge to a global weak solution of the PNP system. The arguments use the renormalization techniques and the results support the PNP system as a model of multi-species charged particles.
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