Solutions to a nonlinear Poisson-Nernst-Planck system in an ionic channel

A limiting one-dimensional Poisson-Nernst-Planck (PNP) equations is considered, when the three-dimensional domain shrinks to a line segment, to describe the flows of positively and negatively charged ions through open ion channel. The new model comprises the usual drift diffusion terms and takes into account for each phase, the bulk velocity defined by (4) including the water bath for ions (see [14]). The existence of global weak solution to this problem is shown. The proof relies on the use of certain embedding theorem of weighted sobolev spaces together with Hardy inequality.