Bounds for systems of coupled advection-diffusion equations with application to the Poisson-Nernst-Plank equations

We consider coupled systems of advection-diffusion equations with initial and boundary conditions and determine conditions on the advection terms that allow us to obtain solutions that can be explicitly bounded above and below using the initial and boundary conditions. Given the advection terms, using our methodology one can easily check if such bounds can be obtained and then one can construct the necessary nonlinear transformation to allow the bounds to be determined. We apply this technique to determine bounding quantities for a number of examples. In particular, we show that the three-ion electroneutral Poisson-Nernst-Planck system of equations can be transformed into a system, which allows for the use of our techniques and we determine the bounding quantities. In addition, we determine the general form of advection terms that allow these techniques to be applied and show that our method can be applied to a very wide class of advection-diffusion equations.