Very weak solutions for Poisson–Nernst–Planck system

We formulate a notion of very weak solution for the Poisson–Nernst–Planck system. The stationary system possesses a local monotonicity formula. Iterative application of the formula reveals improvement in estimates for ion density and potential, and leads to a local boundedness result. Local boundedness extends to steady-state systems for multiple ions and variable coefficients. The formulation applies to the related Keller–Segel system where stationary very weak solutions in two dimensions are regular. Examples illustrate how structure influences this regularity in higher dimensions.