Endpoint bilinear estimates and applications to the two-dimensional Poisson–Nernst–Planck system

Abstract

We study the Cauchy problem of the two-dimensional Poisson–Nernst–Planck (PNP) system in Besov spaces for r ⩾ 2. Our work shows a dichotomy of well-posedness and ill-posedness depending only on r. Specifically, when r = 2, combining the key bilinear estimates in with the heat semigroup characterization of Besov spaces, we prove the well-posedness of the PNP in , while for r > 2 we show that the PNP is ill-posed in in the sense that the difference of the charges must satisfy certain requirements, i.e. either the difference belongs to for r > 2 and the summation belongs to , or the difference belongs to and the summation belongs to for r > 2. Thus our results indicate that the difference of charges plays a crucial role and might provide some instability criterion for numerical analysis.